Writing decimals as fractions To convert a decimal to a fraction, we write the decimal number as a numerator, and we write its place value as the denominator. Example 1: 0.070.070, point, 07 0.0\blueD70.070, point, 0, start color #11accd, 7, end color #11accd is \blueD77start color #11accd, 7, end color #11accd \text{\greenD{hundredths}}hundredthsstart text, start color #1fab54, h, u, n, d, r, e, d, t, h, s, end color #1fab54, end text. So, we write \blueD77start color #11accd, 7, end color #11accd over \greenD{100}100start color #1fab54, 100, end color #1fab54. 0.07=\dfrac{\blueD7}{\greenD{100}}0.07= 100 7 0, point, 07, equals, start fraction, start color #11accd, 7, end color #11accd, divided by, start color #1fab54, 100, end color #1fab54, end fraction But what about repeating decimals? Let's look at an example. Rewrite 0.\overline{7}0. 7 0, point, start overline, 7, end overline as a simplified fraction. Let xxx equal the decimal: \large{x = 0.7777...}x=0.7777...x, equals, 0, point, 7777, point, point, point Set up a second equation such that the digits after the decimal point are identical: \large{\begin{aligned} 10x &= 7.7777...\\ x &= 0.7777... \end{aligned}} 10x x
=7.7777... =0.7777...
Subtract the two equations: \large{9x = 7}9x=79, x, equals, 7 Solve for xxx: \large{ x = \dfrac{7}{9}}x= 9 7 x, equals, start fraction, 7, divided by, 9, end fraction Remember from the first step that xxx is equal to our repeating decimal, so: \large{0.\overline{7}=\dfrac79}0. 7 = 9 7
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