Mathematics notes, chapter 4

Ratios, Rates and Proportions

Textbook Mathematics Form 1 (DLP), pages 74–102

What this chapter covers

Represent the relationship between three quantities in the form a : b : c (4.1)
Identify and find equivalent ratios in numbers, diagrams and daily situations (4.1)
Express ratios of two and three quantities in their simplest form (4.1)
Explain how a rate is related to a ratio, and convert the units of a rate (4.2)
Explain how a proportion is related to ratios, and find an unknown value in a proportion (4.3)
Find the ratio of three quantities from two ratios of two quantities (4.4)
Find a value from a ratio of two or three quantities and the value of one quantity, and find a value related to a rate (4.4)
Solve problems involving ratios, rates and proportions, including making estimations (4.4)
Link percentages with ratios, and find the percentage of a quantity using proportions (4.5)
Solve problems that combine ratios, rates and proportions with percentages, fractions and decimals (4.5)

Marine biologists want to know how many seahorses (Hippocampus kuda) live in a river estuary, but they cannot catch every one. Instead, they tag a first sample, release it, and later count how many tagged seahorses turn up in a second sample. By comparing these numbers they can estimate the whole population. Comparing quantities like this is what ratios, rates and proportions are about.

Photo of a yellow-green seahorse, Hippocampus kuda — The *Hippocampus kuda* seahorse is found in river estuaries and is facing extinction.

Photo: Mathematics Form 1 (DLP) textbook, p. 74.

English	Bahasa Melayu
ratio	nisbah
equivalent ratio	nisbah setara
rate	kadar
proportion	kadaran
percentage	peratusan

4.1Ratios

A ratio compares two or more quantities of the same kind, measured in the same unit. The ratio of a to b is written a : b. A ratio has no units.

Example: 5 000 g to 9 kg = 5 kg : 9 kg = 5 : 9. Change both quantities to the same unit first, then drop the unit.

Ratios of three quantities, a : b : c

Example 1: 0.02 m to 3 cm to 4.6 cm (textbook p. 76)

Same unit (cm): 0.02 m : 3 cm : 4.6 cm = 2 cm : 3 cm : 4.6 cm
Drop the units: = 2 : 3 : 4.6
Remove the decimal (× 10): = 20 : 30 : 46
Divide by 2: = 10 : 15 : 23

Do not write 0.02 : 3 : 4.6. The first quantity is in metres and the others are in centimetres, so the numbers cannot be compared until the units are the same.

Write each in the form a : b : c. a) 2 weeks to 16 days to 1 week b) 0.1 kg to 50 g to 0.25 kg

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a) In days: 14 : 16 : 7
b) In grams: 100 : 50 : 250 = 2 : 1 : 5 (divide by 50)

Equivalent ratios

Three circles: 1 of 2 parts shaded, 2 of 4 parts shaded, 6 of 12 parts shaded. The shaded areas are the same size. — The shaded areas are the same size, so 1 : 2 = 2 : 4 = 6 : 12

Equivalent ratios are ratios that have the same value, like 1 : 2, 2 : 4 and 6 : 12. To find an equivalent ratio, multiply or divide every part by the same number.

Ratio (shaded : whole)	As a fraction	How we get it
1 : 2	12	Start
2 : 4	24	1 × 2 : 2 × 2
6 : 12	612	2 × 3 : 4 × 3

Example 2: which ratios are equivalent to 27 : 45? (textbook p. 77)

Choices: 9 : 15 5 : 3 54 : 90 12 : 56

27 : 45 = 27 ÷ 3 : 45 ÷ 3 = 9 : 15 (equivalent)
27 : 45 = 27 × 2 : 45 × 2 = 54 : 90 (equivalent)
27 : 45 = 27 × 154 : 45 × 154 = 12 : 56 (equivalent)
27 : 45 = 27 ÷ 9 : 45 ÷ 9 = 3 : 5, which is not 5 : 3 (not equivalent)

Order matters. 3 : 5 ≠ 5 : 3. "Boys to girls" and "girls to boys" are different ratios.

