Chapter 5

Chapter 5

Multiplication and Division I: Meaning

5.1 Multiplication as Repeated Addition

Multiplication is not really a basic operation. As the problems in the following activity show, it is possible to solve many multiplication problems by using a simpler operation.

Activity 5.1A

A.Solve the following problems using addition and appropriate units. Draw pictures if it is helpful to do so.

1.Three children are playing a game. Each child gets four cards. How many cards are in use?

2.A rectangular baby quilt is made of four strips each containing six squares. How many squares are in this quilt?

3. Rachel has two pairs of shorts and three T-shirts. Assuming she is indifferent to color coordination, how many outfits does she have?

4.A water bottle has a capacity of 11/2 liters of water. How many liters of water can five of these bottles hold?

B.Answer the following.

1a.Each of the problems in part A involved repeated ___________________ .

b.Each of the problems in part A could have been solved more efficiently using what operation? _____________

c.Thus multiplication can be defined as __________________________________________________________

2. Consider the following sets.

a.There are _____ sets with ______hearts in each set. The union of these sets includes six ____________.

b.In other words, 3 (2 __________) = 6 _______________

c.In this problem, 3 refers to the ___________________ of sets and 2 refers to the ___________ of a set.

3.Reconsider problem #4 in part A. Five referred to the _______________ of bottles and three quarters of a liter referred to the _______________ of a bottle.

4.In these situations, it seems that one of the numbers in a multiplication refers to the _____________________ and the other refers to the _______________________________.

In all of the above problems, answers can be found by using repeated addition. There are so many situations involving repeated addition that this process is called multiplication. (Be warned, however, that repeated addition is not the only meaning of multiplication. We will study another meaning in a later section of this chapter.)

Basic Definition of Multiplication as Repeated Addition

For m a whole number, the product m B is the total number of objects in m disjoint sets, each

containing B elements. m is called the multiplier and B is called the multiplicand.

EMBED Word.Picture.8

m B = B + B + B + . . . + B

m times

The two numbers m and B play two very different roles in this basic meaning of multiplication. The multiplier m is the number of sets while the multiplicand B is the size of the set. The result of a multiplication is called a product. In situations in which multiplication is defined as repeated addition, the multiplicand can be any type of number but the multiplier must be a whole number.

Total = (Number of sets) (Size of the set)

Product = Multiplier Multiplicand

Example 1:Melissa invited all of her running friends over for a morning run followed by brunch. She bought three dozen eggs for the occasion. How many eggs did she buy?

Total number of eggs = 3 sets of 12 eggs = 12 eggs + 12 eggs + 12 eggs = 3 12 eggs = 36 eggs

Of and Times

Notice that of is the word we often use to describe the size of a set. For instance, we might say that a platoon includes three squads of 10 soldiers. This phrasing indicates that the total number can be found by repeated addition, a.k.a., multiplication. IThus the use of the word of can be a signal to multiply. Conversely, times can often be translated as of. For example, 3 times 5 can be interpreted to mean 3 sets of five or 3 fives.

Teaching Tip: Sometimes children are told that of meanstimes. This is a misleading overgeneralization. Of is one of the most common words in the English language and often does not mean times. For example, in the following sentence, Nine of the 12 students in the class passed the test, it would be nonsensical to multiply 9 by 12. It actually makes more sense to say that times often means of.

Factors and Multiples

The multiplier and multiplicand are also called factors. A whole number product is called a multiple of each factor.

Example 2:Consider 3 2 = 2 + 2 + 2 = 6.

a. 3 is the multiplier, 2 is the multiplicand, and 6 is the product.

b. 2 is the size of the set, and 3 is the number of sets.

c. 3 and 2 are factors of 6, while 6 is a multiple of 3 and 2.

Every whole number except 0 has a finite number of whole number factor, but all whole numbers have an infinite number of whole number multiples.

Example 3: Set of factors of 6 = {1, 2, 3, 6}; set of multiples of 6 = {0, 6, 12, 18, . . .}

Teaching Tip: Students often confuse factors with multiples. For instance, a student might say that 3 is a multiple of 6 or that 12 is a factor of 6. Since these are important vocabulary words, teachers need to spend time making sure students learn which is which. Mnemonic devices such as Factors are first or Multiples multiply monotonously may be helpful to some students.

As the next examples indicate, many different notations are used to indicate multiplication.

Example 4:(a) Product of 2 and 3 = 2 times 3 = 2 threes = 2 ( 3 = (2)(3) = 2(3) = 2 * 3 = 2 3

(b) Product of x and y = xy = x y

Units in Repeated Addition

A sum has the same unit as its terms. For example, 3 feet + 3 feet is equal to 6 feet. Similarly, since the basic meaning of a product is the repeated sum of multiplicands, the product has the same unit as the multiplicand.

Example 5:Five yardsticks are placed end to end. How many feet is it from one end to another?

5 3 feet = 3 feet + 3 feet + 3 feet + 3 feet + 3 feet = 15 feet

Activity 5.1B

A.Fill in the blanks, representing the total as a repeated addition. Include units.

Multiplier Multiplicand Total

Ex: Three days a week Heidi walks 13/4 miles. 3 13/4 mi. 13/4 mi. + 13/4 mi. + 13/4 mi. = 51/4 mi

How many miles does she walk every week?

1.Sara has two classes of 20 students. How _____ ________ _______________________________

many students does she have altogether?

2.Peter buys three -gallon bottles of milk. _____ ________ _______________________________

How many gallons of milk has he bought?

B.Answer the following questions.

1a.Find the area of the shaded shape on the centimeter grid to the right. _________

b.What is the shape of the standard unit for measuring area? __________________

2a.Suppose each cube to the right measures 1 cm by 1 cm by 1 cm.

What is the total volume of this set of cubes? _________

b.What is the shape of the standard unit for measuring volume? _______________

Four Major Situations Involving Repeated Addition

1. Distinct Repeated Sets

Example 6: Consider the problem in which each of three children has four cards. How many cards are there altogether?

EMBED Word.Picture.8

We have three sets of four: 3 4 cards = 4 cards + 4 cards + 4 cards = 12 cards.

The most obvious case of repeated sets occurs with a repeating set of physical objects. This physical set may be a hand of cards, a platoon of soldiers, a case of soft drinks, and so on.

2. Arrays

Consider the situation in which Rachel has three T-shirts and two pairs of shorts. The following diagram illustrates one way to determine that Rachel can put together a total of six different outfits.

A horizontal arrangement of objects is called a row and a vertical arrangement is called a column. The above diagram, with 2 rows and 3 columns, is an example of a 2 by 3 array. An R by C array is a set of discrete objects arranged into R rows and C columns. Because the rows of an array are the same size, the total number of elements in an array can be found by repeatedly adding the rows. Since the row size is the same as the number of columns, we have the following generalization.

The total number of elements in an R by C array is R C.

This explains why an R by C array is also described as an R ( C array.

Example 7: This is a 2 ( 5 array, with two rows and five columns.

Total number of elements = 2 5 = 5 + 5 = 10

3. Area and Volume

What is the total number of squares in a baby quilt made of four strips of six squares each?

This is another example of a problem that can be solved by repeated addition. The quilt

consists of four rows, each containing six squares. The total number of squares is equal

to the following: 4 sixes = 6 squares + 6 squares + 6 squares + 6 squares = 24 squares.

This quilt also illustrates why the area of a rectangle can be found by multiplying its length by its width.

Finding the number of squares in a rectangle is analogous to finding the number of elements in an array.

Rectangles as Arrays of Squares

Array with 8 elements Rectangle with an area of 8 squares

Generally speaking, we measure the area of a two-dimensional shape using squares. The squares in a rectangle form an array in which the number of rows corresponds to the length of the rectangle, while the number of columns corresponds to the width. Thus the area of a rectangle is the product of its length and width.

B

Formulas for the areas of other special shapes are derived from this basic area formula.

H

Example 8:The area of a right triangle with legs of length B and H is BH because

its area is half the area of a rectangle with length B and width H.

One special area is not directly derived from the area of a rectangle. The area of a circle is equal to r2, where r is the radius of the circle.

As the following example illustrates, the area of many figures can be found by partitioning the figure.

Example 9:To find the area of the figure given below, partition it as indicated.

6 cm 6 cmArea Half-circle = 0.5 (3.8 cm)2 22.68 cm2

3.8 cm

7.6 cm Area Rectangle = 6 cm 7.6 cm 45.6 cm2

16.8 cm 3.8 6.0 7.0 Area Triangles = 2 (0.5 3.8 cm 7.0 cm) = 26.6 cm2

Area Total = 94.88 cm2

Volume

1

The standard unit for measuring volume is a cube. A cube that measures one unit 1

by one unit by one unit has a volume of one cubic unit. As the following activity

illustrates, the volume of the three-dimensional analog of a rectangle can be found 1

by repeated addition of layers of cubes.

