HESI A2 Ratio and Proportion study guide

The eight procedures a ratio or proportion question needs — writing a ratio in lowest terms, unit rates, scaling, cross-multiplying, splitting a total, telling direct from inverse variation, converting a rate's unit and scale factors — each with a worked example and the trap it sets, plus a setup table, a four-step method and three answer-choice strategies.

What this topic actually asks of you

This topic is where arithmetic becomes a tool. Nothing here is harder than the four operations in basic operations; what changes is that the numbers arrive inside a compare and contrast question, and the comparison has to be set up correctly before any of it can be computed.

That is why this page spends more words on setup than on solving. A proportion is two equal ratios, and almost every error on this topic is a ratio written the wrong way up — the same arithmetic, applied to an inverted relationship.

HESI lists mathematics among the areas the A2 covers; what its owner does not publish is a national content specification, so no page here claims an item count, a clock or a cut score.

The eight procedures

Write a ratio in lowest terms, with its units named, and say whether it is part-to-part or part-to-whole. → example 1
Find a unit rate — amount per one unit — and use it to compare two offers. → example 2
Scale a ratio up or down by multiplying or dividing both sides by the same number. → example 3
Solve a proportion by cross-multiplying when one of the four values is unknown. → example 4
Split a total in a given ratio by adding the parts, dividing, then multiplying back. → example 5
Tell direct from inverse variation before you set the proportion up. → example 6
Convert a rate's unit first, then take the rate. → example 7
Use a scale factor on a drawing, a map or a pair of similar figures. → example 8

The setup table

Most of the difficulty on this topic is deciding what goes where. These are the three shapes a question takes.

Shape	What it looks like	How it is set up
Proportion	two ratios that are equal, one value missing	like over like, units in the same position on both sides
Unit rate	"how much for one?"	amount ÷ count, then multiply by the count you need
Total split	a ratio plus the total	add the parts, divide the total by that sum, multiply each part

Relationship	What happens	What stays constant	Example
Direct	as one goes up, the other goes up	the ratio	more hours → more pay
Inverse	as one goes up, the other goes down	the product	more workers → fewer days

The most useful habit on this topic: write the units beside the numbers. A proportion whose units do not line up is wrong before any arithmetic happens, and the mismatch is invisible if the units only exist in your head.

A four-step method

1. Say what is being compared to what. "Milliliters to milligrams." "Pages to minutes." Two nouns, in that order.

2. Put the same noun in the same place on both sides. If the first ratio is mL over mg, the second is mL over mg. This is the whole trick, and it is the step people skip.

3. Solve, then read the question again. A proportion gives you the fourth value — check that the fourth value is the thing that was asked for.

4. Sanity-check the direction. If the known quantity went up and your answer went down, you either inverted a ratio or the relationship is inverse. Decide which, deliberately.

The eight worked examples

1. Write a ratio in lowest terms, and name it

A solution is made from 45 mL of concentrate and 60 mL of water. Give the ratio of concentrate to water, and concentrate to the whole.

Concentrate to water: 45 : 60, divided by 15 → 3 : 4.
Concentrate to the whole: 45 : 105, divided by 15 → 3 : 7.
Check: 3 + 4 = 7, which is the second ratio's second term.

The trap: reporting 3 : 4 as the fraction of the mixture. Part-to-part and part-to-whole are different questions, and a whole is the sum of the parts.

2. Find a unit rate, then compare

Three pounds of coffee cost $6.48. A five-pound bag costs $10.25. Which is the better buy?

$6.48 ÷ 3 = $2.16 per pound.
$10.25 ÷ 5 = $2.05 per pound.
The five-pound bag is cheaper per pound by 11 cents.

The trap: comparing $10.25 with $6.48. Bags of different sizes cannot be compared until both are priced per one pound.

3. Scale a ratio up or down

2 : 5 = ? : 35.

5 was multiplied by 7 to become 35.
Multiply the other side by the same 7: 2 × 7 = 14.
14 : 35.

The trap: adding rather than multiplying. A ratio expresses relative size; only multiplication (or division) keeps the relationship.

4. Solve a proportion by cross-multiplying

If 2 mL of a solution contains 50 mg of a substance, how many mL contains 175 mg?

Set it up with mL on top both times: 2/50 = x/175.
Cross-multiply: 50x = 2 × 175 = 350.
Divide: x = 350 ÷ 50 = 7 mL.

The trap: writing the second ratio the other way up (50/2 = x/175), which puts milligrams where milliliters belong and gives 4,375. The units were the alarm.

5. Split a total in a given ratio

Two partners split $64 in the ratio 3 : 5. How much is each?

