key stone problems… key stone problems… next set 1 © 2007 herbert i. gross

Key Stone Problems…Key Stone Problems…

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Set 1© 2007 Herbert I. Gross

You will soon be assigned five problems to test whether you have internalized the

material in Lesson 1 of our algebra course. The Keystone Illustrations below are

prototypes of the problems you'll be doing. Work out the problems on your own.

Afterwards, study the detailed solutions we've provided. In particular, notice that several different ways are presented that could be used to solve each problem.

Instructions for the Keystone Problems

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© 2007 Herbert I. Gross

As a teacher/trainer, it is important for you to understand and be able to respond

in different ways to the different ways individual students learn. The more ways

you are ready to explain a problem, the better the chances are that the students

will come to understand.

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The first solution we present is the one we feel is the "simplest". However when it

comes to teaching, there is no "one size fits all" that will help every student. These

other methods are there to be used as supplementary approaches, that is, to be used to help students who do not grasp

the problem at first sight.

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Our first problem illustrates what we mean by a direct computation, and our second illustration illustrates what we

mean by an indirect computation.

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Problem #1

The price of an object is marked “$67 plus tax”. How much must you pay for the

object if the tax is 7% of the marked price?

Keystone Illustrations for Lesson 1

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Answer: $71.69


Solution for Problem 1:

A 7% tax means that for every $1 of the marked price, you have to pay $1.07.

Since the marked price is $67, you would have to pay $1.07 sixty-seven times.

That is, you would have to pay…

67 × $1.07 or $71.69.

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• This is what is meant by a direct computation. In our PowerPoint

presentation, it is what we called the “plain English” model. That is, we start with the marked price, then multiply by $1.07, and the product is our answer.

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Note 1

Other Solutions for Problem 1Using a Formula

The process we used in solving this problem did not depend on the marked price being $67. More specifically, in terms of a formula if we let T denote the price (in dollars) including the 7% tax and M the marked price also in dollars), the formula would be… T = 1.07 × M (1)

In this example we would replace M by 67 to obtain… T = 1.07 × 67 (2)

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So if we wanted to use a calculator we would simply enter the following sequence

of key strokes.

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1.07 × 67 =


71.69

after which the display would show 71.69 .

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• Because the multiplication sign (×), and the letter (x) are easy to confuse, we

usually eliminate the times sign whenever possible by writing, say, 1.07(M) or simply

1.07M rather than 1.07 × M. This is consistent with writing 3 × 1 apple as

3 apples.

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Note 1

Other Solutions for Problem 1

Computing the Tax First

• We could also have approached the problem by first computing 7% of $67 (that is 0.07 × $67) to conclude that the tax was

$4.69, which we would then add to themarked price ($67) to obtain $71.69.

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Computing the Tax First

That is, another way to express equation (2) (T = 1.07 × 67) is:

T = 67 + 0.07(67) (8)

• Notice that equations (2) and (8) are equivalent. More generally…

M + 0.07M = 1M + 0.07M = (1 + 0.07)M = 1.07M

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Other Solutions for Problem 1The “Corn Bread” Model

Whenever we're talking about percent we may think of the total quantity as being a

corn bread that is pre sliced into 100 equally sized pieces. In this way each piece

represents 1% of the whole quantity. So we divide the corn bread that represents the

marked price into 100 equally sized pieces and then to represent the 7% tax we annex

an additional 7 of the equally sized pieces to obtain…

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The “Corn Bread” Model(drawn to scale)

Since the marked price is $67, each of the 100 pieces that represents the marked price represents $67 ÷ 100 or $0.67. Therefore the

tax, which is represented by 7 of these pieces is 7 × $0.67 or $4.69.

The Marked (pretax) Price$67

100 pieces

The Tax?

7 pieces

$4.69

The “Corn Bread” Model

Hence the total cost is…

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Based on the above diagram since there are now 107 pieces, and each piece represents $0.67, the total price is also represented by 107 × $0.67 which again yields $71.69 as the

answer.

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$67 + $4.69

Total Price $71.69

100 pieces + 7 pieces107 pieces × $0.67

/ 107pieces Total Price $71.69

• To add realism to our discussion the above diagram was drawn to scale.

