Ratio, Proportion and Variation
9 minute read
What you will learn
- Define ratio and simplify a ratio
- Divide a quantity in a given ratio
- Solve problems using proportion
- Distinguish direct variation from inverse variation
- Apply these methods to doses, volumes and strengths
Key points
- A ratio compares two quantities of the same kind, such as 1 : 4.
- A ratio of 1 : 4 means 1 part to 4 parts, so there are 5 parts in all. It is not the same as '1 in 4'.
- A proportion states that two ratios are equal. Solve it by cross-multiplying.
- In direct variation, when one quantity doubles the other doubles, for example dose and body weight.
- In inverse variation, when one quantity doubles the other halves, for example concentration and volume when the amount of drug stays the same.
- Always set out the working with units, then check that the answer is the sensible size.
Ratio
A ratio compares two quantities of the same kind. We write it with a colon: 1 : 4 is read "one to four".
Pay attention to the wording:
- 1 : 4 means 1 part to 4 parts. The total is 5 parts.
- 1 in 4 means 1 part in a total of 4 parts. The other part is 3.
Simplifying a ratio
Divide both sides by the same number, just as you would with a fraction.
- 250 : 1000 = 1 : 4 (divide both by 250)
- 15 : 45 : 60 = 1 : 3 : 4 (divide all by 15)
Both quantities must be in the same unit first. To compare 250 mg with 1 g, change the grams to milligrams: 250 mg : 1000 mg = 1 : 4.
Dividing a quantity in a ratio
- Add up the parts.
- Divide the total by the number of parts to find one part.
- Multiply by each share.
Example: share 120 g in the ratio 1 : 2 : 3.
- Parts: 1 + 2 + 3 = 6
- One part: 120 ÷ 6 = 20 g
- Shares: 1 × 20 = 20 g, 2 × 20 = 40 g, 3 × 20 = 60 g
- Check: 20 + 40 + 60 = 120 g ✓
Proportion
A proportion says that two ratios are equal. If a/b = c/d, then cross-multiplying gives a × d = b × c.
There is a quick way to set one up. Write the known pair, then the unknown pair, so that the same kind of quantity is in the same position:
drug in 5 mL = drug in 12 mL
250 mg x
Example 1. 5 mL of a syrup contains 250 mg. How much is in 12 mL?
- 250 mg is to 5 mL as x is to 12 mL
- x = 250 × 12 ÷ 5 = 3 000 ÷ 5 = 600 mg
Example 2. 20 tablets weigh 10 g. What do 150 tablets weigh?
- x = 10 × 150 ÷ 20 = 1 500 ÷ 20 = 75 g
Example 3. A mixture contains 125 mg in each 5 mL. How many millilitres contain 375 mg?
- x = 375 × 5 ÷ 125 = 1 875 ÷ 125 = 15 mL
Example 4. A formula uses 3 g of drug for 30 mL. How much is needed for 150 mL?
- x = 3 × 150 ÷ 30 = 15 g
The unitary method
Another way is to find the amount for one unit, then multiply. In Example 1: 5 mL holds 250 mg, so 1 mL holds 250 ÷ 5 = 50 mg, and 12 mL holds 12 × 50 = 600 mg. You get the same answer.
Variation
Variation describes how one quantity changes when another one changes.
Direct variation
Two quantities vary directly when they rise and fall together in the same ratio. If one doubles, the other doubles.
Example: a dose is 5 mg for every kilogram of body weight.
| Weight | Dose |
|---|---|
| 12 kg | 5 × 12 = 60 mg |
| 18 kg | 5 × 18 = 90 mg |
| 24 kg | 5 × 24 = 120 mg |
The dose divided by the weight is always 5. That fixed value is the constant of variation.
Inverse variation
Two quantities vary inversely when one rises as the other falls. If one doubles, the other halves. Their product stays the same.
Example 1: concentration and volume. The same amount of drug is dissolved in more and more liquid.
- 20 g in 200 mL gives 10%.
- 20 g in 400 mL gives 5%.
- Check the product: 10 × 200 = 2 000 and 5 × 400 = 2 000.
So a quick rule is: new strength = old strength × old volume ÷ new volume.
Example 2: work and time. A batch takes 6 hours using 2 machines. How long with 3 machines?
- More machines means less time, so this is inverse.
- 6 × 2 ÷ 3 = 4 hours
Telling them apart
Ask: if one quantity doubles, what happens to the other?
| Doubles, so the other... | Type | Example |
|---|---|---|
| Doubles | Direct | Dose and body weight |
| Halves | Inverse | Strength and total volume |
Exam focus
- Dividing a total in a ratio, for example 120 g in the ratio 1 : 2 : 3. Add the parts first.
- Proportion questions about doses and volumes: 'If 5 mL contains 250 mg, how much is in 12 mL?'
- Recognising direct and inverse variation, especially concentration and volume.
- Common mistake: treating '1 part to 4 parts' as a total of 4 parts. The total is 5.
Quick check
Test yourself before you move on. Pick an answer to see the explanation.
1.120 g is divided in the ratio 1 : 2 : 3. The largest share is:
2.5 mL of a syrup contains 250 mg of drug. How much drug is in 12 mL?
3.A dose is 5 mg for every kilogram of body weight. How much does a child of 18 kg need?
4.A solution contains 20 g of drug in 200 mL (10% w/v). Water is added to make 400 mL, with the same amount of drug. The new strength is:
5.Mix 1 part of syrup with 4 parts of water. The total number of parts is: