Ratio: meaning, simplifying and comparing: WASSCE Mathematics

A ratio compares two or more quantities. In a Sierra Leonean classroom, the ratio of boys to girls is $3 : 5$ . That means for every $3$ boys, there are $5$ girls. If there are $24$ boys, there are $40$ girls. If there are $9$ boys, there are $15$ girls. The ratio captures the relative size of the two groups without fixing the actual numbers.

Ratios are everywhere: recipes (rice to water $2:1$ ), maps (scale $1:50000$ ), business (profit-sharing $3:2:1$ ), sports (win-loss $4:1$ ). WAEC tests ratio problems in every paper.

This lesson introduces ratios, the standard notation, how to simplify them, how to convert between ratios and fractions/decimals, and how to compare ratios.

Definition

Ratio: A way to compare two or more quantities of the same kind.

Notation: $a : b$ (read " $a$ to $b$ "), where $a$ and $b$ are the two quantities being compared.

Three-way ratio: $a : b : c$ compares three quantities.

Equivalent ratios: Different ratios with the same proportional relationship. $1 : 2 = 2 : 4 = 3 : 6 = 5 : 10$ .

Simplest form: A ratio is in simplest form when the parts share no common factor larger than $1$ .

Learning objectives

By the end of this lesson you should be able to:

explain what a ratio compares
write ratios in different forms and simplify them
express ratios with units in the same units
compare ratios and write them in the form $1 : n$

Reading and writing ratios

Ratios can be written in three ways. Each is equivalent:

Colon notation: $3 : 5$ .
Word form: "three to five."
Fraction form: $\dfrac{3}{5}$ .

The order matters! $3 : 5$ is NOT the same as $5 : 3$ . The first quantity is mentioned first.

Examples

Boys to girls in a class: $3 : 5$ .
For every $3$ boys, $5$ girls.
The boy-fraction of the class is $\dfrac{3}{3 + 5} = \dfrac{3}{8}$ .
The girl-fraction is $\dfrac{5}{8}$ .
Profit-sharing among three partners: $4 : 3 : 2$ .
Total parts $= 4 + 3 + 2 = 9$ .
First partner gets $\dfrac{4}{9}$ , second $\dfrac{3}{9} = \dfrac{1}{3}$ , third $\dfrac{2}{9}$ .
Mixing rice and water in a recipe: $2 : 1$ .
$2$ cups of rice for every $1$ cup of water.
For $4$ cups of water, you'd need $8$ cups of rice.

Simplifying ratios

Divide both parts by their greatest common divisor (GCD).

Example 1

Simplify $12 : 18$ .

GCD of $12$ and $18$ is $6$ . Divide both by $6$ : $12/6 : 18/6 = 2 : 3$ .

Example 2

Simplify $25 : 15$ .

GCD is $5$ . $25/5 : 15/5 = 5 : 3$ .

Example 3

Simplify $36 : 48 : 60$ .

GCD of all three is $12$ . $36/12 : 48/12 : 60/12 = 3 : 4 : 5$ .

Example 4

Simplify $0.6 : 1.2$ .

Multiply both by $10$ to clear decimals: $6 : 12$ . GCD is $6$ . Simplify: $1 : 2$ .

Example 5

Simplify $\dfrac{1}{2} : \dfrac{1}{3}$ .

Multiply both by the LCD ( $6$ ): $\dfrac{6}{2} : \dfrac{6}{3} = 3 : 2$ .

Example 6

Simplify $0.4 : 0.7$ .

Multiply by $10$ : $4 : 7$ . Already in simplest form.

Example 7

Simplify $\dfrac{2}{5} : \dfrac{3}{10} : \dfrac{1}{2}$ .

LCD = $10$ . Multiply all by $10$ : $4 : 3 : 5$ . Already in simplest form.

Worked example

Simplify each ratio.

(a) $20 : 30$ . GCD $= 10$ . $2 : 3$ .

(b) $45 : 60$ . GCD $= 15$ . $3 : 4$ .

(c) $24 : 16 : 8$ . GCD $= 8$ . $3 : 2 : 1$ .

(d) $0.5 : 0.25$ . Multiply by $100$ : $50 : 25$ . Simplify: $2 : 1$ .

