Chapter 11 – Introduction to Algebra (Part II)

Week 9 – Math Skills

Outline Section 11.5 – Translating Verbal Expression

to Mathematical Expressions Section 11.6 – Translating Sentences into

Equations

Section 11.5 – Translating Verbal Expressions into Mathematical Expressions Word problems contain key words that help us

solve them. These keywords translate directly into

mathematical expressions. Keywords for addition

Addition

“More than”“The sum of”“The total of”“Increased by”

• 5 more than x 5 + x• The sum of w and 3 w + 3• The total of 6 and z is 6 + z• x increased by 7 is x + 7

Section 11.5 – Translating Verbal Expressions into Mathematical Expressions Keywords for subtraction

Subtraction

“Less than”“The difference between”“Decreased by”

• 5 less than y y – 5• The difference between w and 3 w – 3• 8 decreased by a 8 - a

Section 11.5 – Translating Verbal Expressions into Mathematical Expressions Keywords for multiplication

Multiplication

“Times”“The product of”“of”“Twice”

• 3 times c 3c• The product of 4 and t 4t• 2/3 of v (2/3)v• twice d 2d

Section 11.5 – Translating Verbal Expressions into Mathematical Expressions Keywords for division

Division

“Divided by”“the quotient of”“the ratio of”

• n divided by 3 n/3• The quotient of z and 4 z/4• The ratio of s to 6 s/6

Section 11.5 – Translating Verbal Expressions into Mathematical Expressions Examples

1. Translate “The sum of 5 and the product of 4 and n” into a mathematical expression.

The sum of 5 5 + The product of 4 and n 4n Put these together 5 + 4n

2. Translate “ the product of 3 and the difference between z and 4” into a mathematical expression.

The product of 3 and 3 • The difference between z and 4 z – 4 Put these together 3(z – 4)

Section 11.5 – Translating Verbal Expressions into Mathematical Expressions Class Examples

1. Translate “The difference between 8 and twice t” into a mathematical expression.

The difference between 8 and 8 - Twice t 2t Put these together 8 – 2t

2. Translate “the quotient of 5 and the product of 7 and x” into a mathematical expression.

The quotient of 5 and 5 ÷ The product of 7 and x 7x Put these together 5 ÷ 7x

Section 11.5 – Translating Verbal Expressions into Mathematical Expressions In some mathematical phrases, we are not

given the name of the variable Before we were given

Translate “the difference between 8 and twice t” We are given the variable here (i.e. t)

Now not given the name of the variable. What to do? Example: Translate “the difference between seven and

twice a number” “a number” could be x, y, z, a, b, c, any variable you choose…

It is just a placeholder The difference between 7 and 7 – Twice a number 2n Put these together 7 – 2n

Section 11.5 – Translating Verbal Expressions into Mathematical Expressions Example

Translate “the total of a number and the square of the number” Choose a variable… The total of a number n + The square of the number n2

Put these together n + n2

Class Example Translate “the product of a number and one-half

of the number” Choose variable… The product of a number and n • One-half of the number ½ n Put these together n • ½ n

Section 11.6 – Translating Sentences into Equations and Solving An equation states that two mathematical

expressions are equal. Keywords in word problems for equal are:

Once we translate the given equation into a mathematical expression, we can find the solution

Equals

EqualsEqual to Is equal toAmounts to Represents

Section 11.6 – Translating Sentences into Equations and Solving Example

1. Translate “three more than twice a number is seventeen”, then solve the equation.

Three more than 3+ Twice a number 2n Is seventeen = 17 Put these together 3 + 2n = 17 Now solve this equation for n

3 – 3 + 2n = 17 – 3; subtract 3 from each side 2n = 14 2n/2 = 14/2; divide both sides by 2 n = 7; is the solution to the equation. We say “the number is 7”

Section 11.6 – Translating Sentences into Equations and Solving Example

1. Translate “a number decreased by 6 equals fifteen” into an equation, then solve.

A number decreased by 6 n - 6 Equals 15 = 15 Put these together n – 6 = 15 Now solve this equation for n

n – 6 + 6 = 15 + 6; add 6 to each side n = 21 We say “the number is 21”

Section 11.6 – Translating Sentences into Equations and Solving Example

1. Translate “eight decreased by twice a number is four” find the number.

Eight decreased by 8 - Twice a number 2n Is four = 4 Putting these together 8 – 2n = 4 Now solve this equation for n

8 – 8 – 2n = 4 – 8 ; subtract 8 from each side -2n = -4 -2n/(-2) = -4/(-2) ; divide both sides by -2 n = 2 We say “the number is 2”

Section 11.6 – Translating Sentences into Equations and Solving Class Example

1. Translate “the product of two and a number is ten” then find the number.

The product of two and a number 2n Is ten = 10 Putting these together 2n = 10 Now solve this equation for n

2n /2 = 10 /2; divided both sides by 2 n = 5 We say “the number is 5”

Section 11.6 – Translating Sentences into Equations and Solving Class Example

1. Translate “three more than one-half a number is nine” find the number.

Three more than 3 + One-half of a number ½ n Is nine = 9 Putting these together 3 + ½ n = 9 Now solve this equation for n

3 – 3 + ½ n = 9 – 3 ; subtract 3 from each side ½ n = 6 ½ n (2) = 6 (2) ; multiply both sides by 2 n = 12 We say “the number is 12”

Final Exam Review What to study?

1. Practice final2. Lecture notes3. Sections in the book