Exercise 5 Page 10

We are told that a student estimates the weight of astronauts on the Moon by multiplying their weight by the decimal 0.16666.... This is a repeating decimal. We want to write the given repeating decimal as a fraction. To do so, we will follow five steps.

1. Assign a variable to represent the repeating decimal.
2. Write an equation by setting the variable and the decimal equal to each other.
3. Multiply both sides of the equation by 10^d, where d is the number of repeating digits in the repeating decimal.
4. Subtract equivalent expressions of the variable and the repeating decimal from each side of the equation.
5. 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

We will use the variable x to represent the given repeating decimal number. We can write an equation by setting this variable equal to 0.16666....

x=0.16666...

Step 3

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

x=0.16666...

MultEqn

LHS * 10^1=RHS* 10^1

x* 10^1=0.16666...* 10^1

ExponentOne

a^1=a

x* 10=0.16666...* 10

Multiply

Multiply

10x=1.6666...

Step 4

We will now subtract x from both sides of the equation. Since x=0.16666..., we will substitute 0.16666... for x on the right-hand side.

10x=1.6666...

SubEqn

LHS-x=RHS-x

10x-x=1.6666...-x

Substitute

x= 0.16666...

10x-x=1.6666... - 0.16666...

SubTerms

Subtract terms

9x=1.5

Step 5

Next, we will solve the obtained equation for x.

9x=1.5

DivEqn

.LHS /9.=.RHS /9.

9x/9=1.5/9

CancelCommonFac

Cancel out common factors

9x/9=1.5/9

SimpQuot

Simplify quotient

x=1.5/9

ExpandFrac

a/b=a * 10/b * 10

x=15/90

ReduceFrac

a/b=.a /15./.b /15.

x=1/6

We found that x is equal to 16. Remember that x is also equal to 0.16666.... By the Transitive Property of Equality, we can conclude that the fraction is equal to the given repeating decimal number. x= 0.16666... x= 1/6 ⇒ 0.16666...= 1/6