How to set up and solve a proportion
TEAS 7 outline: Solve real-world proportion problems
Also covers HESI A2 Math (Ratios and proportions)
- A proportion sets two ratios equal: a/b = c/d.
- Keep the same units in matching spots on both sides.
- Cross-multiply, then divide to get the unknown alone.
- For dosage by weight, convert pounds to kilograms first (divide by 2.2).
- Check the size: a bigger known amount means a bigger unknown.
Key facts
- A proportion is an equation stating that two ratios or rates are equal: a/b = c/d.
- In a true proportion the cross products are equal: if a/b = c/d, then a × d = b × c.
- Keep the same unit in both numerators and the same unit in both denominators, for example mg over kg on both sides.
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- To solve, cross-multiply, then divide both sides by the number multiplying the unknown.
- Scale problems are proportions: drawing (or map) length over real length on both sides.
- For dosage by weight, convert the weight to kilograms first (1 kg is about 2.2 lb; more precisely 2.2046 lb), then set up the proportion.
- Check the answer: if the second known amount is bigger than the first, the unknown should be bigger too.
Quick check1 question · 2 more at the end
A dosage chart lists 7.5 mg of a medication per kilogram of body weight. How many milligrams correspond to a patient who weighs 132 lb? (Use 1 kg ≈ 2.2 lb.)
Correct answer: B.
A per-kilogram rate needs the weight in kilograms, so convert first: 132 ÷ 2.2 = 60 kg. Then set up the proportion 7.5 mg / 1 kg = x mg / 60 kg, so x = 7.5 × 60 = 450 mg. Remember that the number of kilograms is always smaller than the number of pounds.
A This skips the conversion and multiplies 7.5 by 132 lb.
B Correct: 132 ÷ 2.2 = 60 kg, and 7.5 × 60 = 450 mg.
C This divides the weight by the rate (60 ÷ 7.5) instead of multiplying.
D This multiplies 132 by 2.2 (290.4 kg) instead of dividing, then multiplies by 7.5.
Full lesson
Each part starts with its main idea. Open a part when you want the details.
The short answer: A proportion says two ratios are equal, such as a/b = c/d. Write both fractions with the same units in the same positions, cross-multiply (a × d = b × c), then divide to get the unknown alone. Check that the answer makes sense: if one quantity grows, the matching quantity grows by the same factor.
1 What is a proportion and how do you solve one?A proportion is just two fractions that are equal because they describe the same comparison. 1 min
A proportion is just two fractions that are equal because they describe the same comparison.
- Write the ratio you know as a fraction, with units: 3 cups / 4 servings.
- Write the second fraction with the unknown x, units in the same spots: x cups / 10 servings.
- Cross-multiply: top left × bottom right = bottom left × top right.
- Divide to get x alone.
- Check: does the size of the answer make sense?
| Setup | Valid? | Why |
|---|---|---|
| grams/mL = grams/mL | Yes | Same units on top, same units on bottom |
| mL/grams = mL/grams | Yes | Flipped on both sides, still lined up |
| grams/grams = mL/mL | Yes | Each side is first mixture over second mixture, in the same order |
| grams/mL = mL/grams | No | One side is upside down |
To test a proportion, compare cross products: 3/4 = 9/12 because 3 × 12 = 36 and 4 × 9 = 36.
2 Worked example 1: How do you scale a recipe?When a recipe grows, every ingredient grows by the same factor. 1 min
When a recipe grows, every ingredient grows by the same factor.
A soup recipe uses 3 cups of broth for 4 servings. How much broth for 10 servings?
- 3 cups / 4 servings = x cups / 10 servings
- Cross-multiply: 4 × x = 3 × 10, so 4x = 30
- Divide: x = 30 ÷ 4 = 7.5 cups
- Check: 10 is 2.5 times 4, and 3 × 2.5 = 7.5.
Wrong turn: adding instead of scaling (6 more servings, so 3 + 6 = 9 cups) or flipping one side (3/4 = 10/x gives about 13.3 cups).
3 Worked example 2: How do you use a map scale?Scale problems are proportions with map length on top and real length on the bottom. 1 min
Scale problems are proportions with map length on top and real length on the bottom.
On a map, 2 cm stands for 15 km. Two towns are 7 cm apart on the map.
- 2 cm / 15 km = 7 cm / x km
- Cross-multiply: 2x = 15 × 7 = 105
- Divide: x = 105 ÷ 2 = 52.5 km
Mental math shortcut: 7 cm is 3.5 groups of 2 cm, so the distance is 3.5 × 15 = 52.5 km. Wrong turn: 15/2 = 7/x gives under 1 km, which is clearly too small for 7 cm of map.
4 Worked example 3: How do you find a dose by body weight?Convert the weight to kilograms first, then use the per-kilogram rate as one side of the proportion. 1 min
Convert the weight to kilograms first, then use the per-kilogram rate as one side of the proportion.
Math practice: an order reads 3 mg per kg, the person weighs 154 lb, and the liquid has 50 mg in 1 mL. How many mL? (1 kg ≈ 2.2 lb)
- Weight: 154 ÷ 2.2 = 70 kg (calculator is fine here).
- Dose: 3 mg / 1 kg = x mg / 70 kg, so x = 210 mg.
- Volume: 50 mg / 1 mL = 210 mg / y mL, so 50y = 210 and y = 4.2 mL.
- Check: 210 mg is a little more than 4 × 50 mg, so a little more than 4 mL makes sense.
Wrong turn: using 154 as if it were kilograms gives 462 mg, more than double the right amount.
Common mistakes
The traps students fall into most. Read these before you practice.
Watch out: Line up units across the equals sign (mg/kg = mg/kg), not mg/kg = kg/mg; flipping one side gives a wrong answer.
Watch out: Divide pounds by 2.2 to get kilograms, do not multiply; the number of kilograms is always smaller than the number of pounds.
Watch out: Scaling multiplies every amount by the same factor; it does not add the same amount.
Watch out: Cross-multiply diagonally (top left × bottom right), not straight across.
Watch out: Use a per-kilogram rate with the weight in kilograms, never with the weight in pounds.
How to remember it
Matching units in matching spots is the whole secret.
Memory hook: "Same units, same floors." Units upstairs match each other; units downstairs match each other.
Self-check:
- Set up and solve: 4 mL per 20 lb, how many mL for 50 lb?
- Explain why 2 lb is about 0.9 kg, not 4.4 kg.
- Write two different, correct setups for the map example above.
Show answers
- 10 mL.
- Set up: 4 mL / 20 lb = x mL / 50 lb.
- Cross-multiply: 20 × x = 4 × 50, so 20x = 200.
- Divide: x = 200 ÷ 20 = 10 mL.
- Check: 50 is 2.5 times 20, and 4 × 2.5 = 10.
- Divide by 2.2, do not multiply.
- 2 ÷ 2.2 ≈ 0.9 kg.
- A kilogram is bigger than a pound, so you need fewer of them.
- 4.4 comes from multiplying 2 × 2.2, which makes the number bigger instead of smaller.
- Any two of these work:
- 2 cm / 15 km = 7 cm / x km (map over real on both sides).
- 15 km / 2 cm = x km / 7 cm (real over map on both sides).
- 2 cm / 7 cm = 15 km / x km (first over second on both sides).
- Each one cross-multiplies to 2x = 105, so x = 52.5 km.