Exercise 20 Page 399

We want to write the given repeating decimal number as a mixed number. 6.089 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 6.089. x=6.089

   Step 3

   Since the given number has two repeating digits, we will multiply both sides of the equation by 10^2.
   x=6.089

   MultEqn

   LHS * 10^2=RHS* 10^2

   x* 10^2=6.089* 10^2

   CalcPow

   Calculate power

   x* 100=6.089* 100

   Multiply

   Multiply

   100x=608.989

   Step 4

   We will now subtract x from both sides of the equation. Since x=6.089, we will substitute 6.089 for x on the right-hand side.
   100x=608.989

   SubEqn

   LHS-x=RHS-x

   100x - x = 608.989-x

   Substitute

   x= 6.089

   100x-x = 608.989- 6.089

   SubTerms

   Subtract terms

   99x=602.9

   Step 5

   Next, we will solve the obtained equation for x.
   99x=602.9

   DivEqn

   .LHS /99.=.RHS /99.

   99x/99=602.9/99

   CancelCommonFac

   Cancel out common factors

   99x/99=602.9/99

   SimpQuot

   Simplify quotient

   x=602.9/99

   ExpandFrac

   a/b=a * 10/b * 10

   x = 6029/990

   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 = 6029/990

   WriteSum

   Write as a sum

   x=5940+89/990

   WriteSumFrac

   Write as a sum of fractions

   x=5940/990+89/990

   CalcQuot

   Calculate quotient

   x=6+89/990

   AddTerms

   Add terms

   x=6 89990
   We found that x is equal to 6 89990. Remember that x is also equal to 6.089. We can conclude that the mixed number is equal to the given repeating decimal number. 6.089=6 89990