Example 3: a cleaning solution (textbook p. 78)

Puan Habibah mixes 4 cups of vinegar with 8 cups of water. Two equivalent ratios:

4 : 8 = 4 × 2 : 8 × 2 = 8 : 16 (a bigger batch)
4 : 8 = 4 ÷ 2 : 8 ÷ 2 = 2 : 4 (a smaller batch)

Any batch in one of these ratios cleans just as well, because the mixture has the same strength.

Ratios in their simplest form

A ratio a : b is in its simplest form when a and b are whole numbers with no common factor other than 1. Divide by the HCF, or (for fractions) multiply by the LCM of the denominators. For decimals, multiply by 10, 100, … first.

Example 4: simplest form (textbook pp. 78–79)

Ratio	Working	Simplest form
800 g : 1.8 kg	= 800 g : 1 800 g, then ÷ 200 (the HCF)	4 : 9
32 : 24 : 20	÷ 4 (HCF of 32, 24 and 20)	8 : 6 : 5
35 : 710	× 10 (LCM of 5 and 10): 35 × 10 : 710 × 10 = 6 : 7	6 : 7
0.04 : 0.12 : 0.56	× 100 → 4 : 12 : 56, then ÷ 4	1 : 3 : 14

Check your answer: the parts must be whole numbers, and no number except 1 can divide all of them. For example, 4 : 10 is not in simplest form, because 2 still divides both parts.

Express in simplest form. a) 240 g : 1.6 kg b) 30 : 42 : 48 c) 25 : 89 d) 0.09 : 0.12 : 0.24

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a) 240 g : 1 600 g, ÷ 80 = 3 : 20
b) ÷ 6 = 5 : 7 : 8
c) × 45 (LCM of 5 and 9) = 18 : 40, ÷ 2 = 9 : 20
d) × 100 = 9 : 12 : 24, ÷ 3 = 3 : 4 : 8

4.2Rates

A rate is a special ratio that compares two quantities with different units. Example: a car travels 285 km in 3 hours. The rate is 285 km3 hours (distance compared with time).

Situation	Rate	Quantities (units)
Fatin buys 2 kg of mangoes for RM10.	RM102 kg	money (RM) and mass (kg)
A car uses 1 litre of petrol for 12 km.	12 km1 litre	distance (km) and volume (litre)
Karim's pulse is 75 beats per minute.	75 beats1 minute	number of beats and time (minute)

Examples from textbook pp. 81–82.

"Per" means "for each" or "in one". Speed in km/h is read "kilometres per hour". 1.5 kg3 kg is a ratio, not a rate, because both quantities have the same unit.

Converting the units of a rate

To compare two rates, write them in the same units first.

Cherry tomatoes at Stall A, RM15 per kg, and Stall B, RM8 per 500 g — Which stall is cheaper? (Textbook p. 82)

Example 5: cherry tomatoes (textbook p. 82)

Stall B: RM8500 g = RM8 × 2500 g × 2 = RM161 000 g = RM16 per kg
Stall A: RM15 per kg
RM15 < RM16, so Stall A is cheaper.

Example 6: speed and density (textbook p. 82)

a) Convert 5 m/s to km/h.

5 m/s means 5 m in 1 s. In 1 hour (3 600 s): 5 × 3 600 = 18 000 m
18 000 m = 18 000 ÷ 1 000 = 18 km, so 5 m/s = 18 km/h
In one line: 51 000 km ÷ 13 600 h = 51 000 × 3 6001 = 18 km/h

b) A metal has a density of 2 700 kg per m³. Write it in g per cm³.

1 kg = 1 000 g and 1 m³ = 100 cm × 100 cm × 100 cm = 1 000 000 cm³
2 700 kg1 m³ = 2 700 × 1 000 g1 000 000 cm³ = 2.7 g per cm³

1 m³ is not 100 cm³. A cube 1 m long is 100 cm long, 100 cm wide and 100 cm high, so 1 m³ = 1 000 000 cm³. In the same way, 1 m² = 100 × 100 = 10 000 cm².

1 hour = 60 × 60 = 3 600 seconds. To change m/s into km/h, multiply by 3 600 and divide by 1 000 (the same as × 3.6).

a) Object A moves at 25 m per second and object B at 8 km per hour. Which is faster? b) Fertiliser is used at 350 kg per hectare. Write the rate in g per m². (1 hectare = 10 000 m²)

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a) 25 m/s = 251 000 × 3 600 = 90 km/h. 90 km/h > 8 km/h, so A is faster.
b) 350 × 1 000 g10 000 m² = 350 00010 000 = 35 g per m²

4.3Proportions

A proportion states that two ratios or two rates are equal. It can be written as two equal fractions, for example 45 words5 minutes = 540 words60 minutes.

