🚚 Ratios, Proportions and Rates

Part 1 of 7 — Ratios and Rates on the AR Subtest

Topic Overview

What these questions look like

Ratio and rate problems are some of the most common stories on Arithmetic Reasoning (AR). The official description of the subtest lists ratio and proportion by name, and official sample topics include rate × time and proportional rates. You will see wording like this:

• "for every 3 drivers there are 2 mechanics" (a ratio)
• "a machine fills 60 bags in 5 minutes; how many in 20 minutes?" (a proportion)
• "a truck averages 50 miles per hour" (a rate)
• "working together, how long will it take?" (a work rate)
• "which size is the better buy?" (a unit price)

Every one of these is really the same idea: two quantities that change together at a steady rate. Once you find that rate (how much of one thing per one of the other), the rest is multiplying or dividing.

🚚 Ratios, Proportions and Rates

Part 2 of 7 — Ratios and Part-to-Whole Problems

What a ratio says

A ratio compares two amounts. "The ratio of teens to adults is 2 to 3" can be written three ways, and they all mean the same thing:

It means: for every 2 teens there are 3 adults. It does not say there are exactly 2 teens; there could be 20 teens and 30 adults.

Order matters

The ratio is written in the same order as the words. "Teens to adults" puts teens first. "Adults to teens" is $3 : 2$, a different ratio. Flipping the order is the most common ratio mistake on AR, and the flipped ratio is almost always one of the choices.

Simplifying a ratio

Divide both numbers by the same thing, just like reducing a fraction.

Example. A volunteer group has 16 teens and 24 adults. The ratio of teens to adults is

$16 : 24 = \frac{16}{24} = \frac{2}{3}, \text{ so } 2 : 3$

(Divide both by 8.) The ratio of adults to teens is $3 : 2$.

🚚 Ratios, Proportions and Rates

Part 3 of 7 — Proportions and Scale

What a proportion is

A proportion says two ratios are equal. On AR it appears whenever a rate stays the same: the same speed, the same price per item, the same recipe, the same map scale.

Set it up with matching units

Write each ratio with the same unit on top and the same unit on the bottom.

Worked Example. A press stamps 48 parts in 6 minutes. At the same rate, how many parts does it stamp in 25 minutes?

$\frac{48 \text{ parts}}{6 \text{ min}} = \frac{x \text{ parts}}{25 \text{ min}}$

Cross-multiply: if $\frac{a}{b} = \frac{c}{d}$, then $a \times d = b \times c$.

$6 \times x = 48 \times 25 = 1{,}200 \qquad x = 1{,}200 \div 6 = 200$

The unit-rate way gives the same answer: $48 \div 6 = 8$ parts per minute, and $8 \times 25 = 200$ parts.

Estimate: 25 minutes is a little more than $4 \times 6 = 24$ minutes, so the answer is a little more than $4 \times 48 = 192$. ✓

When the unknown is on the bottom

How long does the same press take to stamp 360 parts?

$\frac{48}{6} = \frac{360}{x} \qquad 48x = 6 \times 360 = 2{,}160 \qquad x = 45 \text{ minutes}$

Or with the rate: $360 \div 8 = 45$ minutes.

Key Fact: Use whichever method feels faster. The unit rate is quickest when it divides evenly; cross-multiplying always works.

🚚 Ratios, Proportions and Rates

Part 4 of 7 — Distance, Rate and Time

One formula, three ways to use it

Rate × time is an official AR sample topic. Everything comes from one formula:

$\text{distance} = \text{rate} \times \text{time} \qquad d = r \times t$

The units must match: miles per hour goes with time in hours.

Worked Example. A convoy drives at 52 miles per hour for 2 hours 30 minutes. How far does it go?

• 2 hours 30 minutes is $2\frac{1}{2}$ hours, or 2.5 hours.
• $52 \times 2.5 = 52 \times 2 + 52 \times 0.5 = 104 + 26 = 130$ miles.

Estimate: about 50 mph for 2.5 hours is about 125 miles. ✓

🚚 Ratios, Proportions and Rates

Part 5 of 7 — Work Rates and Unit Prices

Work rates: think "jobs per hour"

If a pump drains a tank in 4 hours, then in one hour it drains $\frac{1}{4}$ of the tank. That fraction is its rate:

$\text{rate} = \frac{1 \text{ job}}{\text{time for the job}}$

When two workers (or machines) work together, their rates add, not their times.

Worked Example. One pump drains a tank in 4 hours. A second pump drains it in 12 hours. How long do they take working together?

• Rates: $\frac{1}{4}$ and $\frac{1}{12}$ of the tank per hour.
• Combined rate: $\frac{1}{4} + \frac{1}{12} = \frac{3}{12} + \frac{1}{12} = \frac{4}{12} = \frac{1}{3}$ of the tank per hour.
• Time together: $1 \div \frac{1}{3} = 3$ hours.

Shortcut for two workers: time together $= \frac{\text{product}}{\text{sum}} = \frac{4 \times 12}{4 + 12} = \frac{48}{16} = 3$ hours.

Reasonableness check: together they must be faster than the faster worker alone. 3 hours is less than 4 hours. ✓ Adding the times (16 hours) or averaging them (8 hours) gives an answer that is slower than one pump alone, which makes no sense.

🚚 Ratios, Proportions and Rates

Part 6 of 7 — Test Strategy and Common Traps

The seven traps behind the wrong choices

Ratio and rate questions have very predictable wrong answers. Learn them and you will recognize them on sight.

Key Fact: Each trap produces a number that really is one of the four choices. Matching a choice proves nothing; a quick check against the story does.

🚚 Ratios, Proportions and Rates

Part 7 of 7 — Mixed Mastery Set

Put it all together

This set mixes every skill from the topic, roughly the way a real AR subtest builds: a few quick ones, a solid middle, then a few that take more steps.

Work each one the way you would at MEPS: last sentence first, find the rate for one, units on every line, an estimate, and a final check. Aim for about 2 to 3 minutes per problem, faster on Set A.