(e) $\dfrac{1}{4} : \dfrac{1}{6}$ . LCD $= 12$ . $3 : 2$ .

(f) $1\dfrac{1}{2} : 2\dfrac{1}{4}$ . Convert: $\dfrac{3}{2} : \dfrac{9}{4}$ . LCD $= 4$ . $6 : 9$ . Simplify: $2 : 3$ .

(g) $50\% : 30\%$ . As decimals: $0.5 : 0.3$ . Multiply by $10$ : $5 : 3$ .

(h) $200$ g $: 0.5$ kg. Convert to same units: $200 : 500$ g. Simplify: $2 : 5$ .

Ratios with units

Important: the quantities being compared must be in the same units.

Example 8

Express $30$ minutes to $1$ hour as a ratio.

Convert to same units. $30$ minutes $: 60$ minutes $= 30 : 60 = 1 : 2$ .

Example 9

Express $500$ g to $2$ kg as a ratio.

Convert to same units. $500$ g $: 2000$ g $= 500 : 2000 = 1 : 4$ .

Example 10

Express $25$ cm to $1$ m as a ratio.

$25$ cm $: 100$ cm $= 25 : 100 = 1 : 4$ .

Example 11

Express $150$ Nigerian naira to $30$ NLe (assuming $1$ NLe = $5$ NGN) as a ratio.

First convert to one currency. $150$ NGN = $30$ NLe (at the given rate).

So $30$ NLe $: 30$ NLe $= 1 : 1$ .

Or if the rate is different (say $1$ NLe = $4$ NGN), $150$ NGN = $37.5$ NLe. Ratio: $37.5 : 30 = 5 : 4$ .

Note: the answer depends on the conversion rate used.

Converting ratios to fractions and percentages

If a ratio is $a : b$ , then $a$ 's share of the total is $\dfrac{a}{a + b}$ .

Example 12

A class has boys to girls in ratio $3 : 5$ . What fraction of the class is boys? girls?

Total parts $= 8$ . Boy-fraction $= \dfrac{3}{8} = 37.5\%$ . Girl-fraction $= \dfrac{5}{8} = 62.5\%$ .

Example 13

A profit of $9000$ NLe is shared among three partners in the ratio $4 : 3 : 2$ . What percentage does each receive?

Total parts $= 9$ . First: $\dfrac{4}{9} \approx 44.4\%$ . Second: $\dfrac{3}{9} \approx 33.3\%$ . Third: $\dfrac{2}{9} \approx 22.2\%$ .

Equivalent ratios

Multiplying or dividing both parts by the same nonzero number gives an equivalent ratio.

Example 14

Find three ratios equivalent to $2 : 3$ .

Multiply by $2$ : $4 : 6$ .
Multiply by $5$ : $10 : 15$ .
Multiply by $10$ : $20 : 30$ .
Divide by $2$ : $1 : 1.5$ (or in fractions, $\dfrac{1}{1} : \dfrac{3}{2}$ , the same thing).

Example 15

Are $4 : 6$ and $6 : 9$ equivalent?

Simplify both: $4 : 6 = 2 : 3$ (divide by $2$ ). $6 : 9 = 2 : 3$ (divide by $3$ ). Both simplify to $2 : 3$ . Equivalent. ✓

Example 16

Are $5 : 8$ and $10 : 15$ equivalent?

Simplify: $5 : 8$ is already in simplest form. $10 : 15 = 2 : 3$ .

NOT equivalent.

The cross-multiplication test

Two ratios $a : b$ and $c : d$ are equivalent iff $ad = bc$ .

Example 17

Test if $2 : 5$ and $6 : 15$ are equivalent.

Cross-multiply: $2 \times 15 = 30$ , $5 \times 6 = 30$ . Equal. Equivalent ✓.

Example 18

Test if $3 : 4$ and $9 : 11$ are equivalent.

$3 \times 11 = 33$ , $4 \times 9 = 36$ . Not equal. Not equivalent.

Finding the missing part of a ratio

If $a : b = c : x$ , then $ax = bc$ , so $x = \dfrac{bc}{a}$ .

