Exercise 8 Page 11

We will write the number 3.7 as a mixed number, and we will fill the boxes while we are finding the result. To do so, we will follow six 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.
6. Write the obtained improper fraction as a mixed number.

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 3.7. x= 3.7

We will fill the first box.

Step 3

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

x=3.7

MultEqn

LHS * 10^1=RHS* 10^1

x* 10^1=3.7* 10^1

ExponentOne

a^1=a

x* 10=3.7* 10

Multiply

Multiply

10x=37.7

We will write 37.7 into the second box.

Step 4

From here we will subtract x from both sides of the equation. Since x=3.7, we will substitute 3.7 for x on the right-hand side.

10x=37.7

SubEqn

LHS-x=RHS-x

10x-x=37.7-x

Substitute

x= 3.7

10x-x=37.7 - 3.7

SubTerms

Subtract terms

9x=34

Let's fill the third box.

Step 5

Next, we will solve the obtained equation for x.

9x=34

DivEqn

.LHS /9.=.RHS /9.

9x/9=34/9

CancelCommonFac

Cancel out common factors

9x/9=34/9

SimpQuot

Simplify quotient

x=34/9

Step 6

Finally, we will write the obtained improper fraction as a mixed number. To do this, we will start by expressing the numerator as a sum.

x=34/9

WriteSum

Write as a sum

x=9+9+9+7/9

WriteSumFrac

Write as a sum of fractions

x=9/9+9/9+9/9+7/9

QuotOne

a/a=1

x=1+1+1+7/9

AddTerms

Add terms

x=3 79

We found that x is equal to 3 79. Remember that x is also equal to 3.7. By the Transitive Property of Equality, we can conclude that the mixed number is equal to the given repeating decimal number. x= 3.7 x= 3 79 ⇒ 3.7= 3 79 Consequently, the figure can be filled as follows.