Exercise 8 Page 399

We want to write the given repeating decimal number as a fraction. Let's rewrite our number a little bit. 0.858585... = 0.85 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

   Let's use the variable x to represent the given repeating decimal number. We can write an equation by setting this variable equal to 0.85. x=0.85

   Step 3

   Since the given number has two repeating digits, we will multiply both sides of the equation by 10^2.
   x=0.85

   MultEqn

   LHS * 10^2=RHS* 10^2

   x* 10^2=0.85* 10^2

   CalcPow

   Calculate power

   x* 100=0.85* 100

   Multiply

   Multiply

   100x=85.85

   Step 4

   We will now subtract x from both sides of the equation. Since x=0.85, we will substitute 0.85 for x on the right-hand side.
   100x=85.85

   SubEqn

   LHS-x=RHS-x

   100x-x=85.85-x

   Substitute

   x= 0.85

   100x-x=85.85- 0.85

   SubTerms

   Subtract terms

   99x=85

   Step 5

   Finally, we will solve the obtained equation for x.
   99x=85

   DivEqn

   .LHS /99.=.RHS /99.

   99x/99=85/99

   CancelCommonFac

   Cancel out common factors

   99x/99=85/99

   SimpQuot

   Simplify quotient

   x=85/99
   We found that x is equal to 8599. Remember that x is also equal to 0.85. Therefore, the fraction that we obtained is equal to the given repeating decimal number. 0.858585... = 85/99