Example 19

If $3 : 5 = 12 : x$ , find $x$ .

$3x = 5 \times 12 = 60$ . $x = 20$ .

Example 20

If $2 : 7 = x : 21$ , find $x$ .

$2 \times 21 = 7x = 42$ . $x = 6$ .

Example 21

If $4 : 6 = x : 15$ , find $x$ .

$4 \times 15 = 6x = 60$ . $x = 10$ .

Three-term ratios

For a ratio $a : b : c$ , you can also find missing parts using cross-multiplication on pairs.

Example 22

If $2 : 3 : 5 = 8 : x : y$ , find $x$ and $y$ .

From $2 : 3 = 8 : x$ : $2x = 24$ , $x = 12$ . From $2 : 5 = 8 : y$ : $2y = 40$ , $y = 20$ .

So the equivalent ratio is $8 : 12 : 20$ .

Comparing ratios

To determine which of two ratios is larger, convert to common form (decimals, fractions with same denominator, or percentages).

Example 23

Which is larger: $3 : 5$ or $4 : 7$ ?

As fractions: $\dfrac{3}{5} = 0.6$ and $\dfrac{4}{7} \approx 0.571$ . So $3 : 5 > 4 : 7$ .

Alternatively: cross-multiply for direct comparison. $3 \times 7 = 21$ and $5 \times 4 = 20$ . Since $21 > 20$ , the ratio $3 : 5$ is larger.

Example 24

Arrange in increasing order: $2 : 3, 3 : 5, 4 : 7$ .

Decimal: $\dfrac{2}{3} \approx 0.667, \dfrac{3}{5} = 0.6, \dfrac{4}{7} \approx 0.571$ .

Increasing: $4 : 7, 3 : 5, 2 : 3$ .

Ratios and proportions in word problems

Many WAEC problems describe a ratio and ask you to find specific quantities.

Example 25

A school has $200$ students. The ratio of boys to girls is $3 : 5$ . How many boys?

Total parts $= 8$ . Each part = $200/8 = 25$ students. Boys = $3 \times 25 = 75$ . Girls = $5 \times 25 = 125$ . Check: $75 + 125 = 200$ . ✓

Example 26

A recipe uses rice and water in the ratio $2 : 3$ . If you use $6$ cups of rice, how much water?

Ratio $2 : 3 = 6 : x$ . $2x = 18$ . $x = 9$ cups of water.

Example 27

Two people share a bonus in ratio $5 : 3$ . The first receives $1000$ NLe more than the second. How much does each receive?

Let first $= 5k$ and second $= 3k$ for some constant $k$ . Difference: $5k - 3k = 2k = 1000$ , so $k = 500$ .

First $= 5 \times 500 = 2500$ NLe. Second $= 3 \times 500 = 1500$ NLe.

Verify: difference $= 1000$ . ✓

Example 28

In a Sierra Leonean village, the ratio of farms growing rice, cassava, and yams is $4 : 3 : 2$ . If there are $36$ farms in total, how many of each type?

Total parts $= 9$ . Each part $= 36/9 = 4$ farms. Rice farms $= 4 \times 4 = 16$ . Cassava farms $= 4 \times 3 = 12$ . Yam farms $= 4 \times 2 = 8$ . Check: $16 + 12 + 8 = 36$ . ✓

Worked example

Solve each ratio problem.

(a) A class has students in ratio $boys : girls = 5 : 7$ . If there are $40$ girls, how many boys? Total students? $5 : 7 = x : 40$ . $7x = 200$ , $x = 28.57$ . Hmm, not integer. Let me reread. Probably $5 : 7 = b : 40$ should give an integer. $b = 200/7$ , not integer. Let me change: assume $40$ is total instead. If the ratio is $5 : 7$ and total is $40$ , each part is $40/12$ , not integer. Let me use a cleaner example: ratio $5 : 7$ and $35$ girls. $5 : 7 = x : 35$ . $7x = 175$ . $x = 25$ boys. Total $= 60$ .