Example 7: writing a proportion (textbook p. 85)

If 10 beans have a mass of 17 g, then 30 beans have a mass of 51 g:
10 beans17 g = 30 beans51 g

On a map, 1 cm represents 50 km, and the 1 180 km Le Tour de Langkawi route is 23.6 cm:
1 cm50 km = 23.6 cm1 180 km

Keep the same order on both sides. If the top of the first fraction is "beans", the top of the second fraction must also be "beans".

Finding an unknown value: three methods

Example 8: electricity costs 43.6 sen for 2 kWh. How much does 30 kWh cost? (textbook pp. 85–86)

Unitary method (find 1 first)	Proportion method	Cross multiplication
2 kWh: 43.6 sen 1 kWh: 43.6 ÷ 2 = 21.8 sen 30 kWh: 30 × 21.8 = 654 sen	43.6 sen2 kWh = x sen30 kWh 2 × 15 = 30, so the top is also × 15 x = 43.6 × 15 = 654	43.62 = x30 2 × x = 43.6 × 30 x = 1 3082 = 654

So 30 kWh costs 654 sen = RM6.54.

Cross multiplication: if ab = cd, then a × d = b × c.

a) There are 200 players in 8 rugby teams of the same size. How many players are in 2 teams? b) Amir does 60 push-ups in 3 minutes. At the same rate, how many can he do in 5 minutes?

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a) 2008 = x2 → 8x = 400 → x = 50 players
b) 1 minute: 60 ÷ 3 = 20 push-ups; 5 minutes: 5 × 20 = 100 push-ups

4.4Ratios, rates and proportions

The ratio of three quantities from two ratios

If p : q and q : r are given, make the common quantity q the same number in both ratios. Then join them.

Example 9: p : q = 7 : 3 and q : r = 3 : 5 (textbook p. 87)

q is already 3 in both ratios, so p : q : r = 7 : 3 : 5.

Example 10: storybooks, reference books and magazines (textbook p. 87)

Storybooks : reference books = 2 : 5 and reference books : magazines = 3 : 2. The LCM of 5 and 3 is 15.

	Storybooks	Reference books	Magazines
First ratio (× 3)	2 × 3 = 6	5 × 3 = 15
Second ratio (× 5)		3 × 5 = 15	2 × 5 = 10
Joined	6	15	10

Storybooks : reference books : magazines = 6 : 15 : 10

Do not just write the numbers side by side (2 : 5 : 2). That only works when the middle quantity is already the same number in both ratios.

a) p : q = 2 : 9 and q : r = 9 : 7. Find p : q : r. b) Donations to P and Q are in the ratio 2 : 3, and to Q and R in the ratio 4 : 1. Find P : Q : R.

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a) q is 9 in both: 2 : 9 : 7
b) LCM of 3 and 4 is 12: 2 : 3 = 8 : 12 and 4 : 1 = 12 : 3, so P : Q : R = 8 : 12 : 3

Finding a value from a ratio

Find the value of one part from the quantity you know, then multiply.

Example 11: flour to water is 5 : 3. There is 480 g of flour. (textbook p. 88)

Bar model: flour is 5 boxes of 96 g making 480 g; water is 3 boxes of 96 g

Unitary method	Cross multiplication (m = mass of water)
5 parts = 480 g 1 part = 480 ÷ 5 = 96 g Water = 3 × 96 = 288 g	53 = 480m 5 × m = 3 × 480 m = 1 4405 = 288 g

Example 12: donations in the ratio 2 : 6 : 5 (textbook p. 88)

The canteen operator, Jaya Bookstore and the PIBG donate in the ratio 2 : 6 : 5. The PIBG donates RM900.

