Exercise 22 Page 418

We are told that the gas mileage of a hybrid car is 3.03 times the gas mileage of a regular car. Knowing that a regular car averages 24 miles per gallon, we want to find the gas mileage of the hybrid car.

Writing the Repeating Decimal as a Fraction

The first step to find the gas mileage is rewriting the given repeating decimal as a fraction. Let's remember how we can write a repeating decimal as a fraction. We will follow five steps.

Assign a variable to represent the repeating decimal.
Write an equation by setting the variable and the decimal equal to each other.
Multiply both sides of the equation by 10^d, where d is the number of repeating digits in the repeating decimal.
Subtract equivalent expressions of the variable and the repeating decimal from each side of the equation.
Solve for the variable. If necessary, write an equivalent fraction so that the numerator and denominator are integers.

Let's do it!

Steps 1 and 2

Let's use the variable x to represent the given repeating decimal number. We can write an equation by setting this variable equal to 3.03. x=3.03

Step 3

Since the given number has one repeating digit, we will multiply both sides of the equation by 10^1.

x=3.03

MultEqn

LHS * 10^1=RHS* 10^1

x* 10^1=3.03* 10^1

ExponentOne

a^1=a

x* 10=3.03* 10

Multiply

10x=30.3

Step 4

We will now subtract x from both sides of the equation. Since x=3.03, we will substitute 3.03 for x on the right-hand side.

10x=30.3

SubEqn

LHS-x=RHS-x

10x-x=30.3-x

Substitute

x= 3.03

10x-x=30.3- 3.03

SubTerms

Subtract terms

9x=27.3

Step 5

Finally, we will solve the equation for x.

9x=27.3

DivEqn

.LHS /9.=.RHS /9.

9x/9=27.3/9

CancelCommonFac

Cancel out common factors

9x/9=27.3/9

SimpQuot

Simplify quotient

x=27.3/9

ExpandFrac

a/b=a * 10/b * 10

x=273/90

We found that x is equal to 27390. Remember that x is also equal to 3.03. The Transitive Property of Equality tells us that the fraction that we got is equal to the given repeating decimal number. x= 3.03 x= 27390 ⇒ 3.03= 273/90

Finding the Gas Mileage of the Hybrid Car

We found that the gas mileage of the hybrid car is 27390 times the gas mileage of a regular car. From the exercise, we know that the regular car averages 24 miles per gallon. We will multiply 24 by 27390 to find the gas mileage of the hybrid car.

24 * 273/90

MoveLeftFacToNum

a*b/c= a* b/c

24 * 273/90

Multiply

6552/90

CalcQuot

Calculate quotient

72.8

Therefore, the gas mileage of a hybrid car is 72.8 miles per gallon.

Hints

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