Exercise 7 Page 11

We will write the number 0.21212121... as a fraction, and we will fill the boxes while we are finding the result. 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

We will use the variable x to represent the given repeating decimal number. We can write an equation by setting this variable equal to 0.21212121.... x= 0.21212121... We will fill the first box.

Step 3

Note that the given number has two repeating digits, so we will multiply both sides of the equation by 10^2.

x=0.21212121...

MultEqn

LHS * 10^2=RHS* 10^2

x* 10^2=0.21212121...* 10^2

CalcPow

Calculate power

x* 100=0.21212121...* 100

Multiply

Multiply

100x=21.212121...

We will write 21.212121... in the second box.

Step 4

From here we will subtract x from both sides of the equation. Since x=0.21212121..., we will substitute 0.21212121... for x on the right-hand side.

100x=21.21212121...

SubEqn

LHS-x=RHS-x

100x-x=21.21212121...-x

Substitute

x= 0.21212121...

100x-x=21.212121... - 0.21212121...

Let's fill the third box.

Next we will simply the equation. 100x-x&=21.212121... - 0.21212121... &⇓ 99x&=21 Therefore, the next box is 21.

Step 5

We will solve the obtained equation for x.

99x=21

DivEqn

.LHS /99.=.RHS /99.

99x/99=21/99

CancelCommonFac

Cancel out common factors

99x/99=21/99

SimpQuot

Simplify quotient

x=21/99

ReduceFrac

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

x = 7/33

We found that x is equal to 733. Remember that x is also equal to 0.21212121... By the Transitive Property of Equality, we can conclude that the fraction is equal to the given number. x= 0.21212121... x= 7/33 ⇓ 0.21212121...= 7/33 Consequently, the figure can be filled as follows.