(b) Two friends shared $4500$ NLe in ratio $2 : 3$ . How much did each receive? Total parts $= 5$ . Each part $= 4500/5 = 900$ . First $= 2 \times 900 = 1800$ NLe. Second $= 3 \times 900 = 2700$ NLe. Verify: $1800 + 2700 = 4500$ . ✓

(c) A line is divided in ratio $3 : 4 : 5$ . If the total length is $24$ cm, find the length of each part. Total parts $= 12$ . Each part $= 24/12 = 2$ cm. First $= 3 \times 2 = 6$ cm. Second $= 4 \times 2 = 8$ cm. Third $= 5 \times 2 = 10$ cm. Verify: $6 + 8 + 10 = 24$ . ✓

(d) Aminata and Mohamed share NLe $7000$ in the ratio $3 : 4$ . How much does each receive? Total parts $= 7$ . Each part $= 7000/7 = 1000$ . Aminata = $3000$ NLe. Mohamed = $4000$ NLe.

(e) Three siblings inherit $36000$ NLe in ratio $5 : 4 : 3$ . How much does each get? Total parts $= 12$ . Each part $= 36000/12 = 3000$ . Siblings: $15000, 12000, 9000$ NLe.

In Sierra Leone

In a Sierra Leonean palm wine cooperative, three vendors split daily proceeds. They contribute palm wine in the ratio $4 : 5 : 6$ . If the total day's sales are $7500$ NLe, how much does each get?

Total parts $= 15$ . Each part $= 7500/15 = 500$ NLe.

Vendor 1: $4 \times 500 = 2000$ NLe. Vendor 2: $5 \times 500 = 2500$ NLe. Vendor 3: $6 \times 500 = 3000$ NLe.

Verify: $2000 + 2500 + 3000 = 7500$ . ✓

This kind of proportional sharing is everywhere in West African business: market vendors splitting profits, family inheritance, kontri organizations. The ratio captures the agreed proportions exactly.

Common mistake

Common errors with ratios:

1. Reversing the order.

$3 : 5$ ≠ $5 : 3$ . The order matters.

2. Mixing units.

Always convert to the same units before forming a ratio. $30$ cm to $1$ m is $30 : 100$ , not $30 : 1$ .

3. Forgetting to simplify.

$20 : 30$ should be $2 : 3$ . WAEC expects simplest form.

4. Confusing ratio with fraction.

The ratio $3 : 5$ means $3$ to $5$ (so the first item is $\dfrac{3}{8}$ of the total, not $\dfrac{3}{5}$ ).

5. Wrong total parts.

For $3 : 5$ , total parts $= 8$ (i.e., $3 + 5$ ), not $5 - 3 = 2$ .

6. Wrong direction in word problems.

"Boys to girls $3 : 5$ , $40$ girls" — the unknown is boys, but make sure you set up the cross-multiplication correctly. $3 : 5 = x : 40$ gives $x = 24$ boys.

WASSCE examination focus

WAEC tests this

Foundational ratio questions in WAEC:

Simplify a ratio. Simplify $24 : 36$ . (Answer: $2 : 3$ .)
Ratio with units. Express $250$ g to $1$ kg as a ratio. (Answer: $1 : 4$ .)
Find the unknown. If $3 : 5 = 9 : x$ , find $x$ . (Answer: $15$ .)
Convert to fraction/percentage. In ratio $2 : 3$ , what fraction is the first part? (Answer: $\dfrac{2}{5}$ .)
Word problem with sharing. Share $1200$ NLe in ratio $3 : 5$ . (Answer: $450$ and $750$ .)

Master the basics — simplification, equivalence, conversion — before tackling word problems.

Practice problems

1. Simplify each ratio. (a) $36 : 24$ (b) $45 : 90$ (c) $48 : 60 : 84$ (d) $\dfrac{1}{2} : \dfrac{1}{4}$ (e) $0.8 : 1.6$ (f) $1\dfrac{1}{2} : 3$

2. Express each as a ratio in simplest form. (a) $30$ minutes to $2$ hours. (b) $400$ g to $1$ kg. (c) $50$ cm to $1.5$ m. (d) $20\%$ to $40\%$ .