One Cubic Inch

Activity 5.1C

1.A solid box has a length of 4 cm, a width of 2 cm, and a height of 3 cm.

________ a. What is the area or the bottom (or top) of this box?

________ b. How many cubic centimeters are in the first layer of this box?

________ c. How many layers does the box have?

________ d. Use the above facts to determine the volume of the box.

2.What is the volume of a box that is 5'' high, 10'' long, and 3'' deep?

______________

3.A cylindrical water tank is 20 feet high. It is known that when the water is one foot deep,

the volume of water in the tank is about 700 cubic feet. What is the capacity of the tank? _____________

[Hint: Think about the volume of each layer.]

The formal name of a typical box is a right rectangular prism. It has rectangular faces

at right angles to each other. A right rectangular prism with length L, width W, and height H 1 1

can be partitioned into a series of identical one unit thick layers. The volume of one of these

layers has the same numerical value as L W, the area of the floor or base of the prism. 1

Since the number of layers corresponds to the height of the solid, the volume of the right

rectangular prism is as follows. 1 W

L

Volume of a right rectangular solid = length width height

Volumes of Solids with Congruent Bases

In general, a prism is any solid with two congruent and parallel polygonal bases connected by parallel lines. This means that the other faces of a prism are parallelograms.

Various Prisms

A prism is a special type of cylinder. A cylinder is any solid with two congruent and parallel bases, not necessarily polygonal, that are connected by parallel lines.

Various Cylinders

Like a prism, a cylinder consists of a series of congruent layers. Thus its volume is the repeated sum of the volume of one layer. The volume of a single layer has the same numerical value as the area of the base of the cylinder; the number of layers corresponds to the height of the cylinder. (The height of a cylinder is the distance between its bases. If the base of a cylinder is horizontal, then its height is vertical.) This yields the following useful formula.

Volume of a Cylinder = Area of its Base Height

Example 10: If the base of a kidney-shaped pool has an area of 40 square feet, then filling it to a depth of one foot will require 40 cubic feet of water. Every additional foot of depth will require another 40 ft3. So filling the pool to a depth of three feet will require 40 ft3 + 40 ft3 + 40 ft3 for a total of 120 ft3.

Example 11:A waste basket is a cylinder that is 2 3 high. Its base has parallel

sides and circular ends. The parallel sides are 10 inches apart and

one foot long. How many gallons of water will this waste basket

hold? There are 231 cubic inches in a gallon.

Find the area of the base. It consists of two half-circles and a rectangle.

The area of a circle is ( r2, where r is the radius. In this situation, the

diameter is 10 and thus the radius is 5. To reduce round-off error,

do not round until the end of the problem.

12

Area of rectangle = 10'' ( 12'' = 120 square inches

Area of two half circles = 2 ( ( ( r2), where radius is 5

10

( 3.14159 ( 52 square inches

( 78.5 square inches

Total area of the base ( 198.5 square inches

Volume of container ( 198.5 square inches ( 27 inches

( 5360.57 cubic inches

( 5360.57 in3 ( 231 in3 per gallon

( 23.2 gallons

4. Cartesian Products

Recall that the number of possible combinations of Rachels shorts and T-shirts was found by pairing each T-shirt with a pair of shorts. In general, the set consisting of all possible ways of pairing elements of a set A with elements of another set B is called a Cartesian product. A Cartesian product can always be illustrated as an array. The number of rows in this array corresponds to the number of elements in set A, designated as NA, and the number of columns corresponds to the number of elements in set B, designated as NB. Thus we have the following.

If C is the Cartesian Product of A and B, then NC = NA NB

Example 12:The license plate of a very small state consists of a letter followed by a single-digit number. How many distinct license plates of this description are possible?

The license plates form an array, partially indicated below.

0 1 2 3 4 5 6 7 8 9

A A0A1A2A3A4A5A6A7A8A9

B B0B1B2B3B4B5B6B7B8B9

. . . . . . . . . .

. . . . . . . . . .

Z Z0Z1Z2Z3Z4Z5Z6Z7Z8Z9

There are 26 rows with each row containing 10 plates. The total number of plates is 26 10 or 260.

A Cartesian product can also be described using a tree diagram, as shown below.

Example 13: Let S represent a pair of Rachels shorts and T represent a T-shirt. The following tree diagram shows the six outfits that result from using these clothes.

S1

S2

T1 T2 T3 T1 T2

T3

S1T1 S1T2S1T3S2T1S2T2 S2T3

As the next activity demonstrates, the idea of a Cartesian product can be extended to more than two sets.

Activity 5.1D

1.Find the volume of a prism that is one foot long with a right triangular base. _________________

The three sides of the base measure 3'', 4'', and 5''.

2.Suppose license plates consist of a letter followed by two digits.

a.List one license plate meeting this description. _________________

b. How many license plates meeting this description start with A? _________________

c.What is the total number of license plates? _________________

3.In Tennessee, license plates consist of three letters followed by three digits.

a.How many license plates are possible in Tennessee? _________________

b.Suppose Tennessee deletes 38 three-letter words from use on license plates. ________________

How many license plates are now possible in Tennessee?

4.Summarize the pattern that occurs when a fraction is multiplied by a whole number in the following.

a.4 1/2 = 1/2 + 1/2 + 1/2 + 1/2 = 4/2 b. 3 4/5 = 3 4 fifths = 12 fifths = 12/5 c. 2 7/3 = 7/3 + 7/3 = 14/3

5.Use the pattern you observed in the previous problem to find the answer to the following word problem.

A chocolate nougat weighs 2/3 ounce. How much do 5 of these nougats weigh?

The set of all possible Tennessee license plates is an example of a general Cartesian product. Just as a license plate is created by choosing letters and digits, an element in a general Cartesian product is formed by choosing elements one at a time from several sets.

N1 N2

N3

N4 . . . Nk

elements elements elements elements

elements

Set 1 Set 2 Set 3 Set 4

Set k

General Cartesian Product

Each element in this Cartesian product contains one element from Set 1, one element from Set 2, and so on. The total number of such elements is found as follows.

Total number of elements in the Cartesian product = N1 N2 ... Nk

Example 14:How many different kinds of pizza can be made if there are five possible toppings from which to

choose?

For each topping, there are two choices, to use the topping or not to use it. Thus there are a total of five sets, each containing 2 choices. So the total number of pizzas is equal to 2 2 2 2 2 or 32.

The next example illustrates a situation in which several sets need to be reconsidered as a single set in order to determine the appropriate number of possibilities.

Example 15:Suppose Tennessee license plates consist of three letters followed by three digits, with 38 three-letter words deleted from use. How many license plates are possible?

Total number of allowable words = 263 - 38 = 17,538.

For each word, there are 103 or 1,000 numbers.

This yields: 17,538 1,000 = 17,538,000 license plates.

Repeated Addition with Rational Numbers as Multiplicands

When the size of a set is not a whole number, using the unit fraction as the main unit leads to an easy process for computing the product.

Example 16:A small measuring cup has a capacity of 3/8 of a liter. How much water will two of these cups hold?

2 3/8 liter = 3 eighths of a liter + 3 eighths of a liter

= 6 eighths of a liter

= 6/8 L (or 3/4 L)

Example 17:I bought three half-gallons of milk today. How many gallons of milk did I buy?

3 1/2 gallon= 1/2 gallon + 1/2 gallon + 1/2 gallon

= 3/2 gallons = 11/2 gallons

As these examples illustrate, we can find the product of a whole number and a rational number by multiplying the number of unit fractions, i.e., the numerator: m N = m N

D D

If a multiplication problem contains mixed numbers, change these mixed numbers to improper fractions to make use of the above property.

Example 18: It takes 12/3 yards of ribbon to make a bow. How much ribbon is needed for four bows?

4 (12/3 yards) = 4 5 thirds of a yard = 20 thirds of a yard = 20/3 yd or 62/3 yards

Compare this to using feet as a unit: 4 5 thirds of a yard = 4 5 feet = 20 feet

5.1 Homework Problems

A.Answer the following.

1a.State the basic definition of multiplication.

b.In situations involving repeated addition, the total can be found by multiplying the ? of sets by the ? of a set.

2.Define the following: (a) multiplicand; (b) multiplier; (c) row; (d) Cartesian product.

3a.List the four general situations leading to repeated addition.

b.Invent and solve your own example for each situation. Do not use the examples given in the text.

4.Show how the area of a 3'' by 5'' rectangle can be found by repeated addition. Use a well-labeled diagram.

5.Show how the number of elements in a 3 by 5 array can be found by repeated addition. Use a labeled diagram.

6.Fill in the blanks. (a) 4 3/5 = 4 ? fifths = 12 ? (b) 3 5/4 = 3 5 ? = 15 ?

7a.Draw a picture to show why 2 3/5 = 6/5. b.Use repeated addition to find 2 3/5 = 6/5.