Bar model: canteen operator 2 boxes, Jaya Bookstore 6 boxes, PIBG 5 boxes making RM900; each box RM180

PIBG: 5 parts = RM900, so 1 part = RM900 ÷ 5 = RM180
Canteen operator: 2 × RM180 = RM360
Jaya Bookstore: 6 × RM180 = RM1 080

Example 13: a 12 cm wire (textbook p. 89)

A 12 cm wire PS is cut into three parts. PQ : QR = 3 : 2 and RS = 2 cm. Find PQ : QR : RS.

Wire P, Q, R, S of total length 12 cm; RS is 2 cm

PR = 12 − 2 = 10 cm, and PQ : QR has 3 + 2 = 5 parts
1 part = 10 ÷ 5 = 2 cm, so PQ = 3 × 2 = 6 cm and QR = 2 × 2 = 4 cm
PQ : QR : RS = 6 : 4 : 2 = 3 : 2 : 1

Check which quantity the given value belongs to. If 24 boys are 3 parts of the ratio 3 : 5, then 1 part is 24 ÷ 3, not 24 ÷ 8. Divide by the total number of parts only when you are given the total.

a) The ratio of the price of a baju kebaya to a baju kurung is 7 : 4. The baju kebaya costs RM84. Find the price of the baju kurung. b) Zanariah, Rusita and Hanifah share the cost of a present in the ratio 5 : 3 : 2. Hanifah pays RM50. How much do Zanariah and Rusita pay altogether?

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a) 7 parts = RM84, 1 part = RM12, baju kurung = 4 × RM12 = RM48
b) 2 parts = RM50, 1 part = RM25; Zanariah and Rusita = (5 + 3) × RM25 = RM200

Values related to a rate

Example 14: Mr Tan's heart beats 420 times in 4 minutes. How many times in 12 minutes? (textbook p. 90)

Double number line: 0, 4, 8, 12 minutes match 0, 420, 840, 1 260 heartbeats

Rate = 420 times4 minutes. Let x be the number of beats in 12 minutes.
4204 = x12 → 4 × x = 420 × 12
x = 5 0404 = 1 260 times

Unitary method: 420 ÷ 4 = 105 beats per minute, and 105 × 12 = 1 260.

a) Six cartons of lime juice cost RM12.25. Find the price of 24 cartons. b) USD1 = RM3.90. Ahmad changes RM200 and gets RM5 back as change. How many US dollars does he receive?

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a) 24 = 6 × 4, so the price is RM12.25 × 4 = RM49.00
b) He changes RM200 − RM5 = RM195. RM195 ÷ RM3.90 = USD50

Solving problems and making estimations

Example 15: estimating a population of snails (textbook p. 91)

A farmer catches 24 snails, marks their shells and releases them. Two weeks later he catches 30 snails, and 5 of them are marked.

marked snails caughtsnails caught = snails marked at firstwhole population
530 = 24x → simplify: 16 = 24x
x = 6 × 24 = 144 snails (an estimate)

This "capture, mark, release and recapture" method gives an estimate, not an exact count, because the second sample is chosen at random. It is the same method used for the seahorses.

A football team played 28 games and won 4 out of every 7. There were no draws. a) How many games did it lose? b) Find the win-loss ratio. c) If this continues, estimate the number of losses when the team has won 20 games.

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a) Wins = 47 × 28 = 16, losses = 28 − 16 = 12
b) 16 : 12 = 4 : 3
c) 43 = 20x → 4x = 60 → x = 15 losses

4.5Relationship between ratios, rates and proportions, with percentages, fractions and decimals

A percentage is a ratio that compares a part with 100. "Per cent" means "per 100": 20% = 20100 = 20 : 100. A ratio of a part to the whole can be written as a fraction, a decimal or a percentage.

Example 16: girls to boys is 3 : 2. What percentage are girls? (textbook p. 94)

A 10 by 10 grid with 60 squares shaded — 3 out of every 5 students are girls: 60 out of every 100

Girls : total = 3 : (3 + 2) = 3 : 5
35 = 3 × 205 × 20 = 60100
Girls = 60% (or 35 × 100% = 60%)

For a percentage, compare the part with the whole. Girls : boys = 3 : 2 does not mean 3 out of 2 or 30%. First add the parts to get the total (5).

