Proportions: Practice

Cross-multiply the known diagonal, then divide by the number diagonally opposite the blank. The known diagonal is $6$ and $10$, and $n$ faces the $4$ across the diagonal.

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

So the missing number is $15$.

The known diagonal is $8$ and $9$, with $n$ under the $8$, opposite the $12$. Multiply the known diagonal and divide by $12$.

$n = \frac{8 \times 9}{12} = \frac{72}{12} = 6$

So the missing number is $6$.

Keep centimeters over kilometers on both sides, with $k$ for the unknown distance: $\frac{3}{45} = \frac{8}{k}$. Cross-multiply the known diagonal $45 \times 8$ and divide by $3$.

$k = \frac{45 \times 8}{3} = \frac{360}{3} = 120$

So $8$ centimeters represents $120$ kilometers.

Check $\frac{8}{12} = \frac{12}{16}$ with cross-products.

$8 \times 16 = 128 \qquad 12 \times 12 = 144$

Since $128 \neq 144$, this proportion is false. The other three have equal cross-products.

Set up servings over cups on both sides, with $c$ for the unknown cups: $\frac{4}{6} = \frac{10}{c}$. Cross-multiply the known diagonal $6 \times 10$ and divide by $4$.

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

So $15$ cups of broth are needed.

Divide by the number diagonally opposite the blank. Here $n$ faces the $6$ across the diagonal, so $6$ is its partner that goes into the division.

$n = \frac{9 \times 14}{6} = \frac{126}{6} = 21$

So you divide by $6$.

The bottom went from $5$ to $15$, a scale factor of $15 \div 5 = 3$. Multiply the top by $3$.

$n = 4 \times 3 = 12$

So the missing number is $12$.

Set up pages over seconds on both sides, with $p$ for the unknown pages: $\frac{9}{6} = \frac{p}{20}$. Cross-multiply the known diagonal $9 \times 20$ and divide by $6$.

$p = \frac{9 \times 20}{6} = \frac{180}{6} = 30$

So it prints $30$ pages.

Simplify $\frac{12}{18}$ by dividing both parts by their greatest common factor, $6$.

$\frac{12}{18} = \frac{12 \div 6}{18 \div 6} = \frac{2}{3}$

So $\frac{2}{3}$ forms a true proportion with $\frac{12}{18}$.

The known diagonal is $6$ and $15$, with $n$ under the $6$, opposite the $9$. Multiply the known diagonal and divide by $9$.

$n = \frac{6 \times 15}{9} = \frac{90}{9} = 10$

So the missing number is $10$.

Set up books over pounds on both sides, with $w$ for the unknown weight: $\frac{3}{9} = \frac{8}{w}$. Cross-multiply the known diagonal $9 \times 8$ and divide by $3$.

$w = \frac{9 \times 8}{3} = \frac{72}{3} = 24$

So $8$ books weigh $24$ pounds.

Check $\frac{10}{15} = \frac{14}{21}$ with cross-products.

$10 \times 21 = 210 \qquad 15 \times 14 = 210$

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