8.Explain why, in situations involving repeated addition, the multiplicand and the product have the same units. Include an example.

9.Which of the following are arrays?

a. b. 1 45 48 c. d.

0 15 32

10.State the number of rows and columns and the total number of elements in each of the arrays in the previous problem.

11.Ron purchases three boxes of light bulbs. Each box contains 6 packages of bulbs, and each package contains two bulbs. Find the total number of light bulbs purchased by using:

a.a series of repeated additions b. multiplication c. a picture d. a tree diagram

12.Use a tree diagram to find the number of different pizzas if there are three types of crusts (thin, medium, or thick), two types of dough (white or whole wheat), and four kinds of topping combinations (plain, pepperoni, super, and vegetarian).

13.The screen on a calculator contains pixels arranged in 62 columns and 48 rows. How many pixels occupy the screen? (A pixel is a single position on the screen. It is either lighted or unlighted.) Draw the beginnings of an array and solve this problem.

14.Ryan now has only 62 toy soldiers, after losing 48 in the woods yesterday.

a.How many toy soldiers did Ryan have before playing with them in the woods?

b.Identify the type of this problem.

15.An auditorium has 100 rows. The first row contains 20 chairs, and each succeeding row contains one more chair than the previous row.

a.How many chairs are in the 100th row? Solve this problem by using an organized table containing at least three rows and finding the pattern.

b.How many chairs are there altogether in the auditorium? [Hint: What is the sum of the chairs in the 1st and 100th row? What is the sum of the chairs in the 2nd and 99th row?]

16.License plates for a certain state contain 4 letters followed by 3 digits.

a.State one possible license plate for this state.

b.How many different license plates are possible?

c.How many license plates starting with LOVE are possible?

d.If 18 four-letter words are eliminated from the possible choices of four-letter combinations, and the use of 000 is eliminated, how many different license plates are possible?

17.Some lottery tickets consist of six digits. What are your chances of winning the lottery if there is only one winning combination of digits?

18.A large bag of mulch is labeled as containing 2 cubic feet of mulch. How many cubic inches of mulch is this? [Hint: One cubic foot is 12'' by 12'' by 12''.]

19.A 10 by 8 patio is to be made with cement. It will be 2'' thick. How much cement is needed?

20.Explain how the area of a right triangle is related to the area of a rectangle with the same base and height. Include a diagram.

21.A clay brick measures 8'' long, 4'' deep, and 3'' high. It is hollow in the middle, with sides and bottom that are 1'' thick. A cubic inch of clay weighs about two ounces. How heavy is this brick?

22.Find the volume of the wedge to the right.

8 cm

3 cm

15 cm

23.A 20' by 30' rectangular swimming pool is 3' 4'' deep at one end

and steadily increases to 8' deep at the other end 30' away.

How many gallons of water does it hold? (There are about

7 gallons of water in one cubic foot.)

_______________________________________________ _______________________________

_________________________________________________________ ____________________

5.2 Division in the Context of Repeated Addition

Like multiplication, division is a derived operation. It is possible to solve many division problems by using more basic operations, as illustrated in the next activity.

Activity 5.2A

A.Show how to solve the following problems using counting, addition, or subtraction. Use pictures or diagrams as appropriate.

1.A kindergarten teacher has one of her children distribute 10 lollipops equally to five children. The child gives one to each child, then another, and another, until they are all gone. How many lollipops does each child get?

2.A class contains 24 children seated at tables in groups of four. How many tables are there?

3.I cut 3 apples in half and gave away all the half-apples, one to each child in the room. How many children are in the room?

B.Travis, Zack, and Chad are playing with toy soldiers. Travis has eight toy soldiers, Zack has six, and Chad has

fourteen. All three boys organize their soldiers into pairs. Then Travis and Zack team up against Chad.

1.Compare the pairs in each army. This situation illustrates that (8 ( 2) + (6 ( 2) is the same as (___ + __) ( 2.

2.Make a generalization using fraction form:A + B =______________________________

C C _________________

A. The Basic Definition of Division

Just as subtraction is the inverse of addition, division is the inverse of multiplication.

BASIC DEFINITION OF DIVISION

Division is the Inverse of Multiplication.

A B = ( is equivalent to B ( ( = A, for B 0.

The first number in a division is called the dividend, the second is the divisor, and the result is the quotient.

Dividend Divisor = Quotient

Example 1: Consider 12 3 = 4.

12 is the dividend, 3 is the divisor, and 4 is the quotient.

12 3 = 4 because 12 = 3 4.

In other words, if we can formulate a problem into the multiplication sentence, A ( = C, then we can find the unknown factor by reformulating the sentence into a division sentence: ( = C A. Notice that the product in the multiplication sentence corresponds to the dividend in the corresponding division sentence.

Example 2: The floor of a right rectangular solid measures 3 m by 2 m, and the solid has a volume of 30 m3.

What is the height of the solid?

V = LWH => 30 = 3 2 H => 30 = 6 H. So H = 30 m3 6 m2 = 5 m.

B. Two Major Interpretations of Division

All situations involving division are equivalent to multiplication problems with a missing factor. However, two quite different situations give rise to division.

1. Division as Partitioning: Total Number of Parts = Size of the Part

The total is known, the number of sets (multiplier) is known, but the size of the set (multiplicand) is unknown.

Example 3:Ten candies were distributed equally to five children. How many candies did each child get?

Solution AThe problem is to determine the size of the set given the number of sets. The solution can be found by partitioning. Ten partitioned into five equal parts yields two candies per part.

Solution BWe have an unknown multiplicand, namely the number of candies given to each child. Thus we have 5 B = 10. By the definition of division, B = 10 5.

Teaching Tip: Young children can partition a set by dealing out the elements in the set like cards in a card game. Later on, such experiences with partitioning can help children understand this basic meaning of division.

Example 4:A pizza has been cut into eight equal pieces, and Anne eats two pieces. If two people share the remaining pizza equally, how much of a pizza will each person eat?

If six pieces are split evenly between two people, each person will get three pieces.

As these examples illustrate, division can be used to find the size of a part given the original quantity and the number of parts into which it is partitioned. This is called the partitioning interpretation of division.

Partitioning Interpretation of Division

For B a natural number, A m can be interpreted to mean

the size of a part when A is partitioned into m equal parts.

A

A ( m

m parts

Units in Partitioning Problems

In situations involving partitioning, the quotient is the size of a part when the dividend is partitioned into the number

of parts specified by the divisor. Hence the quotient, as part of the dividend, has the same unit as the dividend.

Example 5: Sixty feet of rope is cut into 12 pieces of equal length. How long is each piece?

60 feet 12 = 5 feet

2. Division as Repeated Subtraction: Total Size of the Part = Number of Parts

Example 6: A class contains 24 children seated at tables in groups of four. How many tables are there?

* ** * * ** * . . . = 24

* ** ** ** *

Solution AAdd fours until we reach 24: 4 + 4 = 8, 8 + 4 = 12, 12 + 4 = 16, 16 + 4 = 20, 20 + 4 = 24.

We added 6 fours to get 24, so the answer is 6 tables.

Solution B Subtract 4 repeatedly from 24 until we reach 0: 24 - 4 - 4 4 - 4 - 4 - 4 = 0. We had to subtract

six fours, so there are six tables.

Solution CFind a missing multiplier m so that m 4 = 24. That is find m such that m = 24 4.

Division as repeated subtraction occurs in situations where a known quantity has been partitioned into equal parts of a known size. The problem is to determine the number of parts.

Repeated Subtraction Interpretation of Division

For B 0, A B can be interpreted to mean the number of Bs contained in A,

or the number of times B can be subtracted from A.

A

B B B B B B

A B

Number of parts of size B in set A

Stated another way, we have: A - B - B - B . . . - B = 0

Example 7:Since 36 - 9 - 9 - 9 - 9 = 0, we have 36 9 = 4.

Units in Repeated Subtraction

In situations involving repeated subtraction, the quotient is the number of divisors in the dividend. Hence the quotient does not have a reference unit. For this reason we say that the units of the dividend and divisor divide out, just as common factors divide out.

Example 8: How many 200s are in 600?

There are 3 sets of 200s in 600. Thus we can say that in the division of 6 hundred by 2 hundred , the hundreds units divide out.

Example 9: A child arranges six toy soldiers into sets of two soldiers each. How many sets are there?

6 toy soldiers 2 toy soldiers = 3 => There are 3 sets of two soldiers in the set of six soldiers.

C. Rational Numbers in Division

1. Quotients as Rational Numbers

Partitioning whole numbers can lead to parts with fractional sizes. Such problems reveal a surprising connection between quotients and fractions.

Activity 5.2B

A.Three pizzas are to be shared equally among four people. How much pizza does each person get?

1.The problem is to partition 3 pizzas into 4 equal parts and determine the size of the part.

