Decimal Representation Math Example 4

Follow the full solution, then compare it with the other examples linked below.

medium

Convert

0.3\overline{6}

to a fraction in simplest form. (Note: only the

6

repeats.)

1
Let $x = 0.3666\ldots$ Multiply by $10$ : $10x = 3.666\ldots$ ; multiply by $100$ : $100x = 36.666\ldots$
2
Subtract: $100x - 10x = 36.666\ldots - 3.666\ldots = 33$ , so $90x = 33$ .
3
Solve: $x = \dfrac{33}{90} = \dfrac{11}{30}$ .

Answer

0.3\overline{6} = \dfrac{11}{30}

When only part of the decimal repeats, multiply by two different powers of

10

to align the repeating portions, then subtract to eliminate the repeating tail. Reducing the resulting fraction gives the simplest form.

About Decimal Representation

Writing fractions as digits to the right of a decimal point, using place values of tenths, hundredths, thousandths, etc.

Convert [formula] to a decimal by long division, and determine whether it terminates or repeats.

Convert [formula] to a fraction in simplest form.

Write each fraction as a decimal and classify as terminating or repeating: (a) [formula], (b) [formu