Math Core

A ratio compares two quantities by division; a rate is a ratio with units, like miles per hour or dollars per pound. Ratio problems are everywhere in contest math, and nearly all of them fall to one simple idea: think in parts.

The parts method

If red and blue marbles are in the ratio $3 : 5$ , you don't know how many of each there are. You only know the marbles come in "packets" of $3$ red and $5$ blue. So there are $3k$ red and $5k$ blue marbles for some number $k$ , and $8k$ in all.

Ratios as parts

A ratio $a : b$ means the quantities are $ak$ and $bk$ for some positive number $k$ . The total is $(a + b)k$ .

Find $k$ from whatever else the problem tells you (a total, a difference, a new ratio), and then everything else follows.

For example, if the total is $64$ , then $8k = 64$ , so $k = 8$ : there are $24$ red and $40$ blue marbles.

Combining ratios

If $a : b = 2 : 3$ and $b : c = 4 : 5$ , you can't just glue them together, because $b$ is " $3$ " in one ratio and " $4$ " in the other. Scale both so the shared quantity matches. The least common multiple of $3$ and $4$ is $12$ :

$a : b = 8 : 12, \qquad b : c = 12 : 15, \qquad \text{so } a : b : c = 8 : 12 : 15.$

When a ratio changes

If items are added or removed, write each quantity as $ak$ and $bk$ before the change, apply the change, and set up the new ratio as an equation. Only the quantities that change get adjusted.

Common mistake

When a ratio changes, $k$ describes the original amounts. A common mistake is to use the new ratio's parts for the old amounts. Always write the "before" amounts in terms of the "before" ratio.

Rates and unit conversion

A rate like " $6$ inches per minute" is a fraction $\dfrac{6 \text{ in}}{1 \text{ min}}$ . To convert units, multiply by fractions equal to $1$ , such as $\dfrac{60 \text{ min}}{1 \text{ hr}}$ or $\dfrac{1 \text{ ft}}{12 \text{ in}}$ , arranged so unwanted units cancel.

When lengths on a map or model are scaled by a factor, areas are scaled by the square of that factor.

Worked example: Using a total

The ratio of red to blue marbles in a jar is $3 : 5$ . There are $64$ marbles. How many are red?

The marbles come in $8$ parts, so each part is $64 \div 8 = 8$ marbles. Red is $3$ parts: $24$ marbles.

Worked example: Chaining ratios

If $a : b = 2 : 3$ and $b : c = 4 : 5$ , what is $a : c$ ?

From above, $a : b : c = 8 : 12 : 15$ , so $a : c = 8 : 15$ .

Worked example: A changing ratio

In a club, the ratio of boys to girls is $3 : 4$ . After $6$ more boys join, the ratio is $1 : 1$ . How many students were in the club at first?

At first there are $3k$ boys and $4k$ girls. After $6$ boys join, $3k + 6 = 4k$ , so $k = 6$ . The club started with $7k = 42$ students.

Worked example: Converting a rate

A snail crawls $6$ inches per minute. How fast is that in feet per hour?

$\frac{6 \text{ in}}{1 \text{ min}} \cdot \frac{60 \text{ min}}{1 \text{ hr}} \cdot \frac{1 \text{ ft}}{12 \text{ in}} = \frac{360}{12} \ \frac{\text{ft}}{\text{hr}} = 30 \text{ feet per hour}.$

Tip

After finding $k$ , check your answer against every condition in the problem. In the club example, $18$ boys and $24$ girls give $3 : 4$ , and $24$ boys and $24$ girls give $1 : 1$ .

Practice

Practice 1

A recipe uses flour and sugar in the ratio $5 : 2$ . How many cups of sugar go with $15$ cups of flour?