That is, we want to find _____________ ___.

2a.Draw a diagram that shows how to solve this problem by cutting each pizza into four pieces. Shade the pieces to be claimed by the first person.

b.We have 3 pizzas 4 = 12 _______ of a pizza 4 = 3 ___________.

3.Thus 3 4 is equivalent to the rational number _______.

B.Use diagrams to solve the following problems.

1.Adriens will states that one ninth of his estate is to be given to a certain charity and the remaining part of the estate is to be partitioned equally between his two children. What fraction of the estate will each child inherit?

2.Annas will stated that one quarter of her estate was to be given to a certain charity and the remaining part of the estate was to be partitioned equally between her two children. What fraction of the estate did each child inherit?

The above activity illustrates the following relationship between quotients and fractions.

The Connection Between Quotients and Fractions

For any real numbers A and B, with B 0, A B is the same as A/B.

The relationship between A/B and A ( B is not obvious. For instance, consider 3 5 and 3/5. We can interpret 3 5 to mean the size of a part when three units are partitioned into five equal parts; we can interpret 3/5 to mean three of five equal parts of one unit. On the face of it, these seem to be very different problems. They are certainly different processes. Yet, as the following example illustrates, they yield the same result.

Example 10:To partition 3 acres into 5 equal parts:

1 acre 1 acre 1 acre

a.Convert 3 acres into 15 fifths of an acre.

b.15 fifths of an acre 5 = 3 fifths of an acre = 3/5 acre

Thus we have three interpretations for a fraction A/B.

1.A/B can refer to A parts of a unit that has been partitioned into B equal parts.

Example: 3/5 of an acre refers to three parts of an acre that has been partitioned into five equal parts.

2.A/B can refer to the ratio of two quantities, where for every A elements in the first quantity there are B elements in the second quantity.

Example: The ratio of girls to boys in our class is 3/5 means that there are three girls for every five boys.

3.A/B can refer to A divided by B. This interpretation has multiple meanings, including partitioning and repeated subtraction.

Example: If three acres of land are to be shared equally by five heirs to an estate, then each heir receives

3 acres 5 or 3/5 of an acre.

2. Rational Number Dividends and Divisors

What is the meaning of an expression like 3/4 2? This division of a fraction by a whole number can be interpreted as partitioning. Just as with whole numbers, the key to partitioning a fraction into two equal parts is to convert the fraction into a form that includes a multiple of two.

Example 11:Partition 3/4 of a pizza equally between two people.

Cut each of the fourths into two parts. That is, convert 3/4 to 6/8. Now we have:

6 eighths of a pizza 2 = 3 eighths of a pizza = 3/8 pizza

What is the meaning of an expression like 3 3/4 or 3/4 1/8? These divisions can be interpreted in the context of repeated subtractions as the next activity illustrates.

Activity 5.2C

A.Mary is going to make giant three-quarter-pound hamburgers. How many hamburgers can she make with three pounds of hamburger meat?

1.Solve this problem using repeated subtraction.

2.The problem is to find out how many quarter-pounds are in 3 pounds.

a.The division associated with this problem is 3 lbs _____ lb.

b.Convert 3 lbs to quarter-pounds.

c.3 lbs 3/4 lb = ___ quarter-pounds ___ quarter-pounds = _____ (Note that the units cancel out.)

d.So Mary can make ____ hamburgers.

B.Solve the following problems without using standard algorithms.

1.If a strip of metal that is 15 sixteenths of an inch long is cut into pieces that are 3 sixteenths of an inch long, how many of these smaller pieces will there be?

2.If a strip of metal that is 3/4 of a foot long is cut into pieces that are 1/8 of a foot long, how many of these smaller pieces will there be?

3.Suppose a restaurant serves a glass of wine that is 1/8 of a liter. If a box of wine holds 15/6 liters, how many glasses of wine can be poured from this box? Include the fractional part of a glass in your answer. [Hint: Convert to twenty-fourths.]

Understanding the process of dividing a fraction by a fraction is not straightforward. To make sense of these types of division, it is helpful to use the repeated subtraction interpretation of division and a common unit. As the following examples illustrate, this boils down to finding a common denominator.

Example 12:Suppose six acres are divided into three-quarter-acre lots. How many lots will there be?

6 acres = 24 quarter-acres => 6 acres ( 3/4 acre = 24 quarter-acres ( 3 quarter-acres = 8

Example 13:If 21/2 tons of gravel are to be poured into bins each holding half of a ton, how many bins are

needed?

Convert to half-tons: 21/2 tons ( 1/2 ton = 5 half-tons ( 1 half-ton = 5

Fortunately a relatively simple pattern occurs. Following is the explanation for this pattern.

1. Use the Fundamental Property of Fractions to generate equivalent

A/B ( C/D=AD/BD ( BC/BD

fractions with the same denominator.

2. Since AD and BC have the same unit, namely the unit fraction 1/BD, AD/BD ( BC/BD=AD BC

this division can be interpreted to mean How many BCs are in AD?

3.As we shall see, a quotient can be interpreted as a fraction.

AD BC=BC/BD

4.The Shortcut

A/B C/D=AD/BC

Teaching Tip: Sometimes this shortcut is called cross-multiplying. This is a very bad idea. Cross-multiplying more commonly refers to a shortcut used to solve proportions. For instance, the proportion 3/x = 8/5 can be solved by cross-multiplying to obtain the equivalent equation 3 5 = 8x. In contrast, the result of cross-multiplying when dividing fractions is a fraction, not an equation. When different processes are referred to by the same name, students often confuse the results. Thus it is better not to refer to the division of fractions as cross-multiplying. A pedagogically better way of computing the quotient of two fractions, which involves inverting the divisor, will be discussed later in this chapter.

Example 14: Finding 11/2 ( 1/4 using a variety of methods:

(a)Repeated subtraction as visualization: In your minds eye, visualize the number of quarter pieces of pizza in 11/2 pizzas. There are six such pieces.

(b)Formal repeated subtraction: 11/2 - 1/4 - 1/4 - 1/4 - 1/4 - 1/4 - 1/4 = 0 => 11/2 ( 1/4 = 6

(c) Common unit: 11/2 ( 1/4 = 6 fourths ( 1 fourth = 6

(d)Shortcut: 11/2 ( 1/4 = 3/2 1/4 = (3 4)/(2 1) = 6

D. Remainders and Two Useful Theorems

It is a curious fact that inverse operations are often not as well behaved as the original operations. Here is a case in point: multiplying two whole numbers yields a whole number, but dividing two whole numbers can result in a remainder.

Activity 5.2D

1.It takes 15 inches of ribbon to make a certain kind of bow.

a.Suppose Mary has 50 inches of ribbon. How many bows can she make with this ribbon, and how much ribbon will be left over?

b.Specify a length of ribbon that can be used to make bows without having any ribbon left over.

c.Give a general description of the lengths of ribbon that can be used to make bows without having any ribbon left over.

d.Use your calculator to determine how much ribbon will be left over if Mary makes as many ribbons as possible from a roll containing 88 feet of ribbon. Report your answer in inches.

2.The maximum class size for kindergartners in one state is 18. A school has 50 kindergartners. What is the smallest number of kindergarten classes that this school must have?

3.At a practice, a coach divides his team into groups of four girls each. He assigns any remaining players to be referees. If 23 players show up, how many will be referees?

4.Three children steal into the kitchen late one night and find their mothers secret cache of 11 chocolate bars.

a.If the children decide to split the chocolate bars evenly, how many chocolate bars

_____________

does each child get?

b.In the context of this problem, explain the meaning of the remainder of 2 in the equation 11 ( 3 = 3 R 2.

c.Explain what happened to this whole number remainder in this problem.

Division will lead to a left-over when the dividend is not a whole number multiple of the divisor.

Example 15: Twenty-six grapefruits are being packed into boxes that hold six grapefruits each. How many boxes will be filled, and how many grapefruits will be left over?

26 is not a multiple of 6. Instead, 26 = 4 6 + 2. So there will be four full boxes with two grapefruits left over.

26 grapefruits

6 grapefruits 6 grapefruits 6 grapefruits 6 grapefruits 2 g.f.

In general, if A and B are whole numbers, then either (a) A is a whole-number multiple of B or (b) A is the sum of a whole-number multiple of B and a remainder. The remainder has the same unit as the dividend. The relationship between the dividend A, the divisor B, the whole number quotient q and the remainder r is summarized as follows.

The Division Theorem

For any whole numbers A and B, with B 0, A can be written as qB + r,

where q and r are unique whole numbers, with 0 r < B.

A

EMBED Word.Picture.8

q Bs r

This theorem is called the Division Theorem because of the connection between A divided by B and A written as

q B + r. If A consists of q Bs plus a remainder r, then A B is equal to q with a remainder of r.

