Exercise 22 Page 399

We want to write the given repeating decimal number as a mixed number. 11.510 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 11.510. x=11.510

   Step 3

   Since the given number has three repeating digits, we will multiply both sides of the equation by 10^3.
   x=11.510

   MultEqn

   LHS * 10^3=RHS* 10^3

   x* 10^3=11.510* 10^3

   CalcPow

   Calculate power

   x* 1000=11.510* 1000

   Multiply

   Multiply

   1000x=11510.510

   Step 4

   We will now subtract x from both sides of the equation. Since x=11.510, we will substitute 11.510 for x on the right-hand side.
   1000x=11510.510

   SubEqn

   LHS-x=RHS-x

   1000x - x = 11510.510-x

   Substitute

   x= 11.510

   1000x-x = 11510.510- 11.510

   SubTerms

   Subtract terms

   999x=11499

   Step 5

   Next, we will solve the obtained equation for x.
   999x=11499

   DivEqn

   .LHS /999.=.RHS /999.

   999x/999=11499/999

   CancelCommonFac

   Cancel out common factors

   999x/999=11499/999

   SimpQuot

   Simplify quotient

   x=11499/999

   ReduceFrac

   a/b=.a /3./.b /3.

   x = 3833/333

   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 = 3833/333

   WriteSum

   Write as a sum

   x=3663+170/333

   WriteSumFrac

   Write as a sum of fractions

   x=3663/333+170/333

   CalcQuot

   Calculate quotient

   x=11+170/333

   AddTerms

   Add terms

   x=11 170333
   We found that x is equal to 11 170333. Remember that x is also equal to 11.510. We can conclude that the mixed number is equal to the given repeating decimal number. 11.510=11 170333