Converting Between Fractions and Decimals: Additional Practice (Set 2)

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

$\frac{13}{20} = \frac{13 \times 5}{20 \times 5} = \frac{65}{100} = 0.65$

Compute $9 \div 4$. Four goes into $9$ twice with a remainder of $1$, then divide $1.00$.

$\frac{9}{4} = 9 \div 4 = 2.25$

The value is two wholes plus the extra $\tfrac{1}{4} = 0.25$.

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

$16 = 2 \times 2 \times 2 \times 2, \quad 50 = 2 \times 5 \times 5, \quad 15 = 3 \times 5, \quad 40 = 2 \times 2 \times 2 \times 5$

Only $15$ carries a $3$, so $\tfrac{2}{15}$ repeats; the rest terminate.

Convert the fractions: $\tfrac{5}{8} = 0.625$ and $\tfrac{3}{5} = 0.6$. Compare $0.625$, $0.6$, and $0.6$.

$0.625 > 0.6$

So $\tfrac{5}{8}$ is the largest; $0.6$ and $\tfrac{3}{5}$ are equal to each other but smaller.

Compute $5 \div 6$. Six into $50$ goes $8$ times with a remainder of $2$; six into $20$ goes $3$ times with a remainder of $2$, which then returns at every step, so the $3$ repeats.

$\frac{5}{6} = 0.8333\ldots = 0.8\overline{3}$

The $8$ is fixed and only the $3$ falls under the bar.

Convert the fractions: $\tfrac{3}{8} = 0.375$ and $\tfrac{1}{3} = 0.333\ldots$ Compare with $0.4$.

$0.333\ldots < 0.375 < 0.4$

So least to greatest is $\tfrac{1}{3},\ \tfrac{3}{8},\ 0.4$.

Convert the fraction: $\tfrac{2}{3} = 0.666\ldots$ Compare $0.71$, $0.70$, and $0.666\ldots$

$0.71 > 0.70 > 0.666\ldots$

So $0.71$ is the greatest.

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

$\frac{19}{25} = \frac{19 \times 4}{25 \times 4} = \frac{76}{100} = 0.76$

Reduce first: $\tfrac{15}{24} = \tfrac{5}{8}$, whose denominator $8 = 2 \times 2 \times 2$ has only the prime $2$, so the decimal terminates. Compute $5 \div 8$.

$\frac{15}{24} = \frac{5}{8} = 0.625$

The factor of $3$ in $24$ cancelled against the $15$, so there is no repeating block.

Convert the fractions: $\tfrac{1}{4} = 0.25$ and $\tfrac{1}{3} = 0.333\ldots$ Compare with $0.3$.

$0.333\ldots > 0.3 > 0.25$

So greatest to least is $\tfrac{1}{3},\ 0.3,\ \tfrac{1}{4}$.

Three digits follow the point, so $0.104 = \tfrac{104}{1000}$. The greatest common factor of $104$ and $1000$ is $8$.

$\frac{104}{1000} = \frac{104 \div 8}{1000 \div 8} = \frac{13}{125}$

Convert both to decimals so they share a form. By scaling, $\tfrac{4}{5} = \tfrac{8}{10} = 0.8$, and by dividing, $\tfrac{7}{8} = 0.875$.

$0.875 > 0.800$

So $\tfrac{7}{8}$ is larger.