Free ATI TEAS 7 Mathematics guide
October 2026 - Ratios, Rates & Proportions Study Guide for nursing students in
A proportion sets two ratios equal and is solved by cross-multiplying. It is the single most useful skill in the Mathematics section, because dosage, scale, unit conversion, recipe scaling and unit-price problems are all proportions wearing different clothes.
Ratios and rates first
A ratio compares two quantities. A rate is a ratio between quantities in different units.
- A ratio can be written three ways:
3:4,3/4, or "3 to 4". All mean the same thing. - Simplify ratios like fractions. In a class of 12 men and 16 women, the ratio is
12:16, which reduces to3:4. - A rate carries units: 80 km per hour, £1.50 per pack, 2 mg per kilogram.
- A unit rate has 1 in the second position — the "per one" figure. £9.00 for 6 packs is £1.50 per pack.
Part-to-part or part-to-whole
A ratio of 3:4 men to women describes parts against each other. Turned into a fraction of the whole group, men are 3/7 and women 4/7, because there are seven parts in total.
Questions exploit this constantly: given a ratio of 3:4 and a total of 63, each part is 63 ÷ 7 = 9, so there are 27 men and 36 women. Dividing 63 by 3 or by 4 gives an answer that is among the choices and wrong.
Solving a proportion with the units written in
Worked example
A solution contains 12 mg of a drug in every 5 mL. How many millilitres contain 30 mg?
- Write the proportion with units on both sides so they match:
12 mg / 5 mL = 30 mg / x mL. - Check the arrangement: milligrams sit on top on both sides, millilitres underneath on both. If they did not, the setup is wrong.
- Cross-multiply:
12x = 5 × 30, so12x = 150. - Divide:
x = 12.5.
12.5 mL. Writing the units in is the check that catches an inverted proportion before it becomes a wrong answer.
Keep the units in the same positions on both sides of the equals sign. If they do not cancel to leave the unit you want, the proportion is arranged wrongly — and an inverted proportion produces an answer that looks reasonable and is the reciprocal of the truth.
The same skill in four disguises
| Question type | Set up as | Watch for |
|---|---|---|
| Unit price comparison | price ÷ quantity for each option | Different pack sizes; compare per-unit, not per-pack |
| Recipe scaling | ingredient / servings = x / new servings | Scaling only some ingredients |
| Map or model scale | map distance / real distance = x / y | Mixed units — centimetres against kilometres |
| Speed, distance, time | distance / time = x / new time | Minutes given where the rate is per hour |
Terms to know
- Ratio
- A comparison of two quantities, written
3:4,3/4or "3 to 4". - Rate
- A ratio between quantities with different units, such as km per hour.
- Unit rate
- A rate expressed per one unit — the price of one item, the distance in one hour.
- Proportion
- A statement that two ratios are equal.
- Cross-multiplying
- Multiplying each numerator by the opposite denominator to solve a proportion.
- Part-to-whole
- A part compared with the total rather than with another part.
- Scale factor
- The number you multiply by to move between two proportional quantities.
What the TEAS asks most
The same core ideas appear in different wording. If you can answer these without stopping to think, you have what this section requires.
If X gives Y, how much does Z give?
Set up a proportion with matching units and cross-multiply.
Which is the better value?
Reduce both to a unit price. Pack sizes are chosen to make the raw prices misleading.
The ratio is 3:5 and the total is 64. How many are in each group?
Add the parts (8), divide the total by that (8), then multiply each part.
On a map where 1 cm is 5 km, how far apart are two points 3.4 cm apart?
A proportion. Convert units before setting it up, not after.
Key points
- A proportion is two equal ratios, solved by cross-multiplying.
- Write the units into the setup; if they do not match across the equals sign, it is inverted.
- A ratio of
3:4means3/7and4/7of the whole — add the parts before dividing a total. - A unit rate is the "per one" figure, and it is what makes value comparisons possible.
- Dosage, scale, recipes and unit prices are the same operation in different contexts.
Review quiz
4 questions on this topic, each with the reasoning worked through. Pick an answer to see how it went, or open the explanation straight away.
-
Question 1 of 4
A solution contains 8 mg of medication per 10 mL. How many millilitres are needed to deliver 20 mg?
Show the answer and explanation
Correct answer: B. 25 mL
8/10 = 20/xcross-multiplies to8x = 200, giving 25 mL. 16 mL comes from inverting the proportion, 2.5 mL is the answer with a decimal place lost, and 40 mL doubles 20 rather than scaling from the ratio. -
Question 2 of 4
Which is the better value: 6 dressings for £4.80, or 10 dressings for £7.50?
Show the answer and explanation
Correct answer: B. The pack of 10, at £0.75 each
4.80 ÷ 6 = 0.80and7.50 ÷ 10 = 0.75, so the larger pack is cheaper per dressing. The unit rate is the only fair comparison here, since the total prices tell you nothing on their own. -
Question 3 of 4
Staff on a ward are allocated to day and night shifts in a ratio of 5:3. If 48 staff are allocated in total, how many work nights?
Show the answer and explanation
Correct answer: B. 18
The parts add to 8, so each part is
48 ÷ 8 = 6staff, and nights take three parts:3 × 6 = 18. 30 is the day figure, 16 comes from dividing 48 by 3 as though the ratio had three parts in total, and 9.6 comes from dividing by 5 — both errors that skipping the "add the parts" step produces. -
Question 4 of 4
On a floor plan, 2 cm represents 5 metres. A corridor measures 7 cm on the plan. How long is it?
Show the answer and explanation
Correct answer: B. 17.5 m
2/5 = 7/xcross-multiplies to2x = 35, sox = 17.5metres. 14 m comes from doubling 7, 10 m from multiplying 2 by 5, and 2.8 m from dividing 7 by 2.5 — all of them arrangements of the right numbers in the wrong proportion.
Put this TEAS topic into practice
Build a set on proportions and use the rationale on every question to reinforce what you just reviewed.
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