Mathematics 8 · Handsworth 2026–27 · Unit 3

Ratios, Rates, Proportions & Percent

Ms. Ras Mulinta
Handsworth Secondary
Guided notes

This unit is everywhere in real life — recipes, maps, sale prices, the better deal at the grocery store. A ratio compares amounts. A rate compares amounts with different units. A proportion says two ratios are equal, and once you can solve one you can scale anything up or down. Percent is just a ratio out of 100. Take your time. Each idea has a worked example, then a few problems to try. Answers are at the end of each section so you can check yourself right away.

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The toolkit words we will use

A ratio compares two amounts of the same kind — 3 cups flour to 2 cups sugar, written 3 : 2 or 3⁄2. A rate compares two amounts with different units — 120 km in 2 hours. A unit rate is a rate with a denominator of 1 — 60 km in 1 hour, written 60 km/h. A proportion is a statement that two ratios are equal, like 3⁄2 = 6⁄4. A percent is a ratio out of 100; "per cent" means "per hundred."
Reading a ratio three ways
A class has 12 girls and 16 boys. The ratio of girls to boys is 12 : 16.
Simplify by dividing both parts by their common factor 4: 12 : 16 = 3 : 4.
3 : 4  (also written 3⁄4 or "3 to 4")

1 · Ratios & rates compare, then simplify

Simplify a ratio the same way you simplify a fraction: divide both parts by a common factor until they share no more. To find a unit rate, divide so the second amount becomes 1 — this tells you "how much for one." Unit rates make two deals easy to compare.
Worked example — simplify a ratio
A recipe uses 8 cups of flour to 6 cups of sugar. Write the ratio in simplest form.
8 : 6 — both divide by 2.
= 4 : 3
Worked example — find a unit rate
A car travels 240 km in 4 hours. Find the unit rate (km per hour).
Divide both by 4 so time becomes 1 hour: 240 ÷ 4 = 60.
= 60 km/h

Check yourself

  1. Simplify the ratio 10 : 15.
  2. A printer prints 90 pages in 3 minutes. Find the unit rate in pages per minute.
  3. You earn $54 for 6 hours of babysitting. Find your pay per hour.
Answers
1. 10 : 15 = 2 : 3    2. 90 ÷ 3 = 30 pages/min    3. 54 ÷ 6 = $9/hour

2 · Proportional reasoning & solving proportions two equal ratios

A proportion sets two ratios equal: a⁄b = c⁄d. When one number is missing, find it by scaling (multiply or divide both parts of one ratio by the same number) or by cross-multiplying (a × d = b × c), then dividing. Both give the same answer — pick whichever the numbers make easy.
Worked example — scaling up
3 apples cost $2. At the same rate, how much do 12 apples cost?
3⁄2 = 12⁄n. From 3 to 12 you multiply by 4, so multiply the price by 4 too.
2 × 4 = 8.
= $8
Worked example — cross-multiplying
Solve 5⁄8 = n⁄24.
Cross-multiply: 8 × n = 5 × 24, so 8n = 120.
Divide both sides by 8: n = 120 ÷ 8 = 15.
n = 15

Check yourself

  1. Solve 2⁄5 = n⁄20.
  2. Solve 7⁄n = 21⁄9.
  3. 4 muffins use 6 eggs. How many eggs for 10 muffins?
Answers
4. From 5 to 20 is ×4, so n = 2×4 = 8    5. Cross-multiply: 21n = 63, n = 3    6. 4⁄6 = 10⁄n → 4n = 60, n = 15 eggs

3 · Percents a ratio out of 100

A percent means "out of 100." To find a percent of an amount, change the percent to a decimal (divide by 100, move the point two places left) and multiply. 25% of 80 = 0.25 × 80 = 20. Percents connect three ways: 50% = 0.5 = 1⁄2.
Worked example — percent of an amount
Find 15% of 60.
15% = 0.15.   0.15 × 60 = 9.
= 9
Percents less than 1: 0.5% means half of one percent. As a decimal, 0.5% = 0.005 (the 5 sits two places right of where the 5 in 5% would). Percents greater than 100: 150% = 1.5, which is more than the whole. Fractional/decimal percents: 12.5% = 0.125 = 1⁄8.
Worked example — a percent less than 1
Find 0.5% of 200.
0.5% = 0.005.   0.005 × 200 = 1.
= 1
Worked example — a percent greater than 100
Find 150% of 40.
150% = 1.5.   1.5 × 40 = 60.
= 60
Worked example — decimal percent
Find 12.5% of 64.
12.5% = 0.125.   0.125 × 64 = 8.
= 8

Check yourself

  1. Find 20% of 90.
  2. Find 0.25% of 800.
  3. Find 200% of 35.
  4. Find 2.5% of 40.
Answers
7. 0.20 × 90 = 18    8. 0.0025 × 800 = 2    9. 2.00 × 35 = 70    10. 0.025 × 40 = 1