However, if one can visualize what is contained in the diagram, there is no

need to draw it to scale. For example, suppose that we know that the marked price is $67, we could have written…

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Note 1

The Marked (Excluding Tax) Price100 pieces

$67.00

The Tax7 pieces

?

• Since the marked price is $67, each of the 100 pieces that represents the marked price

represents $67 ÷ 100 or $0.67. Therefore the tax, which is represented by 7 of these

pieces is 7 × $0.67 or $4.69. Hence the total cost is $67 + $4.69 or $71.69.

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Note 1

The Marked (Excluding Tax) Price100 pieces

$67.00

The Tax7 pieces

?$4.69

The Marked Price + Tax Price

$67.00 + $4.69 = $71.69

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= Total Price

• In summary, once we recognized that 100 pieces represented $67, elementary

arithmetic would have told us that since the total cost was represented by 107 pieces, it

would be 107 × $0.67 regardless of whatscale we had used.

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Note 1

The Marked (Excluding Tax) Price

100 pieces

The Tax

7 pieces

Total Price 107 pieces

107 × $.67 = $71.69

A rate is usually expressed as a phrase consisting of two noun phrases separated

by the word “per”. For example...• miles per hour (speed)• miles per hour per hour (acceleration)• dollars per person• apples per lawyer

In doing computations the word “per” is replaced by “ ÷ ”. For example, miles per

hour means miles ÷ hours ormileshours

Brief Review of Constant Ratesnext


If an automobile traveling at a constant speed goes 60 miles in 2 hours, its speedis 60 miles ÷ 2 hours (60 miles/2 hours ) or

30 miles per hour. If the speed remains constant the answer will be 30 miles per

hour no matter what time interval we use. To generalize this idea, if the automobile

travels m miles in h hours, we divide m by h to find the speed. In this case we know

that the speed is 30 miles per hour.Hence, m/h = 30 or m = 30 × h

Examplenext


To most students a phrase such as “miles per hour” seems less threatening than the

equivalent fraction form “miles/hours”. However, being able to switch from one form to

another can be very advantageous. For example, given a rate such as 3/7 of a mile per

minute, we can paraphrase it by recognizing that in 7 minutes it travels 3/7 of a mile 7 times,

or 3 miles. More visually…

Psychological Notenext


37

milesminutes

(which we read as 3 miles per 7 minutes).

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In other words, 3 miles per 7 minutes is the same rate as 3/7 of a

mile per minute. When we say 3 miles per 7 minutes, we do not see

any fractions.

Psychological Notenext


We are quite comfortable with our understanding that 6 ÷ 2 = 3. What might not be as obvious is that if 6 and 2 modify

the same noun, the answer remains the same. For example, 6 apples ÷ 2 apples = 3 (not 3 apples!). Namely, 6 apples ÷ 2 apples

means the number we have to multiply2 apples by to obtain 6 apples as the

product.

A Note On Canceling “Common” Rates

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Clearly 3 × 2 apples = 6 apples.

Do not confuse 6 apples ÷ 2 apples with 6 apples ÷ 2.

Namely, 6 apples ÷ 2 = 3 apples because 2 × 3 apples = 6 apples.

So in the language of common fractions 6 apples/2 apples = 3

and6 apples/2 = 3 apples.

A Note On Canceling “Common” Ratesnext


In other words if the same noun occurs in numerator and denominator, we may

“cancel” it in the same way that we can cancel the same numerical factor if it occurs

in both the numerator and denominator.This idea can play an important role when

we deal with problems that involve constant rates.

A Note On Canceling “Common” Rates

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Using Ratio and ProportionFrom the given information, we know that for every $1 of the marked price, the price

including tax will be $1.07. This is a constant rate problem and the rate is…

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$1.07 after tax (3)$1.00 pre tax

and because “dollars” is in both the numerator and denominator we may write this as…

1.07 after tax1.00 pre tax

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1.00 pre tax1.07 after tax

or equivalently

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• Observe the adjective/noun theme here.

