We are given the cost of making a computer component. Notice that the cost is an approximation of a repeating decimal. Let's rewrite this decimal as repeating decimal.
≈ 2.161616 ⇒ 2.161616...
We will write the number as a fraction and a mixed number. To write the number as a fraction we will follow five steps.
Write an equation by setting the variable and the decimal equal to each other.
Multiply both sides of the equation by 10^d, where d is the number of repeating digits in the repeating decimal.
Subtract equivalent expressions of the variable and the repeating decimal from each side of the equation.
Solve for the variable. If necessary, write an equivalent fraction so that the numerator and denominator are integers.
Then we will write the obtained improper fraction as a mixed number. 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 2.161616....
x=2.161616...
Step 3
Notice that the given number has two repeating digits, so we will multiply both sides of the equation by 10^2.
We found that x is equal to 21499 as a fraction. Remember that x is also equal to 2.161616.... By the Transitive Property of Equality, we can conclude that the improper fraction is equal to the given repeating decimal number.
x= 2.161616... x= 214/99 ⇓ 2.161616...= 214/99
Writing the Mixed Number
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.
Here we found that x is equal to 2 1699. Recall again that x is also equal to 2.161616.... We will use the Transitive Property of Equality to conclude that the mixed number is equal to the given repeating decimal number.
x= 2.161616... x= 2 1699 ⇓ 2.161616...= 2 1699