4 · Financial literacy — best buys unit price wins

When two sizes or two stores offer the same item, the best buy is the one with the lowest unit price — the cost for one unit (one gram, one millilitre, one item). Find each unit price by dividing price by the amount, then compare. Lowest price per unit is the better deal.
Worked example — best buy
Brand A: 500 g of pasta for $2.00. Brand B: 750 g for $2.70. Which is the better buy?
Brand A unit price: 2.00 ÷ 500 = $0.004 per gram.
Brand B unit price: 2.70 ÷ 750 = $0.0036 per gram.
$0.0036 is less than $0.004, so Brand B costs less per gram.
Brand B is the better buy.

Check yourself

  1. A 2 L bottle of juice costs $3.20. Find the unit price per litre.
  2. Store X: 6 granola bars for $4.50. Store Y: 10 bars for $7.00. Which has the lower price per bar, and what is it?
Answers
11. 3.20 ÷ 2 = $1.60 per litre    12. X: 4.50 ÷ 6 = $0.75/bar; Y: 7.00 ÷ 10 = $0.70/bar → Store Y is lower at $0.70/bar

Three-term ratios same rule, three parts

A ratio can compare three amounts, written a : b : c. Simplify it exactly as you simplify a two-term ratio, by dividing every part by a common factor. The factor has to divide all three parts, so 9 : 15 : 24 will not take a 2.
Worked example, simplify a three-term ratio
Write 9 : 15 : 24 in lowest terms.
All three parts divide by 3.
= 3 : 5 : 8
Worked example, sharing in a ratio
Three cousins share $90 in the ratio 2 : 3 : 4.
Count the parts first: 2 + 3 + 4 = 9 parts. One part is 90 ÷ 9 = $10.
= $20, $30, $40  (and they add back to $90)

Check yourself

  1. Write 15 : 25 : 35 in lowest terms.
  2. Write 24 : 36 : 60 in lowest terms.
  3. A paint mix uses white, blue and black as 180 mL : 120 mL : 60 mL. Write it in lowest terms.
  4. Share $140 in the ratio 2 : 2 : 3.
Answers
1. 3 : 5 : 7    2. 2 : 3 : 5    3. 3 : 2 : 1    4. 7 parts, one part $20, so $40, $40, $60

Scale and similar shapes a proportion in disguise

A scale tells you what one unit on the drawing stands for in real life. Written as a ratio, 1 : 40 means the real object is 40 times the model, so model to real you multiply, and real to model you divide. Similar shapes keep the same ratio of width to height, so an enlargement is just a proportion with one part missing.
Worked example, a map scale
A map uses 1 cm to 25 km. Two towns are 275 km apart. How far apart on the map?
Divide the real distance by what 1 cm stands for: 275 ÷ 25.
= 11 cm
Worked example, a scale written as a ratio
A model is built to a scale of 1 : 40 and is 15 cm long. How long is the real object?
Real is 40 times the model: 15 × 40 = 600 cm.
= 600 cm, which is 6 m
Worked example, similar shapes
A photo 8 cm wide and 12 cm tall is enlarged to 20 cm wide. How tall is it?
Width to height stays the same: 8⁄12 = 20⁄h, so 8h = 240.
h = 240 ÷ 8 = 30 cm

Check yourself

  1. A scale of 1 : 500 shows a path 4 cm long. What is the real length in metres?
  2. A 1 : 20 model boat is 35 cm long. What is the real length in metres?
  3. A drawing uses 1 cm to 1.5 m. A hall is 12 m long. How long is it on the drawing?
  4. Two similar rectangles: the first is 5 by 8, the second is 15 wide. How tall is it?
Answers
1. 4 × 500 = 2000 cm = 20 m    2. 35 × 20 = 700 cm = 7 m    3. 12 ÷ 1.5 = 8 cm    4. 15 ÷ 5 = 3, so 8 × 3 = 24

If you want to go further

Notice that a unit rate and a unit price are the same idea — "how much for one" — and that finding one is really solving a proportion in disguise. Try this: a percent like 15% is itself a unit rate, 15 per 100. So "15% of 60" is just scaling the ratio 15⁄100 down to a denominator of 60. Once you see every problem in this unit as "two ratios, find the missing piece," they all become the same problem.

Mathematics 8 · Unit 3 — Ratios, Rates, Proportions & Percent · Ms. Ras Mulinta · Handsworth Secondary 2026–27 · BC Math 8 content: numerical proportional reasoning (rates, ratio, proportions, percent), percents less than 1 and greater than 100 (decimal and fractional percents), and financial literacy (best buys). Pairs with the Unit 3 practice set and answer key.