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Practice sheet · Medicine Path · Lens 1: Math

Rates, half-life & tests

Practice for The Math of Medicine: units, rates, percents, spread, herd immunity, half-life, tests, risk and ranges (pp.8–16), plus Algebra A1–A2 (pp.17–18).
Warm-upCoreLevel Up
Name Date Time I spent
Remember

All numbers here are made-up maths. Never work out or change a real dose: follow the label, or ask a pharmacist or doctor. General information, not medical advice.

Rate = amount ÷ time. Half-life: each one multiplies by ½. Herd immunity ≈ 1 − 1 ÷ R₀. For tests, count people in a table, not percents.

Warm-up

One step

Units, rates and percents.
1

Units

Change 2.5 g to mg, and 750 mL to litres.

2

Drip rate

A made-up IV bag holds 900 mL and runs over 6 hours. What is the rate in mL/h, and in mL per minute?

3

Blood types

About 37% of Canadians are O-positive and 8% are O-negative. About how many of 600 donors are each type?

4

Spread

In a made-up outbreak, each case infects 3 others. How many new cases are there in round 4?

Core

Two or three steps

Make a table when you can.
5

Caffeine limit

Health Canada suggests teens have no more than 2.5 mg of caffeine per kg per day. What is the limit for a 60 kg teen? Would one 80 mg drink stay under it? Two?

6

Herd immunity

Use the simple model 1 − 1 ÷ R₀ (p.12). What share must be immune if R₀ = 8? If R₀ = 2.5?

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Medicine Path · Unizon · PracticeRates, half-life & test results
7

Half-life

A made-up 320 mg amount has a 5-hour half-life. How much is left after 15 hours? After 20 hours?

8

Positive test

10,000 people; 2% have a made-up illness. A test finds 95% of those who have it, but is wrongly positive for 4% of those who don’t. What share of positives are true?

9

Risk headline

A made-up study: 8 in 1,000 people get ill without a medicine, 6 in 1,000 with it. Find the relative drop and the absolute drop.

10

Normal range

Five made-up temperatures: 36.5, 36.9, 37.1, 36.7 and 36.8 °C. Find the mean, the median and the range.

Level Up

Equations

A1–A2 (pp.17–18) and half-life powers.
11

Emptying bag

A made-up bag holds V = 1,200 − 150t mL after t hours. When is 300 mL left?

12

Two bags

Bag A: V = 600 − 40t. Bag B: V = 900 − 90t. When do they hold the same volume, and how much?

13

Halving power

A made-up 400 mg amount follows A = 400 × (½)ⁿ after n half-lives. When is 25 mg left? With a 6-hour half-life, how many hours is that?

14

Clinic poster

A poster is 3 dm longer than it is wide, with an area of 28 dm². Solve w(w + 3) = 28.

Make your own problem

Your turn to be the public-health analyst

  1. Pick an idea from the book: a rate, a half-life, a test, a risk.
  2. Write a question with made-up numbers (never real doses).
  3. Solve it, and explain what the answer does and does not tell you.

My problem, and my worked answer:

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Medicine Path · Unizon · PracticeRates, half-life & test results

Try every problem first, then check here. Wrong answers are useful: find the step where your work and the key split apart.

The key never prints with Print my work or Print blank. Open it to get its own Print the answer key button.

Answer key

Worked solutions. Every dose and illness here is made-up maths. General information, not medical advice.

Warm-up
1
2,500 mg · 0.75 L × 1,000 for g → mg. ÷ 1,000 for mL → L.
2
150 mL/h · 2.5 mL/min 900 ÷ 6 = 150. 150 ÷ 60 = 2.5. Keep the “per what” in the unit.
3
≈ 222 · ≈ 48 0.37 × 600 = 222. 0.08 × 600 = 48.
4
81 3⁴ = 3 × 3 × 3 × 3 = 81, not 3 × 4.
Core
5
150 mg · one: yes · two: no 2.5 × 60 = 150. 80 < 150, but 2 × 80 = 160 > 150. Labels give real amounts.
6
87.5% · 60% 1 − 1 ÷ 8 = 0.875. 1 − 1 ÷ 2.5 = 1 − 0.4 = 0.6. Real thresholds depend on more than this model.
7
40 mg · 20 mg 15 h = 3 half-lives: 320 → 160 → 80 → 40. One more: 20.
8
≈ 32.6% Ill: 200, test finds 190. Healthy: 9,800, false positives 0.04 × 9,800 = 392. 190 ÷ (190 + 392) = 190 ÷ 582 ≈ 0.326. Most positives are false, so doctors re-test.
9
relative 25% · absolute 0.2 points (8 − 6) ÷ 8 = 25% lower. 0.8% → 0.6%: 0.2 percentage points, or 2 fewer in 1,000.
10
mean 36.8 · median 36.8 · range 0.6 °C 184.0 ÷ 5 = 36.8. Sorted: 36.5, 36.7, 36.8, 36.9, 37.1. 37.1 − 36.5 = 0.6.
Level Up
11
6 hours 1,200 − 150t = 300 → 150t = 900 → t = 6.
12
6 h, both 360 mL 600 − 40t = 900 − 90t → 50t = 300 → t = 6. 600 − 240 = 360.
13
n = 4 · 24 hours (½)ⁿ = 25 ÷ 400 = 1/16 = (½)⁴. 4 × 6 = 24 h.
14
4 dm × 7 dm w² + 3w − 28 = 0 → (w + 7)(w − 4) = 0 → w = 4 (not −7). Check: 4 × 7 = 28.
Make your own: write the unit with every number (mg, mL/h, %), and make a table of people for test questions.
Medicine Practice · Medicine Path · Unizon · Not medical adviceAnswer key