Proportions: Additional Practice (Set 1)

Simplify $\frac{20}{24}$ to $\frac{5}{6}$ (divide by $4$), then scale to a denominator of $18$. The bottom $6$ becomes $18$, a factor of $3$.

$n = 5 \times 3 = 15$

So the missing number is $15$. Cross-multiplying agrees: $\frac{18 \times 20}{24} = \frac{360}{24} = 15$.

The known diagonal is $6$ and $18$, with $n$ under the $6$, opposite the $27$. Multiply the known diagonal and divide by $27$.

$n = \frac{6 \times 18}{27} = \frac{108}{27} = 4$

So the missing number is $4$.

Keep model centimeters over real meters on both sides, with $m$ for the model length: $\frac{3}{4} = \frac{m}{30}$. Cross-multiply the known diagonal $3 \times 30$ and divide by $4$.

$m = \frac{3 \times 30}{4} = \frac{90}{4} = 22.5$

So it is $22.5$ centimeters on the model.

Check $\frac{8}{14} = \frac{12}{21}$ with cross-products.

$8 \times 21 = 168 \qquad 14 \times 12 = 168$

The cross-products match, so this proportion is true. The other three do not balance.

Simplify $\frac{8}{12}$ to $\frac{2}{3}$ (divide by $4$), then scale to a denominator of $30$. The bottom $3$ becomes $30$, a factor of $10$.

$n = 2 \times 10 = 20$

So the missing number is $20$. Cross-multiplying agrees: $\frac{30 \times 8}{12} = \frac{240}{12} = 20$.

The known diagonal is $15$ and $40$, with $n$ under the $15$, opposite the $25$. Multiply the known diagonal and divide by $25$.

$n = \frac{15 \times 40}{25} = \frac{600}{25} = 24$

So the missing number is $24$.

Keep original over enlarged on both sides, with $h$ for the enlarged height: $\frac{4}{10} = \frac{6}{h}$. Cross-multiply the known diagonal $10 \times 6$ and divide by $4$.

$h = \frac{10 \times 6}{4} = \frac{60}{4} = 15$

So the enlarged height is $15$ inches.

The number $18$ is not a whole-number multiple of $4$, so cross-multiply. Here $n$ is under the $18$, opposite the $7$.

$n = \frac{7 \times 18}{4} = \frac{126}{4} = 31.5$

So the missing number is $31.5$.

Set up kilometers over minutes on both sides, with $d$ for the unknown distance: $\frac{9}{40} = \frac{d}{100}$. Cross-multiply the known diagonal $9 \times 100$ and divide by $40$.

$d = \frac{9 \times 100}{40} = \frac{900}{40} = 22.5$

So the runner goes $22.5$ kilometers.

Simplify $\frac{21}{14}$ to $\frac{3}{2}$ (divide by $7$). With the new bottom $2$ scaled to $10$ (a factor of $5$), the top becomes $3 \times 5 = 15$.

$n = 15$

Cross-multiplying agrees: $\frac{21 \times 10}{14} = \frac{210}{14} = 15$.

The known diagonal is $1.5$ and $10$, with $n$ over the $1.5$, opposite the $6$. Multiply the known diagonal and divide by $6$.

$n = \frac{1.5 \times 10}{6} = \frac{15}{6} = 2.5$

So the missing number is $2.5$.

First find the flour. Set up sugar over flour: $\frac{2}{5} = \frac{14}{f}$, so $f = \frac{5 \times 14}{2} = 35$. The finished proportion is $\frac{14}{35}$, which simplifies back to the recipe ratio.

$\frac{14}{35} = \frac{14 \div 7}{35 \div 7} = \frac{2}{5}$

So $\frac{14}{35}$ is the correct check.