Example 16: The following statements convey the same information:

a.242 = 5 43 + 27

b.242 contains 5 forty-threes with 27 left over.

c.242 43 is equal to 5 with a remainder of 27.

It is common (at least in elementary school) to indicate a whole-number quotient and remainder using the R notation, as illustrated in the next example. Note that R does not indicate addition.

Example 17: 14 5 = 2 R 4 means that 14 = (2 5) + 4. In other words, 14 contains 2 fives with 4 left over.

Another useful theorem related to division is illustrated in the following example.

Example 18:Bridge is a card game involving exactly four players. Marge is organizing a bridge party at her retirement community. First eight people sign up, so Marge prepares two tables for four. Then another 12 people sign up, so Marge prepares three more tables for a total of five tables. Obviously, if all 20 people had signed up at the same time, Marge would also have prepared five tables. This illustrates the following fact: 20 = 12 + 8 = 12 + 8

4 4 4 4

In general we have the following result.

Quotient of a Sum Property

If A, B, and C are real numbers with C ( 0, then A + B = A + B

C C C

This is called the Quotient of a Sum Property because it states that the quotient of a sum (A + B) is the same as the sum of the quotients A/C and B/C.

Teaching Tip: Many students find the Quotient of a Sum Property rather strange when it is read from left to right. Just ask them to read the property from right to leftin this direction the property should be very familiar!

See how the Quotient of a Sum Property plays a role in the next example.

Example 19:Forty-one acres are to be divided into eight lots of equal size. What will be the size of each lot?

Since 41 acres = 8 5 acres + 1 acre, each lot will include 5 acres. If the remaining acre is partitioned equally among the eight lots, each lot will increase by an eighth of an acre. Thus the total size of each lot will be 51/8 acres.

Summary: 41 acres/8 = 40 acres/8 + 1 acre/8 = 5 acres + 1/8 acre = 51/8 acres

As this example shows, a quotient can be expressed as a non-whole number that includes the remainder as a fractional part of the divisor.

If A = qB + r, then A ( B = qB + R = qB + r = q + r

B

B B B

Example 20:387 ( 8 = (48 8 + 3) ( 8 = 48 8 + 3 = 48 8 + 3 = 48 +3 = 48

8 8 8 8

The concept of whole number quotients also applies to problems involving fractional dividends and divisors. In such cases, be careful to interpret the remainder correctly.

Example 21:Suppose three and a quarter liters of acid is being poured into half-liter containers.

a. How many containers will be filled? Include fractional parts.

Compute the answer using the shortcut: 31/4 liters ( 1/2 liters = 13/4 ( 2/1 = 13/2 = 61/2

This means that 61/2 containers will be filled.

b.How many full containers will there be, and how much acid will be left over?

Since 31/4 ( 1/2 = 61/2, there will be six full containers. The left-over acid would fill 1/2 of a half liter container, so there is 1/4 of a liter of left-over acid.

Remember that the fractional part of a quotient is equal to the remainder divided by the divisor. To find the remainder in terms of original units, multiply the fractional part of the quotient by the divisor.

Finding Whole Number Remainders from Quotients in Decimal Form

If a calculator is used to find a quotient, the answer is usually expressed in decimal form. The whole number quotient q is clearly identifiable as the whole number part of this decimal. One way to find the whole number remainder is to use the relationship between A, B, q, and r: A = qB + r. Solving this for r yields the following equation: r = A qB. In other words, find r by subtracting q Bs from A.

Example 22:242 43 = 5.6279069 => 242 = 5 43 + r => r = 242 5 43 = 27

Described in another way: When we compute 242 43 as 5.62, we have determined that there are five 43s in 242, plus a remainder. To find the remainder, subtract the five 43s from 242.

Another way to find the whole number remainder r is to recognize that the fractional part of the decimal represents the ratio of r to the divisor. Thus r can be found by multiplying this fractional part by the divisor. Avoid rounding errors by using all the digits provided by your calculator for the fractional part.

Example 23:242 43 = 5.6279069 => r = 43 0.6278069 = 27

Situations Involving Whole Number Quotients and Remainders

While there are many division situations in which the answer is a non-whole number quotient, there are many division situations in which the answer must be a whole number. These situations usually involve units that are indivisible, i.e., units that cannot be partitioned into smaller units.

Example 24:The organizer of the schools May Day event decides to form six rows of chairs for the audience. She wants the same number of chairs in each row. There are eighty-seven chairs available. How

many chairs should be in each row?

Find 87 6 = 14 r 3. This means that 87 = 14 6 + 3. Put 14 chairs in each row, with three chairs left over.

Example 25:The sixth grade is scheduled to see the play The Lion King, but the bus has broken down. Parents with minivans are being recruited to take all 87 sixth graders to the play. If each minivan carries six

passengers (not including the driver), how many parents with minivans need to be recruited?

Since 87 = 14 6 + 3, we can fill up 14 vans and part of another van. This means we need 15 vans to take all 87 sixth graders to the play. (Alternately, line up 14 parents with minivans and one parent with a sedan.)

As the above examples illustrate, sometimes the quotient is rounded up and sometimes it is rounded down to find the appropriate answer to a question. Use common sense to decide which way to round.

Sometimes the remainder plays the starring role in a division problem! That is, sometimes the relevant part of a division is not the quotient but the remainder. Consider the next examples.

Example 26:January 1, 2002 fell on a Tuesday. On what day did January 31, 2005 fall?

Starting with January 1, every seven days there will be another Tuesday. January 29 will fall on a Tuesday because it is 28 days after January 1. Thus January 31 will fall on a Thursday.

Example 27:December 25, 2005 falls on a Sunday. On what day will December 25, 2009 fall?

There are 365 days in most years and 365 = 52 7 + 1. This means that a year consists of 52 full weeks plus a day. That extra day, the remainder in the division 365 ( 7, means that from one 365-day year to the next, every date moves forward one day. So December 25, 2006 will fall on a Monday and December 25, 2007, will fall on a Tuesday. The year 2008 is a leap year with 366 days, the extra day occurring on February 29. This means that all dates after February 29 move forward two days from the previous year. Thus December 25, 2008 will fall on Thursday. December 25, 2009, will fall on a Friday.

Teaching Tip: An efficient way to identify leap years, which normally occur when the year is divisible by four, is to use the following property: a whole number is divisible by four if and only if the last two digits are divisible by four. For example, 2036 will be a leap year because 36 is divisible by 4.

Various examples in this section have illustrated four effects of the remainder. These are summarized below.

Four Possible Effects of the Remainder

1.Eliminate the remainder. Round the quotient down to the nearest whole number.

2.Round the quotient up to the next whole number.

3.Retain the remainder as the answer.

4.Include the remainder in the answer as a fractional part of the divisor.

Teaching Tip: Students have been known to lose track of the existence of whole number quotients and remainders in later grades because they become so accustomed to using calculators that yield only decimal quotients. Their memories can be jogged by working problems that require whole number answers, not decimal answers.

Summary

Division is defined as the inverse of multiplication. From an understanding of multiplication as finding a total given a number of repeated sets, there arise two understandings of division. The first is to find the size of the repeated set. The second is to determine the number of these repeated sets. Complications occur because of the backwards nature of division, especially as it relates to the existence of remainders and the behavior of rational numbers.

5.2 Homework Problems

A.Concepts

1.Definitions, Properties, and Vocabulary

a.State the basic definition of division.

b.Use the basic definition of division to rewrite A ( = ( as a multiplication sentence.

c.Rewrite the following multiplication sentence as a division sentence: 4 ( ( = 2/3

2a.Use the basic definition of division to rewrite 8 ( 0 = ( as a multiplication sentence.

b.Explain why this multiplication sentence, and hence the division sentence, has no solution.

3.Identify the divisor, dividend, and quotient in the following division sentence: 6 ( 1/3 = 18.

4.List three numbers in each of the following sets.

a.Multiples of 12

b.Factors of 12

c.Numbers divisible by 12

5.Justify your answers to the following.

a.Is 24 a multiple of 8? b. Is 24 divisible by 8? c. Is 24 a factor of 8?

d.Is 0 a multiple of 8? e. Is 0 divisible by 8? f. Is 0 a factor of 8?

6.Why can division always be interpreted as the process of finding an unknown factor?

7.Which of the following can be interpreted as A ( B, for B ( 0?

a.A/Bb. A : Bc. Number of Bs in Ad. (, where A ( ( = B

8.Explain the meaning of 5/6 using:

a.the basic definition of an elementary fraction;

b.division interpreted as partitioning;

c.division interpreted as repeated subtraction, with a whole number quotient and remainder.

9.The Division Theorem

a.For any two whole numbers A and B, A can be written as a ? of Bs plus a ? .

b.Show this relationship for A = 17 and B = 3.

c.Show this relationship for A = 6 and B = 17.

d.If A = cB + d, describe A ( B.

e.Fill in the blanks: 37,893 = ? ( 87 + ? and 37,893 ( 87 = ? R ?

