Basic Ratio and Proportionality

A ratio compares two quantities of the same kind, such as two lengths, two counts, or two amounts of money, but never one of each. The ratio $A:B = 5:3$ means $A$ and $B$ are made of parts, $5$ parts to $3$ parts.

If you know the ratio and one of the quantities, you can find the other by scaling.
Suppose $A:B = 2:7$ and $A = 6$.
Here $A$ is the $2$-part multiplied by $3$, because $6 = 2 \times 3$.
So $B = 7 \times 3 = 21$.

If you know the ratio and the total, first add the parts.
Suppose $A:B = 3:5$ and $A + B = 96$.
The parts total $3 + 5 = 8$, so one part is $96 \div 8 = 12$.
Then $A = 3 \times 12 = 36$ and $B = 5 \times 12 = 60$.

Some questions change a quantity and ask how the ratio shifts.
Suppose $A:B = 1:3$ with $B = 15$, so $A = 5$. If $k$ is added to $A$ while $B$ stays the same and the ratio becomes $2:5$, set up $\frac{5 + k}{15} = \frac{2}{5}$.
Cross-multiplying: $5(5 + k) = 30$, so $25 + 5k = 30$, giving $k = 1$.

Ratios also compare shapes. In two similar figures the matching sides are in the same ratio, but their areas are in the ratio of the sides squared.
Two similar squares with sides $2$ cm and $6$ cm have a side ratio of $1:3$, so their area ratio is $1^2 : 3^2 = 1 : 9$.

The ratio $A:B = 7:2$, and $A = 21$. What is $B$?
$A$ is the $7$-part multiplied by $3$, since $21 = 7 \times 3$.
So $B = 2 \times 3 = 6$.

The ratio $A:B = 4:9$, and $A + B = 130$. Find $A$ and $B$.
The parts total $4 + 9 = 13$, so one part is $130 \div 13 = 10$.
Then $A = 4 \times 10 = 40$ and $B = 9 \times 10 = 90$.

The ratio $A:B = 1:2$ with $B = 12$, so $A = 6$. An amount $k$ is added to $A$ while $B$ stays the same, and the ratio becomes $2:3$. Find $k$.
Set up $\frac{6 + k}{12} = \frac{2}{3}$.
Cross-multiply: $3(6 + k) = 24$, so $18 + 3k = 24$, giving $k = 2$.

Two similar squares have sides $3$ cm and $12$ cm. What is the ratio of their areas?
The side ratio is $3:12 = 1:4$, and areas go as the square of the sides.
So the area ratio is $1^2 : 4^2 = 1 : 16$.