Add the parts: 3 + 5 = 8.
One part: 64 ÷ 8 = $8.
Multiply back: 3 × 8 = $24, and 5 × 8 = $40.
Check: 24 + 40 = 64.

The trap: taking 3 : 5 to mean $3 and $5, then scaling the wrong way. The ratio describes parts of the whole, not fractions of the total (3/5 of $64 is a different question).

6. Direct or inverse — decide before you set it up

Two questions, one of each kind.

Direct. 4 m of fabric costs $10. What do 9 m cost? 10/4 = x/9, cross-multiply → 4x = 90 → $22.50. More fabric, more money.
Inverse. 6 workers finish a wall in 8 days. How long for 4 workers at the same rate? 6 × 8 = 48 worker-days, and 48 ÷ 4 = 12 days. Fewer workers, more days.
Check the direction in both: 9 m costs more than 4 m did, and 4 workers take longer than 6 did.

The trap: setting up every question as a direct proportion. Ask first whether the second quantity should grow or shrink; if it should shrink, the proportion is inverse.

7. Convert a rate's unit first

A machine prints 90 pages in 2.5 minutes. What is its rate in pages per hour?

Pages per minute: 90 ÷ 2.5 = 36 pages per minute.
Minutes per hour: 60.
36 × 60 = 2,160 pages per hour.

The trap: multiplying 90 by 60 without first finding the per-minute rate. A rate is per one unit; find it, then scale it.

8. Use a scale factor

On a map, 1 in represents 25 mi. Two towns are 3.6 in apart on the map. How far apart are they?

Scale factor: 25 mi per inch.
3.6 × 25 = 90 mi.
Check: 3.6 in is nearly four times 1 in, and 4 × 25 = 100, so 90 mi is right where it should be.

The same move works on similar figures: sides 4 and 6 in one figure, a side of 10 in the other → 4/6 = 10/x → 4x = 60 → x = 15.

The trap: setting the scale the wrong way up (1/25 instead of 25/1) and reporting 0.144 in. The answer to "how far" is a distance, not a fraction of an inch.

The traps, collected

A ratio written upside down — it survives correct cross-multiplication.
Part-to-part reported as part-to-whole (3 : 4 offered where the question wanted 3 : 7).
Units dropped from the setup, so a mismatch has nowhere to show itself.
Comparing totals instead of unit rates across differently sized packages or batches.
Scaling by adding rather than multiplying both sides of a ratio.
Direct assumed where the relationship is inverse, so the answer moves the wrong way.
A rate used before it is reduced to one unit, which is where per-minute and per-hour get confused.
A scale factor inverted, turning a distance into a fraction.

Answer-choice strategies: three moves for when the setup will not come

1. Find the unit rate and read the choices. A question about "which is the better value" is answered by two divisions, and the choices bracket the answer.

$10.25 for 5 lb is a little over $2 a pound — a choice of $2.05 is right and $3.42 is not.

2. Test the direction of the answer. Before solving, ask whether the unknown should be larger or smaller than the number you know. Half of the wrong choices fail this.

If 2 mL holds 50 mg, then 175 mg needs more than 2 mL — a choice of 0.57 mL is already gone.

3. Estimate with a round proportion. Round one number to make the division mental, then pick the choice nearest the estimate.

90 pages in 2.5 min is about 36 a minute, and 36 × 60 is about 2,000 an hour.

How to practice this

Write the two nouns before the two numbers. "mL per mg" going in, "mL per mg" coming out. It costs five seconds and prevents the inverted ratio.
Re-derive one worked example from memory the next day, then compare it with the page. If the setup matches, the method is yours.
Practice the inverse direction on purpose. It is the shape that feels wrong and is right.
Keep units in every line until the last one, where the answer carries the unit the question asked for.
Then move to dosage calculation, where a proportion is the whole method rather than a topic.

Where to go next

The applied topic — Dosage Calculation: proportions against a label, a weight and a time.
Back to the arithmetic — Basic Operations: fractions, decimals and percents, the three forms every ratio takes.
Units — Measurement and Conversion: the factors a rate is built from.
The subject's own page — Ratio and Proportion lists what is published for this subject.

Sources. The procedures above are written for this course, in our own words: a proportional method is a fact and no publisher's prose is reproduced here. The publisher's own account of the areas the exam covers is quoted from its HESI A2 page. Accessed 2026-10-23.

HESI A2 is a trademark of Elsevier Inc. Elk Ridge Academy is an independent study resource and is not affiliated with, endorsed by, or sponsored by Elsevier Inc. Exam names are used descriptively only.