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Namely… $1.07$1.00

$1.00$1.07

≠

However… $1.07after tax$1.00 pre tax

$1.00 pre tax$1.07 after tax

=

Note 1next

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So if the pre-tax cost is $67.00, and if we let A stand for the after-tax cost, the rate is also given by… A after tax

67.00 pre tax (4)

However since the rate (ratio) is constant, the rates expressed in expressions (3) and (4) are the same. In other words…

1.07 after tax1.00 pre tax

A after tax 67.00 pre tax

= (5)

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or… A after tax per pre tax 67.00

1.07 after tax per pre tax 1.00

=

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And since the nouns on both sides of equation (5) are the same, we may rewrite the equation as…

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1.071.00

A 67

=

We can then multiply both sides of equation (6) by 67 to obtain the same result we obtained using other methods. Namely:

A = 67 × 1.07 = 71.69

Note 1next

= 1.07 (6)

An Alternative Way to View Ratio and Proportion

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• While an expression such as $1.07after tax$1.00 pre tax

looks like a fraction, we may read it as

$1.07$1.00

after tax dollars per pre tax dollars.

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Note 1

•In this context, using our adjective/noun theme we may view equation (5) as our

noun phrase being “after-tax dollars per pre-tax dollars” And since the nouns onboth sides of the equation are the same, the adjectives must also be the same.

Hence:

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1.071.00

A67

=

And this is equivalent to the result we obtained using ratio and proportion.

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Note 1

• For students who are comfortable with “filling-in-the-blank” questions, we may

reword this question in the form…

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$67 before tax = $___ after tax (7)

Since the blank” is modifying “after tax”, in order to use our adjective/noun theme

the noun on the left hand side of the equation must also be “after tax”.

Note 1

To obtain this form we use a “cute” way of multiplying by 1. Namely, since $1.07 after tax is equivalent to $1.00 before tax,

we may view

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$1.07 after tax$1.00 before tax

to obtain the equivalent expression…

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as being another name for 1. Hence we may multiply the left

hand side of equation (7) by$1.07 after tax$1.00 before tax

$67 before tax 1


×

Note 1

and after canceling the common denominations, we see that the left side of

equation (7) is equivalent to $67 × 1.07 after tax; whereupon we obtain…

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$67 before tax 1


× = $_____after tax67

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Note 1


The “Function Machine” (Computer Program Model):

Using this method the input is the marked price, the output is the total cost and theprogram is “Multiply by 1.07”. That is…

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Marked Price

input program output

Total Cost× 1.07

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So in this case we obtain:

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Marked Price


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$67Total Cost

$71.69× 1.07$67


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Problem #2

The sales tax on an object is 7% of the marked price. How much was the marked

price if the price including the tax was $74.90?

Answer: $70.00

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Keystone Illustrations for Lesson 1


Solution for Problem 2

In terms of the “plain English” model, the answer was $74.90 after multiplying the marked price by 1.07. That is, we pick a

number and then multiply it by 1.07. So in terms of the calculator it would seem that the sequence of key strokes should be:

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? × 1.07 = 74.90


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Using a Formula

The formulas we use in both Problem 1 and Problem 2 are the same. That is, usingthe same notation as before, the formula here is also T = 1.07 × M (9)

However in Problem 2, it is T that is replaced by $74.90 and we thus obtain…

74.90 = 1.07 × M (10)

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Using a Formula

In this case 74.90 was obtained after we multiplied by 1.07. Therefore to paraphrase the indirect equation ...

74.90 = 1.07 × M (10)

into an equivalent form that can be solved by a direct computation, we would have to rewrite it as…

74.90 ÷ 1.07 = M (11)

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• Reading comprehension is very important. For example in using the

formula T = 1.07 × M, it is important to know whether a given number (such as 74.90 or

67) represents T or whether it represents M.

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• A main point here is that equation (10) involves an indirect computation (algebra)

while the equivalent equation (11) involves a direct computation (arithmetic).

Note 2

Other Solutions for Problem 2The “Corn Bread” Model

The corn bread model in this instance is the same as it was in our solution to problem 1, except now the corn bread consists of the

total cost.