10.Fill in the blanks.

a.If 27 4 = 63/4, then 27 = ? 4 + ?.

b. If 473 = 8 56 + 25, then 473 ? = 8 + 25/?.

11.Which of the following are equivalent to 56 = 9 6 + 2?

a.56 9 = 6 R 2 b. 56 6 = 9 R 2 c. 56 9 = 62/56 d. 56 9 = 62/9 e. 56 6 = 9 + 2

12.The Quotient of a Sum Theorem

a.State the sum that is the same as (x + y)/z.

b.According to the Quotient of a Sum Theorem, 96/3 is the same as 90/3 + ? .

c.Determining the number of threes in 96 is the same as determining the number of threes in 90 and adding this to the number of threes in ? .

d.The Quotient of a Sum Theorem states that first adding A and B and then dividing the sum by C is the same as first dividing A by C and dividing B by C and then ?.

B.Division as Partitioning

1.Describe the meaning of 6 ( 2 in terms of partitioning.

2.Identify which of the following three quantities is unknown in a partitioning problem:

total, size of repeated set, number of repeated sets.

3.Write and solve a story problem that involves partitioning for each of the following conditions:

a.The dividend is three fifths.

b.The quotient is three fifths.

c.The dividend is 0.

d.The divisor is 0.

4a.Identify which of the following three quantities have the same units in a partitioning problem: total, size of repeated set, number of repeated sets.

b.Explain why these two quantities have the same unit. Include an example.

5.Use the partitioning interpretation of division to explain why A ( A = 1, for A ( 0.

6a.For division interpreted as partitioning: (total) (number of parts) = ? .

b.What type of number occurs as the divisor in a partitioning problem, and why?

7a.A B can be interpreted as the process of partitioning a set of size A into B parts and finding ? .

b.Using this interpretation, we have 8 people 2 = ? . Justify your answer.

C.Division as Repeated Subtraction

1.Describe the meaning of 6 ( 2 in terms of repeated subtraction.

2.Identify which of the following three quantities is unknown in a repeated subtraction problem:

total, size of repeated set, number of repeated sets.

3.Write and solve a story problem that involves repeated subtraction for each of the following conditions:

a.The dividend is three fifths.

b. The quotient is three.

c.The dividend is 0.

d. The divisor is 0.

e.The divisor is 1/3.

4a.Identify which of the following three quantities have the same units in a repeated subtraction problem: total, size of repeated set, number of repeated sets.

b.Explain why these two quantities have the same unit. Include a word problem as an illustration.

5.Use the repeated subtraction interpretation of division to explain why A ( A = 1, for A ( 0.

6a.A B can be interpreted as the process of finding how many times B must be subtracted from A to get ? .

b.Using this interpretation, we have 6 feet 3 feet = ? because ? .

7a.Use the repeated subtraction interpretation of division to explain why 8 tenths 2 tenths = 4.

b.Explain why A/B ( C/B = A ( C in terms of repeated subtraction and the common unit of the dividend and divisor.

8.Invent a story for each of the following and find the answers.

a.18 lbs 3 lbs = ? b. 18 lbs 3 = ?

9.Which of the following can be computed by determining M ( 2?

a.What number should I multiply 2 by to get M?

b.What is the size of a part if M is partitioned into two parts of equal size?

c.How many twos are in M?

d.If M is partitioned into halves, how many halves will there be?

D.Rational Numbers and Division

1.Rational Divisors

a.Invent a story that can be solved by finding 31/3 2/3.

b.Draw a labeled diagram that illustrates how to find the solution.

2.Rational Dividends

a.Invent a story that can be solved by finding 41/2 3.

b.Draw a labeled diagram that illustrates how to find the solution.

3.Explain why 15/8 3/8 is the same as 15 3 using the repeated subtraction interpretation of division and unit fractions.

4.Rational Quotients

a.Use a diagram to illustrate how to divide two pizzas evenly among three people.

b.Fill in the blanks with appropriate unit fractions: 5 6 = 30 ? 6 = 5 ?

c.Suppose 4 units are partitioned into M equal parts. Describe the size of a part.

5.Find 11/2 3/8 by the following methods.

a.repeated subtraction

b. common denominators

c. a third method of your own choosing

6.Which of the following can be computed by determining M ( 1/2?

a.What number should I multiply 1/2 by to get M?

b.What is the size of a part if M is partitioned into two parts of equal size?

c.How many twos are in M?

d.If M is partitioned into halves, how many halves will there be?

E.Remainders

1. Basics

a.Under what circumstances will division of whole numbers include a nonzero remainder?

b.When the remainder is 0, the dividend must be a (multiple/factor/term/product) of the divisor.

c.A remainder in a division problem can be considered as a fractional part of the ? .

2.Find the whole number quotient and remainder for the division 4379 ( 35.

3a.List the four possible effects of a remainder on the answer of a division problem.

b.Invent a word problem for each of these four effects.

F.Problem Solving

1.The teacher decides to organize his class of 22 students into teams of four children each, with the leftover children working with her. How many teams will there be, and how many children will be working with the teacher?

2.If a 73/5 acre lot is to be divided equally into 6 lots, what will be the size of each lot?

3.I cut oranges into fourths and gave a piece to each of 22 children. How many whole oranges did I use?

4.Twenty-five children are going on a field trip in vans holding 7 children each. How many vans are needed?

5.January 1, 2004 falls on a Thursday. Determine the day of the week for January 1, 2012.

6.The 15th day of a certain year falls on a Thursday. On what day of the week will the 327th day of the year fall?

7.A construction company is paving a 21/4 mile stretch of freeway at the rate of 200 yards a day. How long will it take to complete the job?

8.The Martian year is almost exactly 687 days. Suppose Martians have seven-day weeks like we do. If the Martian year of 2005 started on a Monday, on what day of the week would the Martian year of 2006 fall?

9.On Venus the year is a little over 224 days. Suppose Venutians have five-day weeks (Monday through Friday), with leap years that occur every three years and contain two extra days. The Venutian year of 2005 started on a Monday and is a leap year.

a.On what day of the week will the Venutian year of 2006 start?

b.On what day of the week will the Venutian year of 2009 start?

10.The water in a tank weighs 668.75 pounds. One cubic foot of water weights 62.5 pounds. How many cubic feet of water does the tank hold?

11.A manufacturer had a roll of 750 yards of linen goods that he cut into pieces 27 inches long to make dish towels. He sold the towels at $4.80 a dozen.

a.If he sold all the towels, what was his revenue? [Hint: Revenue is the amount of money taken in.]

b.If the cost of producing and cutting the roll of linen goods was $380, what was the profit per towel?

12.A chemistry professor is preparing for a lab with 18 students. Each pair of students will need a tenth of a liter of a 40% nitric acid solution for the days experiment. How much of this acid must the professor prepare?

13.A 31/4 yard strip of steel is to be used to make pieces that are a half foot long. How many pieces can be made, and how much steel will be left over?

14.An estate worth one and a half million dollars is to be shared equally among five heirs. How much does each heir inherit?

15.Eight and two thirds miles of interstate are to be paved in 20 days. How much road should be paved each day, on average? Report your answer in feet.

16.How many nails weighing 3/8 of an ounce can be made from a third of a pound of metal?

____________________________________________________________________________________________

______________________________________________________________________________________ ______

5.3 Multiplication as a Means of Comparison

Besides repeated addition, multiplication has a second major meaning. This is illustrated in the following activity.

Activity 5.3A

A.Jerry, Nick, and Melissa went running one Saturday morning. Melissa ran twice as far as Jerry. Nick ran two thirds as far as Jerry. Let Jerrys, Nicks and Melissas distances be represented by J, N, and M respectively.

1.Write an equation expressing the relationship between J and M.

______________

2.Suppose Jerry ran 12 miles.

a.Use a diagram to determine how far Nick ran.

b.Write an equation expressing the relationship between J and N.

______________

3.In the last thirty years, there has been a 200% increase in the price of bread.

a.____________________________________ is 200% of ___________________________________________

b.Label three sets in the following diagram: the old price,

the increase, and the new price.

c. If a loaf of bread cost 50 thirty years ago, how much does it cost now? Label the diagram _____________

appropriately to find the answer.

4.Suppose an employee gets one tenth off the sticker price.

a.____________________________________ is 1/10 of _____________________________________________

b.How much will an employee pay for an item with a sticker price of $60? Label the following diagram using the information in this problem and figure out the answer. Label three sets: the sticker price, the discount, and the discounted price.

Multiplication in Comparison Situations

In the above problems, multiplication is used to describe the relationship between two quantities. In such situations, the product is not a total but an amount that is described relative to a base of comparison. The multiplier indicates how many or how much of the base is necessary to generate the described amount.

Described Amount = m Base of Comparison

Example 1: Melissa ran twice as far as Jerry.

