Converting Between Fractions and Decimals: Practice

The denominator $40$ divides a thousand, since $40 \times 25 = 1000$, so multiply the top and bottom by $25$.

$\frac{9}{40} = \frac{9 \times 25}{40 \times 25} = \frac{225}{1000} = 0.225$

Dividing $9 \div 40$ gives the same result.

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

$30 = 2 \times 3 \times 5, \quad 40 = 2 \times 2 \times 2 \times 5, \quad 15 = 3 \times 5, \quad 12 = 2 \times 2 \times 3$

Only $40$ is built from just $2$s and $5$s, so $\tfrac{11}{40}$ terminates; the others carry a $3$ and repeat.

Compute $11 \div 8$. Eight goes into $11$ once with a remainder of $3$, then divide $3.000$ in the usual way.

$\frac{11}{8} = 11 \div 8 = 1.375$

The whole part is $1$ because $\tfrac{11}{8}$ is one whole plus the extra $\tfrac{3}{8} = 0.375$.

Convert the fractions to decimals: $\tfrac{1}{3} = 0.333\ldots$ and $\tfrac{2}{5} = 0.4$. Compare with $0.3$.

$0.3 < 0.333\ldots < 0.4$

So least to greatest is $0.3,\ \tfrac{1}{3},\ \tfrac{2}{5}$.

Three digits follow the point, so $0.875 = \tfrac{875}{1000}$. The greatest common factor of $875$ and $1000$ is $125$.

$\frac{875}{1000} = \frac{875 \div 125}{1000 \div 125} = \frac{7}{8}$

Two digits follow the point, so $0.32 = \tfrac{32}{100}$. The greatest common factor of $32$ and $100$ is $4$.

$\frac{32}{100} = \frac{32 \div 4}{100 \div 4} = \frac{8}{25}$

Keep the whole part and convert only the fraction. Scale $\tfrac{3}{5}$ to tenths by multiplying the top and bottom by $2$.

$2\tfrac{3}{5} = 2 + \frac{3 \times 2}{5 \times 2} = 2 + \frac{6}{10} = 2.6$

Convert $\tfrac{5}{8}$ to a decimal by computing $5 \div 8$.

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

So the board is $0.625$ metre long.

Three digits follow the point, so $0.024 = \tfrac{24}{1000}$. The greatest common factor of $24$ and $1000$ is $8$.

$\frac{24}{1000} = \frac{24 \div 8}{1000 \div 8} = \frac{3}{125}$

Convert the fraction: $5 \div 9 = 0.555\ldots = 0.\overline{5}$. Compare with $0.55$.

$0.555\ldots > 0.55$

The repeating $5$s keep going past the second place, so $\tfrac{5}{9}$ is larger.

The denominator $25$ divides a hundred, since $100 = 25 \times 4$, so multiply the top and bottom by $4$.

$\frac{17}{25} = \frac{17 \times 4}{25 \times 4} = \frac{68}{100} = 0.68$

Convert each to a decimal: $\tfrac{5}{8} = 0.625$ and $\tfrac{2}{3} = 0.666\ldots$ Compare with $0.6$.

$0.6 < 0.625 < 0.666\ldots$

So least to greatest is $0.6,\ \tfrac{5}{8},\ \tfrac{2}{3}$.