Converting Between Fractions and Decimals: Additional Practice (Set 1)

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

$\frac{3}{40} = \frac{3 \times 25}{40 \times 25} = \frac{75}{1000} = 0.075$

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

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

$16 = 2 \times 2 \times 2 \times 2, \quad 15 = 3 \times 5, \quad 20 = 2 \times 2 \times 5, \quad 25 = 5 \times 5$

Only $15$ carries a $3$, so $\tfrac{8}{15}$ repeats; the rest terminate.

Three digits follow the point, so $0.144 = \tfrac{144}{1000}$. The greatest common factor of $144$ and $1000$ is $8$.

$\frac{144}{1000} = \frac{144 \div 8}{1000 \div 8} = \frac{18}{125}$

The denominator $16 = 2 \times 2 \times 2 \times 2$ has only the prime $2$, so the decimal terminates. Compute $1 \div 16$, or scale by $625$ since $16 \times 625 = 10000$.

$\frac{1}{16} = \frac{625}{10000} = 0.0625$

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

$\frac{21}{40} = \frac{21 \times 25}{40 \times 25} = \frac{525}{1000} = 0.525$

Convert the fraction: $7 \div 9 = 0.777\ldots = 0.\overline{7}$. Compare with $0.77$.

$0.777\ldots > 0.77$

The repeating $7$s continue past the second place, so $\tfrac{7}{9}$ is larger.

Three digits follow the point, so $0.408 = \tfrac{408}{1000}$. The greatest common factor of $408$ and $1000$ is $8$.

$\frac{408}{1000} = \frac{408 \div 8}{1000 \div 8} = \frac{51}{125}$

Keep the whole part and convert only the fraction. Compute $7 \div 8 = 0.875$.

$3\tfrac{7}{8} = 3 + 0.875 = 3.875$

The denominator $8$ is built from $2$s, so the decimal terminates.

A fraction terminates when its denominator in lowest terms is built only from $2$s and $5$s. Factor each denominator.

$18 = 2 \times 3 \times 3, \quad 21 = 3 \times 7, \quad 125 = 5 \times 5 \times 5, \quad 33 = 3 \times 11$

Only $125$ is built from just $5$s, so $\tfrac{7}{125}$ terminates; the others repeat.

Convert $\tfrac{11}{20}$ by scaling to a hundred, since $100 = 20 \times 5$.

$\frac{11}{20} = \frac{11 \times 5}{20 \times 5} = \frac{55}{100} = 0.55$

So the time is $0.55$ minute.

The denominator $16 = 2 \times 2 \times 2 \times 2$ has only the prime $2$, so the decimal terminates. Compute $13 \div 16$, or scale by $625$ since $16 \times 625 = 10000$.

$\frac{13}{16} = \frac{13 \times 625}{16 \times 625} = \frac{8125}{10000} = 0.8125$

Reduce first: $\tfrac{14}{35} = \tfrac{2}{5}$, whose denominator $5$ has only the prime $5$, so it terminates. Scale to tenths by multiplying by $2$.

$\frac{14}{35} = \frac{2}{5} = \frac{4}{10} = 0.4$

The factor of $7$ in $35$ cancelled against the $14$, so there is no repeating block.