Set up and solve ratios, proportions and rates, including unit rates, scaling, direct and inverse variation, and unit conversion (Number and Quantity, Integrating Essential Skills).

What this topic is asking

Ratios, proportions and rates are among the most frequently tested ideas on the ACT, appearing both in the Number and Quantity area and throughout the Integrating Essential Skills questions. The skill is to set up the relationship correctly (which quantity pairs with which) and then solve, usually by cross-multiplying a proportion or by finding a unit rate.

Ratios and proportions

A ratio can be written $a : b$ or $\frac{a}{b}$. A proportion is two equal ratios.

To share a quantity in a ratio, add the ratio parts to get the number of equal shares, divide the total by that, then multiply out. For example, sharing \40$in the ratio$3 : 5$gives$3 + 5 = 8$shares, each worth$\frac{40}{8} = \$5$, so the parts are \15$and$\$25$.

Rates and unit rates

A rate compares quantities with different units. A unit rate expresses "per one" of something.

Unit rates are also how you compare value: the cheaper option is the one with the smaller price per unit, found by dividing price by quantity.

Unit conversion

Many ACT rate problems require a unit conversion, handled by multiplying by a fraction equal to 1. To convert 90 minutes to hours, multiply by $\frac{1 \text{ hour}}{60 \text{ minutes}}$: $90 \times \frac{1}{60} = 1.5$ hours. To convert a speed from feet per second to feet per minute, multiply by $\frac{60 \text{ seconds}}{1 \text{ minute}}$. Setting the conversion as a fraction so the unwanted unit cancels keeps you from multiplying when you should divide.

Direct and inverse variation

Two further patterns appear:

• Direct variation: $y = kx$. As $x$ grows, $y$ grows in proportion, and the ratio $\frac{y}{x} = k$ stays constant. If $y = 12$ when $x = 3$, then $k = 4$ and $y = 4x$.
• Inverse variation: $y = \frac{k}{x}$. As $x$ grows, $y$ shrinks, and the product $xy = k$ stays constant. If $y = 6$ when $x = 2$, then $k = 12$ and $y = \frac{12}{x}$.

Spotting which one a problem describes ("doubles when", "halves when") tells you whether to hold a ratio or a product constant.

Why setup is everything

In a ratio or rate problem, the arithmetic is easy; the score depends on setting up the right relationship. The two reliable habits are aligning like units in a proportion (flour with flour) and labelling the unit rate with its units (pages per minute, miles per hour). With those in place, cross-multiplication or a single multiplication finishes the job, and a quick estimate confirms the size is sensible.

Try this

Q1. If 3 pens cost \4.20$, how much do 7 pens cost at the same rate? [1 point]

• Cue. Unit rate \frac{4.20}{3} = \1.40$per pen;$7 \times 1.40 = \$9.80$.

Q2. $y$ varies directly with $x$, and $y = 20$ when $x = 5$. Find $y$ when $x = 8$. [1 point]

• Cue. $k = \frac{20}{5} = 4$, so $y = 4x = 4(8) = 32$.