Ratio and Proportion

Ratio questions look like four different question types and are really one question type with four wordings. Learn the total-parts method and the whole family collapses.

Parts to a whole

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A flight of 40 trainees has men and women in a ratio of 5 to 3. How many women are in the flight?

The ratio 5 to 3 does not mean 5 men and 3 women. It means that for every 5 men there are 3 women, which makes 8 parts in total.

• Total parts: 5 plus 3 is 8.
• One part: 40 divided by 8 is 5 trainees.
• Women: 3 parts, so 3 times 5 is 15.

Check it against the other group: men are 5 parts, or 25, and 25 plus 15 is 40. That check costs two seconds and catches almost every error on this question type.

Three steps, every time:

1. Add the parts.
2. Divide the real total by the total parts. That is the value of one part.
3. Multiply by the parts you were asked about.

The three-term row is the point of the table. A ratio with three parts works exactly the same way, and the test does ask them.

The fraction that a ratio is not

In a ratio of 5 to 3, the first group is:

• 5/8 of the whole - five parts out of eight total parts.
• 5/3 of the other group - which is a comparison between the two groups, not a fraction of anything.

Both are true statements and they answer different questions. The wrong answer on this question type is treating 5/3 as a fraction of the total, or treating 5/8 as the relationship between the groups.

Read what the question compares to. "What fraction of the flight is women" needs the total on the bottom. "How many times as many men as women" needs the other group on the bottom.

Equivalent ratios and proportions

A proportion is two equal ratios, and the standard move is cross-multiplication.

A scale drawing uses 3 inches for every 25 feet. A corridor measures 7.5 inches on the drawing. How long is the real corridor?

Set it up so the same unit is on top on both sides:

3 inches over 25 feet equals 7.5 inches over x feet.

Cross-multiply: 3 times x equals 25 times 7.5, so 3x is 187.5 and x is 62.5 feet.

The setup is the whole risk. If inches are on top on the left, inches must be on top on the right. Mixing them produces an answer that is off by a factor of the ratio squared, and it will look plausible.

A faster route when the numbers cooperate: 7.5 inches is 2.5 times 3 inches, so the real length is 2.5 times 25 feet, which is 62.5. Scaling the whole ratio by one factor beats cross-multiplication when you can see the factor.

Ratios that change

The nastier version gives you a ratio, changes the population, and asks for the new ratio.

A motor pool has trucks and cars in a ratio of 4 to 7. There are 44 vehicles. If 6 more trucks arrive, what is the new ratio of trucks to cars?

Get the actual counts first. Total parts 11, one part is 44 divided by 11, which is 4. So 16 trucks and 28 cars.

Six more trucks makes 22 trucks and 28 cars. The ratio 22 to 28 both divide by 2, giving 11 to 14.

Never try to adjust the ratio directly. Convert to counts, change the counts, convert back and reduce.

Rates written as ratios

"Three out of every five" and "3 to 2" describe the same population in different words, and they are easy to mix up.

"Out of" puts the total on the right. "To" puts the other group on the right. That one word changes the answer, and the test writes both.

What you can skip

Across the 57 questions on this topic:

• Inverse proportion by name never appears here; the inverse cases live in the work and speed lessons.
• Three-part ratios come up about three times. Add the parts and share the total exactly as with two.

Where people lose points

Using a ratio term as a fraction of the total. The 5 in "5 to 3" is 5/8 of the whole, not 5/3 and not 5/5.

Forgetting to add the parts. Dividing the real total by a single term instead of by the sum is the most common arithmetic error here.

Reversing the order. "Ratio of officers to enlisted" is officers on top. The reciprocal is a choice.

Not reducing. A ratio answer is expected in lowest terms. If your answer is 22 to 28 and the choices show 11 to 14, that is not a different answer.

Cross-multiplying a proportion that was set up with mismatched units. Write the unit next to each number in the setup. It takes three seconds and it is the only reliable defense.

Work one in under a minute

A maintenance section mixes coolant and water in a ratio of 2 to 5. How much coolant is in 63 gallons of mixture?

Total parts: 2 plus 5 is 7. One part: 63 divided by 7 is 9. Coolant is 2 parts, so 18 gallons.

Check: water is 5 parts, or 45, and 18 plus 45 is 63.

Where this leads

Ratio is the backbone of scaling, of mixture problems, and of gear and pulley questions on Mechanical Comprehension, which are ratio questions about metal.