Let J = Jerrys distance and M = Melissas distance: J ______________

We have M = 2 J.

M _____________________________

Example 2:A 200% increase means that the increase is two times the original price. If the original price was 50, then the increase is 2 50, or 100. The new price will be 50 + 100 or $1.50.

Teaching Tip: Especially when an increase is over 100% of the original value, students may forget to add the increase to the original price to find the final value. Warn them to be extra careful when they are working with these types of problems.

Rational Number Multipliers

If the multiplier is a whole number, multiplication in comparison situations is similar to repeated addition. In the above example, for instance, 2 J still means J + J. Unlike repeated addition, however, multipliers in comparison situations can be non-whole rational numbers. As the next example illustrates, the meaning of these multipliers is directly based on the meaning of elementary fractions.

Example 3:Nick ran two thirds as far as Jerry. This means that Nicks distance N is two thirds of Jerrys distance J, or two of three equal parts of Jerrys distance.

J ______________

N = 2/3 of J

N __________

Since 2/3 plays exactly the same role in this example as 2, the multiplier 2 did in the previous example, it seems reasonable to interpret 2/3 of J as multiplication. For instance, if Jerry ran 12 miles, then 2/3 J means to partition 12 into three equal parts and select two of these equal parts: 2/3 12 = (12 3) 2 = 8.

In general, for any positive rational number N/D, N/D B means N/D of B, where N/D is interpreted as an elementary fraction. That is, N/D B means N of D equal parts of B: N/D B = (B D) N.

Example 4:If Y = 7% X, then Y is 7/100 of X, or seven of one hundred equal parts of X.

Example 5: The guests ate two thirds of a box of 24 candies. How many candies did they eat?

Solution ATo find 2/3 of 24, first partition 24 into three equal parts. This yields 8 candies in each part, with 16 candies in two parts. The guests ate 16 candies.

Solution B2/3 24 candies = 2/3 of 24 candies = 2 (24 candies 3) = 2 8 candies = 16 candies

As the next activity illustrates, this process does not always yield a whole number.

Activity 5.3B

A.John ordered eight pizzas for a party. His guests ate two-thirds of all the pizza. How much pizza did they eat?

1.Solve this problem by partitioning the pizzas below and shading the pizzas that were eaten.

2.2/3 of 8 pizzas = 2/3 of ____ thirds of a pizza = 16 ___________________________ = 51/3 ____________

B.Four fifths of the City Council of Newton voted in favor of a tax increase. Of those in favor of a tax increase, two thirds indicated that the increase should be more than 1%. What fraction of the city council was in favor of a tax increase over 1%?

1.Suppose the large rectangle to the right represents the Newton City Council.

a.Shade the area representing those who voted in favor of a tax increase.

b.Stripe the area representing those who favored an increase of more than 1%.

c.Use this diagram to find the answer to the question.

______________

2.Symbolically:

(1) The problem is to find _____ of _____ of the city council.

(2) Convert the base so that its numerator is a multiple of 3: 4/5 = 12/____.

3a.Solve the following problem by using fifteenths as the unit.

2/3 4/5 = 2/3 of 12/15 = 2/3 of 12 _______________ = 8 ________________ or 8/____

b.The pattern that occurs indicates the following shortcut: 2/3 4/5 = (2 4)/(___ ___)

Teaching Tip: Fractions such as 4/5 can be written as either four-fifths or four fifths. The use of two separate words emphasizes fifths as the primary unit; the use of a hyphenated word emphasizes 4/5 as a single unit.

Parts of Parts

As the last problem in the above activity illustrates, it is common to describe parts of parts using multiplicative comparisons. This leads to expressions such as 2/3 of 4/5 of the City Council. How much is 2/3 of 4/5? The following example shows several ways of determining the answer, all involving the identification of fifteenths as the key unit.

Example 6:Four fifths of the class passed the test. Of those who passed, two thirds made at least a B. What

fraction of the class made at least a B?

Students making at least a B = 2/3 of those who passed

= 2/3 of 4/5 of the class

= (2/3 4/5) of the class

Solution A Use the Fundamental Property of Fractions to convert 4/5 to an equivalent fraction with a numerator that is a multiple of three: 2/3 4/5 = 2/3 of 4/5 = 2/3 of 12/15 = 2/3 of 12 fifteenths = 8 fifteenths.

Solution BUse a one-dimensional line segment partitioned into five equal parts. Partition each of these parts into three parts and identify 2/3 of the small parts within 4/5 of class.

4/5 of class

4/5 = 12/15

|_ __|_ __|__ _|_ __| |

|_._ _|_ _ _|_ _ _|_ _ _| |

2/3 of 12/15 = 8/15

Solution CUse a two-dimensional area diagram. Use vertical lines to partition the rectangle into five equal parts and then use horizontal lines to partition 4/5 into thirds. Extend the horizontal lines to partition the entire rectangle into thirds in order to determine the size of the smallest part relative to the whole.

4/5 of the whole

the whole

2/3 of 4/5 of the whole = 8/15 of the whole

Partitioning a quantity into five parts and then partitioning each of these five parts into three parts

creates a total of 15 parts. As the diagram illustrates, 2/3 of 4/5 includes 8 of these 15 parts, or 8/15.

The above example indicates that there is a surprisingly simple way to compute the product of two fractions: simply multiply the numerators and multiply the denominators: A . C = A C.

B D B D

Thus, for example, we can compute 2/3 4/5 as follows: 2/3 4/5 = (2 4)/(3 5) = 8/15. The justification for this easy shortcut is not the least bit obvious. The two-dimensional area diagram may be the most straightforward way to verify that this shortcut works, but the most fundamental explanation is based on understanding that 2/3 4/5 means

2/3 of 4/5, and recognizing that 2/3 of 4/5 is the same as 2/3 of 12 fifteenths.

Teaching Tip: A good algorithm for computing the quotient of rational numbers can be obtained by combining two patterns. We have just noted that A/B D/C = AD/BC. Previously we found that A/B C/D = AD/BC. So we have:

A C = A . D = AD

B D B C BC

Since D/C is the inverse of C/D, this rule can be summarized as follows: To divide fractions , invert the second fraction and multiply. Be sure to stress to students that the second fraction, not the first, is to be inverted.

Multiplication with Decimals and Percents

If the multiplier m is between 0 and 1, m is often expressed

in percent form. While the form of the multiplier has no effect

on the meaning of the comparison, the use of percent (which

means hundredths) as a unit makes the use of grid paper

almost a necessity for drawing an illustrative diagram.

Example 7: A is 3/4 of B => A = 3/4 B

=> A = 75% B

To compute answers, convert percents to decimal form and use the rules for decimal multiplication. (Justifications for these rules will be discussed later.)

Example 8:Becky invested 60% of her bonus in bonds and put the rest in her savings account. If her bonus was $2500, how much money did she put in her savings account?

Amount invested in bonds= 60% of B, where B is the bonus B

=> Amount left in savings = 40% of bonus

= 0.4 $2500 bonds savings

= $1000

60% of B 40% of B

Identifying the Components of Multiplicative Comparisons

To understand a multiplicative comparison, it is very important to identify the described amount and the base of comparison. As the next activity illustrates, this is not as easy to do as one might think.

Activity 5.3C

1.State the amount being referred to by the number in the following situations.

a.Alexandrias salary now is three times what it was at her part-time position. __________________________

b.One-third of my salary is used to pay my rent. ___________________________

c.Hamilton County has a 9.25% sales tax. ___________________________

2.For each of the above situations, describe the base to which the described amount is being compared.

a.___________________________ b. ___________________________ c. ___________________________

3.Suppose a real estate agent earns a 10% commission for selling a house. Fill in the following blanks.

______________________________________ is 10% of __________________________________________

4.Suppose you buy an item at a 1/4 off sale. Fill in the following boxes and blanks with either original price, sale price, or discount.

a. b. ________________ = 1/4 _____________________

c. _________________ = 3/4 ______________________

5.The newspaper reported that the price of gasoline jumped 9% from August 1 to August 2.

a.Identify each of the three amounts, F, G, and H in the following diagram as either price on August 1, price on August 2, or price increase.

F _______________________

F G

G _______________________

H

H __________________________

b.Fill in the following blanks with either price on August 1, price on August 2, price increase, or an

appropriate percent.

(1) is 100% of F.

(2) is 9% of

(3) is ___________ % of ____________________________________

Here are some pointers for identifying the components of a multiplicative relationship

1.Described Amount is (___)% of Base of Comparison => A = m B

A multiplicative relationship can always be phrased in the above form, which corresponds directly to the equation A = m B.

Example 9: Garys commission is one tenth of the selling price. Selling Price

=> commission = 1/10 selling price C

All Students

Example 10:Forty percent of the students are women.

=> The number of women is 40% of the students.

