Converting Between Fractions and Decimals: Practice

The bar means divide, so compute $1 \div 8$. Since $8$ does not go into $1$, divide into $1.000$.

$\frac{1}{8} = 1 \div 8 = 0.125$

Eight into $10$ tenths goes once, into $20$ hundredths twice, into $40$ thousandths five times, ending the division.

Compute $3 \div 8$, dividing into $3.000$.

$\frac{3}{8} = 3 \div 8 = 0.375$

Eight into $30$ goes $3$ times, into $60$ goes $7$ times, into $40$ goes $5$ times, with no remainder left.

Compute $1 \div 3$. Three into $10$ goes $3$ times with a remainder of $1$, and that remainder of $1$ returns at every step, so the $3$ repeats forever.

$\frac{1}{3} = 0.\overline{3}$

The overline marks the digit that repeats, so this is not the same as the terminating $0.3$.

Three digits follow the point, so the denominator is a thousand: $0.125 = \tfrac{125}{1000}$. Reduce by dividing the top and bottom by $125$, their greatest common factor.

$\frac{125}{1000} = \frac{125 \div 125}{1000 \div 125} = \frac{1}{8}$

Compute $5 \div 8$, dividing into $5.000$.

$\frac{5}{8} = 5 \div 8 = 0.625$

Eight into $50$ goes $6$ times, into $20$ goes $2$ times, into $40$ goes $5$ times, ending the division.

Compute $2 \div 3$. Three into $20$ goes $6$ times with a remainder of $2$, which returns at every step, so the $6$ repeats forever.

$\frac{2}{3} = 0.\overline{6}$

The overline shows the $6$ continues without end, unlike the terminating $0.6$.

Two digits follow the point, so $0.04 = \tfrac{4}{100}$. The greatest common factor of $4$ and $100$ is $4$.

$\frac{4}{100} = \frac{4 \div 4}{100 \div 4} = \frac{1}{25}$

Each fraction is in lowest terms, so factor each denominator and look for a prime other than $2$ or $5$.

$8 = 2 \times 2 \times 2, \quad 6 = 2 \times 3, \quad 4 = 2 \times 2, \quad 20 = 2 \times 2 \times 5$

Only $6$ has the extra prime $3$, so $\tfrac{1}{6}$ repeats; the rest terminate.

Compute $7 \div 8$, dividing into $7.000$.

$\frac{7}{8} = 7 \div 8 = 0.875$

Eight into $70$ goes $8$ times, into $60$ goes $7$ times, into $40$ goes $5$ times, ending the division.

One digit follows the point, so $0.6 = \tfrac{6}{10}$. The greatest common factor of $6$ and $10$ is $2$.

$\frac{6}{10} = \frac{6 \div 2}{10 \div 2} = \frac{3}{5}$

The denominator $20$ divides a hundred, since $100 = 20 \times 5$, so multiply the top and bottom by $5$.

$\frac{1}{20} = \frac{1 \times 5}{20 \times 5} = \frac{5}{100} = 0.05$

Two decimal places are needed, so a placeholder zero holds the tenths place.

Compute $1 \div 6$. Six into $10$ goes once with a remainder of $4$; six into $40$ goes $6$ times with a remainder of $4$, which then returns at every step, so the $6$ repeats after the first tenth.

$\frac{1}{6} = 0.1666\ldots = 0.1\overline{6}$

Only the $6$ is under the bar, since the first digit after the point is a $1$.