Proportions: Practice

The number $60$ is not a whole-number multiple of $8$, so cross-multiply. Here $n$ is over the $60$, opposite the $8$.

$n = \frac{3 \times 60}{8} = \frac{180}{8} = 22.5$

So the missing number is $22.5$.

The known diagonal is $9$ and $40$, with $n$ under the $9$, opposite the $15$. Multiply the known diagonal and divide by $15$.

$n = \frac{9 \times 40}{15} = \frac{360}{15} = 24$

So the missing number is $24$.

Keep centimeters over kilometers on both sides, with $k$ for the real distance: $\frac{2}{25} = \frac{7.5}{k}$. Cross-multiply the known diagonal $25 \times 7.5$ and divide by $2$.

$k = \frac{25 \times 7.5}{2} = \frac{187.5}{2} = 93.75$

So the cities are $93.75$ kilometers apart.

Cross-multiply the known diagonal $2.5 \times 10$, with $n$ over the $10$, opposite the $4$.

$n = \frac{2.5 \times 10}{4} = \frac{25}{4} = 6.25$

So the missing number is $6.25$.

Set up cookies over cups on both sides, with $k$ for the unknown cookies: $\frac{24}{3} = \frac{k}{2}$. Cross-multiply the known diagonal $24 \times 2$ and divide by $3$.

$k = \frac{24 \times 2}{3} = \frac{48}{3} = 16$

So the baker can make $16$ cookies.

Simplify $\frac{10}{14}$ to $\frac{5}{7}$ (divide by $2$), then scale to a denominator of $21$. The bottom $7$ becomes $21$, a factor of $3$.

$n = 5 \times 3 = 15$

So the missing number is $15$. Cross-multiplying agrees: $\frac{21 \times 10}{14} = \frac{210}{14} = 15$.

Set up buckets over minutes on both sides, with $t$ for the unknown time: $\frac{14}{10} = \frac{35}{t}$. Cross-multiply the known diagonal $10 \times 35$ and divide by $14$.

$t = \frac{10 \times 35}{14} = \frac{350}{14} = 25$

So it takes $25$ minutes.

A true proportion has equal cross-products, the diagonal products. The diagonals pair the top of each side with the bottom of the other.

$a \times d = b \times c$

Multiplying straight across or adding does not follow from the proportion.

The known diagonal is $18$ and $20$, with $n$ under the $18$, opposite the $24$. Multiply the known diagonal and divide by $24$.

$n = \frac{18 \times 20}{24} = \frac{360}{24} = 15$

So the missing number is $15$.

The number $14$ is not a whole-number multiple of $4$, so cross-multiply. Here $n$ is under the $14$, opposite the $9$.

$n = \frac{9 \times 14}{4} = \frac{126}{4} = 31.5$

So the missing number is $31.5$.

The known diagonal is $2.4$ and $5$, with $n$ over the $2.4$, opposite the $8$. Multiply the known diagonal and divide by $8$.

$n = \frac{2.4 \times 5}{8} = \frac{12}{8} = 1.5$

So the missing number is $1.5$.

The known diagonal is $12$ and $12$, with the blank under the left $12$, opposite the $18$. Multiply the known diagonal and divide by $18$.

$\frac{12 \times 12}{18} = \frac{144}{18} = 8$

So the blank is $8$. Check: $\frac{12}{8} = \frac{18}{12}$ because $12 \times 12 = 144$ and $8 \times 18 = 144$.