Ratios and percentages – Arithmetic (Senior 4 Basic Mathematics) | Ganzaa

Arithmetic

Lessons

Senior 4 · Basic Mathematics

Arithmetic

5 lessons

Lesson 2 of 5

Ratios and percentages

25 min

Comparing by division

The 2022 Population and Housing Census counted 13 246 394 residents of Rwanda: 6 429 326 males and 6 817 068 females. Saying "there were 387 742 more females" compares by subtraction. Statisticians prefer to compare by division: there were about 94 males for every 100 females, and the City of Kigali held about 13% of the whole population. "94 for every 100" is a ratio; "13%" is a percentage. Both describe a part in relation to something else, which is how newspapers, budgets, recipes and building plans speak.

Definition

Ratio: a comparison of two or more quantities of the same kind by division, written $a : b$ and read " $a$ to $b$ ". It can also be written as the fraction $\frac{a}{b}$ . A ratio does not change when all its terms are multiplied or divided by the same non-zero number, so it can be put in simplest form.

Proportion: an equality of two ratios, $a : b = c : d$ , which holds exactly when $a d = b c$ .

Example: $450 \text{ g} : 1.2 \text{ kg} = 450 : 1200 = 3 : 8$ (same units first, then divide by 150).

Worked example 1: a ratio from the census

Using the 2022 census, express the number of males to females as a ratio in the form $n : 100$ (the sex ratio), and find the percentage of the population that was female.

Solution

\frac{6\,429\,326}{6\,817\,068} \times 100 \approx 94.3, \quad \text{so males : females} \approx 94.3 : 100 .

\frac{6\,817\,068}{13\,246\,394} \times 100\% \approx 51.5\% \text{ female}.

Source: NISR, 2022 census main indicators, Tables 1 and 4.

Sharing in a given ratio

To share an amount in the ratio $a : b : c$ , add the terms to find the number of equal parts, find the value of one part, then multiply. Three partners, Mukamana, Habimana and Uwera, invested 400 000, 600 000 and 1 000 000 Frw in a shop, a ratio of $2 : 3 : 5$ . They agree to share a profit of 750 000 Frw in the same ratio:

There are $2 + 3 + 5 = 10$ parts, each worth $\frac{750\,000}{10} = 75\,000$ Frw. Check: $150\,000 + 225\,000 + 375\,000 = 750\,000$ . ✓

Worked example 2: a mix and a map

(a) A common concrete mix uses cement, sand and gravel in the ratio $1 : 2 : 4$ by volume. How much of each is needed for $2.1\ \mathrm{m^3}$ of dry mix? (b) On a map with scale $1 : 50\,000$ , a road measures 7.4 cm. How long is the real road?

Solution

$1 + 2 + 4 = 7$ parts, one part $= \frac{2.1}{7} = 0.3\ \mathrm{m^3}$ : cement $0.3\ \mathrm{m^3}$ , sand $0.6\ \mathrm{m^3}$ , gravel $1.2\ \mathrm{m^3}$ .
Every 1 cm on the map is 50 000 cm on the ground: $7.4 \times 50\,000 = 370\,000$ cm $= 3.7$ km.

Percentages

Definition

Percentage: a ratio with 100 as the second term; $p\% = \frac{p}{100}$ . So $18\% = \frac{18}{100} = 0.18$ .

$p %$ of a quantity $Q$ is $\frac{p}{100} \times Q$ .
$A$ as a percentage of $B$ is $\frac{A}{B} \times 100\%$ .
Percentage change $= \frac{\text{new} - \text{old}}{\text{old}} \times 100\%$ (positive for an increase, negative for a decrease).
Multiplier: increasing by $p %$ multiplies by $1 + \frac{p}{100}$ ; decreasing by $p %$ multiplies by $1 - \frac{p}{100}$ .

Example: 15% of 32 000 Frw is $0.15 \times 32\,000 = 4\,800$ Frw.

Worked example 3: population growth, 2012 to 2022

The 2012 census counted 10 515 973 residents and the 2022 census 13 246 394. Find the percentage increase.

