Exercise 18 Page 399

We want to write the given repeating decimal number as a mixed number. 4.1 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

   Let's use the variable x to represent the given repeating decimal number. We can write an equation by setting this variable equal to 4.1. x=4.1

   Step 3

   Since the given number has one repeating digit, we will multiply both sides of the equation by 10^1.
   x=4.1

   MultEqn

   LHS * 10^1=RHS* 10^1

   x* 10^1=4.1* 10^1

   ExponentOne

   a^1=a

   x* 10=4.1* 10

   Multiply

   Multiply

   10x=41.1

   Step 4

   We will now subtract x from both sides of the equation. Since x=4.1, we will substitute 4.1 for x on the right-hand side.
   10x=41.1

   SubEqn

   LHS-x=RHS-x

   10x - x = 41.1-x

   Substitute

   x= 4.1

   10x-x = 41.1- 4.1

   SubTerms

   Subtract terms

   9x=37

   Step 5

   Next, we will solve the obtained equation for x.
   9x=37

   DivEqn

   .LHS /9.=.RHS /9.

   9x/9=37/9

   CancelCommonFac

   Cancel out common factors

   9x/9=37/9

   SimpQuot

   Simplify quotient

   x=37/9

   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=37/9

   WriteSum

   Write as a sum

   x=36+1/9

   WriteSumFrac

   Write as a sum of fractions

   x=36/9+1/9

   CalcQuot

   Calculate quotient

   x=4+1/9

   AddTerms

   Add terms

   x=4 19
   We found that x is equal to 4 19. Remember that x is also equal to 4.1. We can conclude that the mixed number is equal to the given repeating decimal number. 4.1=4 19