Exercise 6 Page 10

We want to write the given repeating decimal number as a mixed number. 2.3181818... 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 2.3181818.... x=2.3181818...

   Step 3

   Since the given number has two repeating digits, we will multiply both sides of the equation by 10^2.
   x=2.3181818...

   MultEqn

   LHS * 10^2=RHS* 10^2

   x* 10^2=2.3181818...* 10^2

   CalcPow

   Calculate power

   x* 100=2.3181818...* 100

   Multiply

   Multiply

   100x=231.8181818...

   Step 4

   We will now subtract x from both sides of the equation. Since x=2.3181818..., we will substitute 2.3181818... for x on the right-hand side.
   100x=231.8181818...

   SubEqn

   LHS-x=RHS-x

   100x-x=231.8181818...-x

   Substitute

   x= 2.3181818...

   100x-x=231.8181818... - 2.3181818...

   SubTerms

   Subtract terms

   99x=229.5

   Step 5

   Next, we will solve the obtained equation for x.
   99x=229.5

   DivEqn

   .LHS /99.=.RHS /99.

   99x/99=229.5/99

   CancelCommonFac

   Cancel out common factors

   99x/99=229.5/99

   SimpQuot

   Simplify quotient

   x=229.5/99

   ExpandFrac

   a/b=a * 2/b * 2

   x=459/198

   ReduceFrac

   a/b=.a /9./.b /9.

   x = 51/22

   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=51/22

   WriteSum

   Write as a sum

   x=22+22+7/22

   WriteSumFrac

   Write as a sum of fractions

   x=22/22+22/22+7/22

   QuotOne

   a/a=1

   x=1+1+7/22

   AddTerms

   Add terms

   x=2 722
   We found that x is equal to 2 722. Remember that x is also equal to 2.318. By the Transitive Property of Equality, we can conclude that the mixed number is equal to the given repeating decimal number. x= 2.3181818... x= 2 722 ⇕ 2.3181818...= 2 722