=> number of women = 40% of the students Women Students

2.% (Amount) or (Amount) rate

In many situations, the described amount is stated before or after the multiplier, with the multiplier expressed in percent form. The base, often unspecified, is usually a total or the original amount.

Example 11:The state has an 8% sales tax. If the sticker price is $30, how much is the tax?

Sales tax = 8% of sticker price = 0.08 $30 = $2.40

Example 12:The store gives a 15% employee discount.

employee discount = 15% original price

3.Part-Whole: Part = m Whole

a.Described Part

A part of a set is often described relative to the size of the set (the whole).

Example 13:One fourth of 40 students were sick. How many students were sick?

Number of sick students = 1/4 of total number of students

= 1/4 of 40

10 10 10 10

= 10

Total Number of Students

It is particularly common to describe a decrease relative to the original amount.

Decrease

Example 14:The size of the class decreased by a third when the instructor

enforced the prerequisites.

Remaining Students

Decrease = 1/3 of Original

Original Class

It is common to describe decreases using percents without stating the base of comparison. The original amount is always the base of comparison for a percent decrease.

Example 15:An 8% decrease in the price of gasoline means that the decrease is 8% of the old price.

b.The Other Part

With the part-whole model, we get two for the price of one. For example, if we know that 1/4 of the students are sick, then we also know that (1 - 1/4) or 3/4 of the students are not sick. If the multiplier is in percent form, we find the multiplier for the other part by subtracting from 100%. (100% is equal to 1.)

Describing the Other Part of a Set

If A = 25% of B, then the other part = 75% of B.

A Other Part

25% of B 75% of B

100% of B

Example 16:At a 25% off sale, what is the sale price of an item originally priced at $34.95?

Let P represent the original price. Note that P is 100% of itself.

Sale price = Original Price - Discount

= 100% of P - 25% of P

= 75% of P

25% P 75% P

= 0.75 $34.95

= $26.21

100% P

5.Expanding Amounts

a.The Increase

In a situation in which the size of a set increases, the increase is often described relative to the original amount.

Example 17:The value of a stock increases by 150%. If it used to be worth $6 a share, how much was the

increase and how much is the stock worth now?

Increase = 150% of old value

= 1.5 $6.00

old value increase

= $9.00

New Value = $6 + $9 = $15

New Value

The original amount is always the base of comparison for a percent increase.

Teaching Tip: Some students are disconcerted by the possibility that a percent may be larger than 100%. This may be due to associating percents exclusively with the part-whole type of comparison. When a part is compared to a whole, the percent certainly cannot exceed 100%. However, there are many types of comparisons in which the described amount can be larger than the base of comparison. For instance, an increase can exceed the original amount. In these situations the multiplier is larger than 100%.

b.The New Amount

We also get two for the price of one in increase situations because the new amount is the union of the old amount and the increase. This means that the new amount can be described in terms of the old amount by adding the percent increase to 100%.

The Relationship Between the New Amount N and the Original Amount B

B Increase

100% of B X% of B

New Amount

N = (100% + X%) of B

Example 18:Tuition has increased by 15%. If the tuition was $4000, what is the new tuition?

Tuition increase= 15% old tuition (T)

Old Tuition Increase

New Tuition = old tuition + increase

= 100% T + 15% T 100% T 15% T

= 115% T

= 1.15 $4000

115% T

= $4600

Reporting Sensible Answers

There are some situations in which non-whole numbers do not make sense as answers. In such situations, round the

answer to the nearest whole number.

Example 19: A teacher reported that two thirds of her class had done well on the year-end standardized tests. This teacher has 25 students. How many of her students did well on the tests?

Number of students who did well = 2/3 of 25 = 16.666...

About 17 students did well on the tests.

5.3 Homework Problems

A.Basic Concepts

1.Invent a word problem for the expression 3 ( 6 based on the following meanings of multiplication.

a.Repeated addition

b.Means of comparison

2. Consider the expressions 2/3 ( 6 and 2/3 of 6.

a.What is the relationship between these two expressions?

b.Explain the meaning of 2/3 of 6. Include a labeled diagram.

c.Invent and solve a comparison word problem that is solved by computing 2/3 ( 6.

3.Which of the following are true in situations involving multiplicative comparisons?

a.The described amount is never more than the base of comparison.

b.The described amount must be a part of the base of comparison.

c.The described amount can be a whole number multiple of the base of comparison.

d.If one part of a set is 10% of the set, then the other part must be 90% of the set.

e.If a set increases in size by 10%, then the original set is 90% of the enlarged set.

f.If a set decreases in size by 10%, then the shrunken set is 90% of the original set.

g.In comparison situations, the amount is always described explicitly.

h.In comparison situations, the base of comparison is always described explicitly.

4.Fill in the blanks.

a.If A is 2/3 of B and B is 1/4 of C, then A is ? of C.

b.If A is 20% of B and B is 150% of C, then A is ? % of C.

c.If A = 0.4 B and B = 0.8 C, then A is ? C.

5.In the following diagrams, the base of comparison B is represented by heavily outlined rectangles. Describe the shaded area in terms of B in each of these situations.

a.

b.

c. d.

66%B

6.In the following diagrams there are two bases of comparison, B1 and B2. B2 is represented by the largest rectangle and B1 is represented by the next largest rectangle. A is represented by the shaded area. For each diagram, estimate the multiplier for the following multiplicative relationships: (a) A and B1; (b) B1 and B2;

(c) A and B2. [Hint: Extend lines and draw extra lines to make good estimates.]

Example: (a) A is 1/2 of B1 (B1 is striped.)

(b) B1 is 1/3 of B2.

(c) A is 1/6 of B2.

a.

b.

c. d.

7.Suppose Y has the following length: . If possible, accurately draw the following lengths.

a.a length that is twice the length of Y b. a length that is 2 units longer than Y

c.a length that is one fourth the length of Y d. a length that is a fourth of a unit less than Y

e.a length that is 50% more than Yf. a length that is 25% less than Y

8a.Explain the meaning of 3/5 of a number M without making reference to multiplication.

b.What is the meaning of A/B M, where A/B is a positive rational number?

c.A/B 23 can be computed by dividing 23 by ? and multiply the result by ? .

9.Explain why 1/5 of 3 is the same as 3 5, with the latter interpreted as partitioning.

10.Which of the following are equivalent to 3/5 B?

a.3 of 5 equal parts of B

b. 3 (B 5)

c. B 3/5

d. Partitioning B into 5 equal parts and selecting three parts

11. Find the following products of rational numbers using unit fractions and the definition of elementary fractions.

a.2 6/5 = 2 ? fifths = ? fifths

b. 1/3 of 7 feet = 1/3 of 21 ? of a foot = ?

c.1/5 10/11 = 1/5 of ___ elevenths = ?

d. 1/6 5/3 = 1/6 of 30 ? = ?

12.Develop examples to show that of does not necessarily mean times, while times usually means of.

13.Use each of the following methods to find 1/4 1/3.

a.Creating an equivalent fraction with a numerator that is a multiple of 4

b.Partitioning a one-dimensional line segment

c.Partitioning a two-dimensional rectangle

14.Write a word problem for which it makes no sense to report 1/3 53 as 172/3.

15.Show how to find 3/5 of 10 sevenths using discrete sets.

16.Six long distance runners get a take-out order of six pizzas for dinner. When they get home, they find that they were shortchanged one pizza. They divide these five pizzas equally among themselves. Which of the following expressions can be used to determine how much pizza each runner gets?

a.6 ( 5 b. 1/5 of 6 c. 5 ( 6

d. 30 sixths ( 6

e. 1/6 of 5

B.For each of the following:

(a)Identify all described amounts A and their bases of comparison B.

(b)Write the corresponding multiplication equations of the form A = m B.

(c)Draw and label a picture illustrating the situation.

(d)Write multiplication equations for the other part or the new quantity.

1.The sales tax rate in Hamilton County, Tennessee, is 9.25%.

2.A shirt is on sale for 1/4 off.

3.Two fifths of the class are women.

4.The price of gas went up 10% this week.

5.The price of gas went down 10% last week.

6.Three quarters of the students at the university are undergraduates. Of these, one third are Asian.

7.In 1997, 23.4% of all pregnancies ended in abortion, with 55.4% of these abortions occurring within the first eight weeks of pregnancy.

C.Solve the following problems.

1.Adrian ran three fourths as far as Paula. Paula ran 24 miles. How far did Adrian run?

2.Alison makes $60,000 more than Larry and her salary is three times his. What is their combined salary?

3.An employee gets a 10% discount on merchandise.

a.What is the discount for an item marked $79.95?

b.Determine the price the employee will pay for an item marked $147.99 by doing a single multiplication.

4.A companys stock lost 9/10 of its value when the company went bankrupt.

a.If the stock used to be worth $20 per share, how much is it worth now?

b.If the stock is now worth $20 per share, how much was it worth befo