Example 17: Zakaria saves 30% of his pocket money (textbook p. 94)

Savings : pocket money = 30 : 100 = 3 : 10

Red : green apples	Fraction of red apples	Decimal	Percentage
3 : 5 (total 8 parts)	38	3 ÷ 8 = 0.375	0.375 × 100% = 37.5%

a) The ratio of students who wear spectacles to those who do not is 1 : 3. What percentage do not wear spectacles? b) 35% of a hard disk is full. Find the ratio of the full part to the empty part.

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a) 31 + 3 = 34 = 75%
b) Empty = 100% − 35% = 65%; 35 : 65 = 7 : 13

Finding a percentage using proportions

Write a proportion with the percentage over 100: partwhole = x100

Example 18: percentages by proportion (textbook pp. 95–96)

a) 2 of 8 ribbons are blue: 28 = x100 → 8x = 200 → x = 25, so 25% are blue.

b) Puan Jorana spends RM15 of her RM300: 15300 = y100 → divide 300 by 3 to get 100, so y = 15 ÷ 3 = 5, she spends 5%.

Example 19: is it really 45% off? (textbook p. 96)

A shirt is labelled "45% price reduction". The original price is RM85.00 and the scanner shows RM57.80.

57.8085.00 = p100 → 85p = 5 780 → p = 68. The selling price is 68% of the original price.
Discount = 100% − 68% = 32%
32% is less than 45%, so the discount does not match the label.

a) 18 of the 45 students in a class are girls. What percentage are girls? b) Shoes cost RM45 and are sold at a 25% discount. How much does a buyer save?

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a) 1845 = x100 → 45x = 1 800 → x = 40, so 40%
b) x45 = 25100 → 100x = 1 125 → x = 11.25, so the buyer saves RM11.25

Solving problems

Example 20: renting apartments (textbook pp. 97–98)

Apartment	Rent per month	Deposit
Two-room (Ben)	RM450	1 month's rent
Three-room (Farid)	RM550	2 months' rent

First payment: Ben = 450 + 450 = RM900; Farid = 550 + 2 × 550 = RM1 650
Ben : Farid = 900 : 1 650 = 6 : 11 (÷ 150)
Ben earns RM3 750: 4503 750 × 100 = 12% of his salary goes on rent.
Farid earns RM5 000: 5505 000 × 100 = 11% of his salary goes on rent.

An enlargement to 150% means every length is multiplied by 150100 = 32. The ratio of a new length to the old length is always 3 : 2.

Right-angled triangle PQR with PQ = 3 cm and QR = 4 cm, and its enlargement P'Q'R' with P'Q' = 4.5 cm and Q'R' = 6 cm — P'Q' : PQ = 4.5 : 3 = 3 : 2 and Q'R' : QR = 6 : 4 = 3 : 2

A book that costs RM25 is sold at a 30% discount in a shop. Online, the same book costs RM20 with a 15% discount. Which is the better buy?

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Shop: 70% of RM25 = 70100 × 25 = RM17.50
Online: 85% of RM20 = 85100 × 20 = RM17.00
Buying online is better, because RM17.00 < RM17.50 (a bigger percentage discount does not always mean a lower price).

Summary

	Ratio	Rate	Proportion
Meaning	Compares 2 or 3 quantities in the same unit	Compares 2 quantities with different units	Two ratios or two rates are equal
Written as	a : b or a : b : c (no units)	A fraction with units, e.g. km/h, RM per kg	ab = cd
Example	800 g : 1.8 kg = 4 : 9	5 m/s = 18 km/h	43.6 sen2 kWh = 654 sen30 kWh
Percentage?	Yes: part : whole as a fraction, decimal or %	No	Used to find a percentage: partwhole = x100

Remember	Key fact
Units	Change to the same unit before writing a ratio; 1 m³ = 1 000 000 cm³, 1 h = 3 600 s
Equivalent ratios	Multiply or divide every part by the same number; order matters (3 : 5 ≠ 5 : 3)
Simplest form	Whole numbers with HCF 1: clear fractions (× LCM) and decimals (× 10, 100) first
Three quantities	Make the common middle quantity equal using the LCM, then join
Value from a ratio	Find 1 part from the quantity you know, then multiply
Unknown in a proportion	Unitary, proportion or cross multiplication: ab = cd → ad = bc
Percentage	Part : whole, then × 100%; discount % = 100% − (selling price as % of original)