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The Tax

7 pieces

The Marked (pretax) Price

100 pieces

That is, 100 pieces represent the marked price, and 7 pieces represent the tax. So our picture, becomes…

Total Price ($74.90)107 pieces

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The “Corn Bread” Model

Hence the total cost consists of 107 pieces, collectively worth $74.90.

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The Tax

7 pieces

The Marked (pretax) Price

100 pieces

Since the pieces are of equal size, the size of each piece isgiven by $74.90 ÷ 107 = $0.70.

100 × $0.70 = $70.

Because the marked price is represented by 100 of

these pieces, the marked price is 100 × $0.70 or $70.



Using this method the input is the marked price, the output is the total cost and theprogram is “Multiply by 1.07”. That is…

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Marked Price


Total Cost× 1.07

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So undoing the program would look like this…

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Marked Price


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Total Costx 1.07


outputprograminput

÷


In this problem the total cost (that is, the output) is $74.90; and so we see that…

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Marked Price


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$70

Total Cost

$74.90÷ 1.07 $74.90

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Using Ratio and ProportionIn this exercise we know that for every $1 of the marked price, the price including tax will

be $1. This is a constant rate problem and the rate is:

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$1.07 pre tax (12) $1.00 after tax

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• Observe that when we used this method for solving Problem 1, rather than using expression (12) we used the expression

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The reason is that it is computationally simpler to put the “unknown” in the numerator (thus avoiding the “cross multiplication” algorithm which is usually done by rote).

$1.07after tax$1.00 pre tax .

Note 2

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• In problem 1 the “unknown” was the “after tax” while in problem 2 the unknown” is the

“pre tax”.

$P pre tax$79.40 after tax . (13)

Note 2

So if the after-tax cost is $74.90 and if we let P stand for the pre-tax cost, the rate is

also given by…

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• However since the rate (ratio) is constant, the rates expressed in

expressions (12) and (13) are the same. In other words…

$P pre tax $79.40 after tax

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$1.00 pre tax $1.07 after tax (14)

=

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• and since the nouns on both sides of equation (14) are the same we may

rewrite the equation as.

P79.40

Note 2next

1.00 1.07 (15)

=

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• We can then multiply both sides of equation (7) by 74.90 to obtain the same result we obtained using other methods, namely…

P = 74.90 ×

Note 2next

=1.001.07

74.90 1.07

= 74.90 ÷ 107

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In trying to solve this problem it might have been tempting to take 7% of $74.90 and call this the tax. However the $74.90

was obtained after we added the sales tax. For this reason it was better to view formula (1) as T = 1.07 × M rather than as

T = M + 0.07M, since the $74.90 represents the entire cost (that is,

including the 7% tax).

Final Caution

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There are times when algebraic means are either too cumbersome or else

nonexistent for solving certain types of problems. In such cases, we often use trial and error (sometimes referred to as

numerical analysis) to solve the problem.

Trial and Error or Estimation

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We saw in part (a) that if the pre-tax cost of the object was $67, the after-tax cost

was $71.69. Since in part (b) the after-tax cost was $74.90, we know that the pre-tax

cost has to be greater than $67.


Example

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As another example, suppose there was a collection of 40 coins consisting solely of nickels and dimes, the value of which was $3.40, and you wanted to know how

many of the coins were dimes.


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A rather simple first step is to notice that if all 40 coins were dimes the value would have been $4.00; and if all 40 coins had

been nickels, the value would have been $2.00. In other words, before you are even

told what the total value is you would know that it had to be more than $2.00 but less

then $4.00

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The fact that the total value (i.e., $3.40) is closer in value to $4 than to $2 tells us

that there are more dimes than nickels. As a guess we might assume that there are

30 dimes and 10 nickels. If this had been the case the total value would have been

$3.00 + $0.50 or $3.50. Since this exceeds the total value, we would need more

nickels and fewer dimes.

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If we had assumed that there had been 25 dimes and 15 nickels, the total value would have been $2.50 + $0.75 or $3.25. And since

this is less than the given total value we know that we need more dimes and fewer

nickels.Combining our previous two steps we see that we need more than 25 dimes but less

than 30. Continuing this way we would eventually hit upon the correct answer.

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key stone problems… key stone problems… next set 1 © 2007 herbert i. gross

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