Exercise 9 Page 11

We want to write the given repeating decimal number as a fraction. 0.233333... To do so, 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 0.233333.... x=0.233333...

   Step 3

   Since the given number has one repeating digit, we will multiply both sides of the equation by 10^1.
   x=0.233333...

   MultEqn

   LHS * 10^1=RHS* 10^1

   x* 10^1=0.233333...* 10^1

   ExponentOne

   a^1=a

   x* 10=0.233333...* 10

   Multiply

   Multiply

   10x=2.333333...

   Step 4

   We will now subtract x from both sides of the equation. Since x=0.233333..., we will substitute 0.233333... for x on the right-hand side.
   10x=2.333333...

   SubEqn

   LHS-x=RHS-x

   10x-x=2.333333...-x

   Substitute

   x= 0.233333...

   10x-x=2.333333... - 0.233333...

   SubTerms

   Subtract terms

   9x=0.1

   Step 5

   Finally, we will solve the obtained equation for x.
   9x=0.1

   DivEqn

   .LHS /9.=.RHS /9.

   9x/9=0.1/9

   CancelCommonFac

   Cancel out common factors

   9x/9=0.1/9

   SimpQuot

   Simplify quotient

   x=0.1/9

   ExpandFrac

   a/b=a * 10/b * 10

   x = 1/90
   We found that x is equal to 190. Remember that x is also equal to 0.233333.... Therefore, the Transitive Property of Equality tells us that the fraction that we obtained is equal to the given repeating decimal number. x= 0.233333... x= 1/90 ⇕ 0.233333...= 1/90