Solution

\frac{13\,246\,394 - 10\,515\,973}{10\,515\,973} \times 100\% = \frac{2\,730\,421}{10\,515\,973} \times 100\% \approx 26.0\% .

The population grew by about 26% in ten years. (This is not 2.6% a year: lesson 5 shows why yearly growth compounds.)

Worked example 4: reverse percentages and successive changes

(a) A phone costs 29 500 Frw including an 18% tax. What was the price before tax? (b) A trader raises a price by 20%, then later lowers the new price by 20%. What is the overall change?

Solution

The price with tax is 118% of the price before tax: $1.18 \times P = 29\,500$ , so $P = \frac{29\,500}{1.18} = 25\,000$ Frw.
Multipliers: $1.20 \times 0.80 = 0.96$ . The final price is 96% of the first, a 4% decrease overall, not 0%.

Warning

Two classic errors. (1) In a reverse percentage, taking 18% off the final price: $29\,500 - 0.18 \times 29\,500 = 24\,190$ Frw is wrong, because the 18% was calculated on the original price, not on 29 500. Divide by the multiplier instead. (2) Mixing up percent and percentage points: if a pass rate rises from 40% to 50%, it rose by 10 percentage points, which is a 25% increase ( $\frac{10}{40}$ ).

Tip

Think in multipliers. "Up 7%" is " $\times 1.07$ ", "down 15%" is " $\times 0.85$ ", "including 18% tax" is " $\times 1.18$ ", and undoing a change is dividing by its multiplier. Successive changes simply multiply.

Cognito: Percentage Increase and Decrease (Multiplier Method). Watch how one multiplication replaces 'find the percentage, then add or subtract'.

✦

Practice

Try each one before opening the answers. They get harder as you go.

Write in simplest form: (a) $45 : 60$ (b) $2.5\ \text{kg} : 750\ \text{g}$ (c) $\frac{1}{2} : \frac{1}{3}$ .
Share 96 000 Frw between two sisters in the ratio $5 : 3$ .
A recipe for 6 people uses 900 g of beans. How much is needed for 10 people?
Of the 13 246 394 residents counted in 2022, 3 701 245 lived in urban areas. What percentage is that, to one decimal place?
A school's enrolment rose from 640 to 728 students. Find the percentage increase.
After a 12% discount a radio costs 35 200 Frw. Find the original price.
A price rises by 10% and then by another 10%. A student says the total rise is 20%. Is she right? Explain.

Show answersHide answers

(a) $3 : 4$ (b) $2500 : 750 = 10 : 3$ (c) multiply both by 6: $3 : 2$ .
8 parts of 12 000 Frw: 60 000 Frw and 36 000 Frw.
$\frac{900}{6} \times 10 = 1500$ g, that is 1.5 kg.
$\frac{3\,701\,245}{13\,246\,394} \times 100\% \approx 27.9\%$ .
$\frac{88}{640} \times 100\% = 13.75\%$ .
$P \times 0.88 = 35\,200$ , so $P = 40\,000$ Frw.
No: $1.1 \times 1.1 = 1.21$ , a 21% rise, because the second 10% is taken of the already higher price.

Quick check

Key points

A ratio compares quantities of the same kind by division; simplify it after putting both in the same units.
To share in $a : b : c$ , find one part $= \frac{\text{total}}{a + b + c}$ .
Percentage change $= \frac{\text{new} - \text{old}}{\text{old}} \times 100\%$ ; work with multipliers, and divide by the multiplier to reverse a change.
Successive percentage changes multiply; they do not add.

Sources and credits

Written by Ganzaa: the explanations, the diagram, the worked examples, the practice items and the questions. The shop, recipe and price figures are illustrative.
Data: population by sex (2012 and 2022) and by residence, Tables 1, 4 and 7 of Fifth Population and Housing Census, Rwanda 2022: Main Indicators, National Institute of Statistics of Rwanda (NISR), CC BY 4.0; the ratios and percentages were calculated by Ganzaa.
Video: GCSE Maths - Percentage Increase and Decrease (Multiplier Method), Cognito, embedded from YouTube.

1Comparing by division

2Sharing in a given ratio

3Percentages

Comparing by division

Sharing in a given ratio

Percentages