3. Find the missing value. (a) $4 : 5 = x : 25$ . (b) $7 : 3 = 28 : x$ . (c) $1 : 2 = x : 10 = 7 : y$ .

4. Are these ratios equivalent? Show your working. (a) $6 : 8$ and $15 : 20$ . (b) $3 : 4$ and $9 : 13$ .

5. A class has $24$ boys and $36$ girls. What is the ratio of boys to girls in simplest form?

6. A school spends $30000$ NLe in the ratio $5 : 7$ between equipment and salaries. How much for each?

7. Aminata, Mohamed, and Foday split a $48000$ NLe inheritance in the ratio $3 : 4 : 5$ . How much does each receive?

8. A recipe uses sugar, flour, and butter in ratio $1 : 3 : 2$ . If you use $300$ g of flour, how much sugar and butter?

9. A bonus of $12000$ NLe is shared between two people. The first receives $2000$ NLe more than the second. In what ratio is the bonus shared?

10. Two roads have lengths in ratio $2 : 5$ . If the shorter is $40$ km, find the longer.

11. Convert each ratio to a fraction and a percentage. (a) $3 : 7$ . (b) $1 : 4$ . (c) $4 : 6$ .

12. Which is larger: $2 : 5$ or $3 : 8$ ?

13. Arrange in decreasing order: $1 : 4, 2 : 9, 1 : 5$ .

14. A mixture of palm wine and water is in ratio $4 : 1$ . If the mixture is $25$ litres, how much palm wine and how much water?

Worked solutions

1. (a) GCD $= 12$ . $3 : 2$ . (b) GCD $= 45$ . $1 : 2$ . (c) GCD $= 12$ . $4 : 5 : 7$ . (d) LCD $= 4$ . $2 : 1$ . (e) ×10: $8 : 16 = 1 : 2$ . (f) $\dfrac{3}{2} : 3 = 3 : 6 = 1 : 2$ .

2. (a) $30 : 120 = 1 : 4$ . (b) $400 : 1000 = 2 : 5$ . (c) $50 : 150 = 1 : 3$ . (d) Cancel %: $20 : 40 = 1 : 2$ .

3. (a) $4x = 100$ , $x = 20$ . (b) $3 \times 28 = 84$ , $7x = 84$ , $x = 12$ . (c) From $1 : 2 = x : 10$ : $x = 5$ . From $1 : 2 = 7 : y$ : $y = 14$ .

4. (a) Simplify: $6 : 8 = 3 : 4$ . $15 : 20 = 3 : 4$ . Equivalent ✓. (b) $3 : 4$ already simplest. $9 : 13$ — cross-multiply: $3 \times 13 = 39$ , $4 \times 9 = 36$ . Not equivalent.

5. $24 : 36$ . GCD $= 12$ . $2 : 3$ .

6. Total $= 12$ parts. Each part $= 30000/12 = 2500$ . Equipment: $5 \times 2500 = 12500$ NLe. Salaries: $7 \times 2500 = 17500$ NLe.

7. Total parts $= 12$ . Each part $= 48000/12 = 4000$ . Aminata: $12000$ , Mohamed: $16000$ , Foday: $20000$ NLe.

8. Flour is $3$ parts $= 300$ g. So $1$ part $= 100$ g. Sugar = $100$ g. Butter = $2 \times 100 = 200$ g.

9. Let parts be $5k$ and $3k$ (first vs second). Total $5k + 3k = 8k = 12000$ , $k = 1500$ . First $= 7500$ , second $= 4500$ . Difference $= 3000$ . Doesn't match. Let me redo. Let first $= a$ , second $= b$ , $a + b = 12000$ , $a - b = 2000$ . Solve: $a = 7000$ , $b = 5000$ . Ratio: $7000 : 5000 = 7 : 5$ .

10. $2 : 5 = 40 : x$ . $2x = 200$ , $x = 100$ km.

11. (a) $\dfrac{3}{10}$ of $30\%$ . (b) $\dfrac{1}{5} = 20\%$ . (c) $\dfrac{4}{10} = 40\%$ .

12. Cross-multiply: $2 \times 8 = 16$ , $5 \times 3 = 15$ . $16 > 15$ , so $2 : 5 > 3 : 8$ .

13. Decimal: $\dfrac{1}{4} = 0.25$ , $\dfrac{2}{9} \approx 0.222$ , $\dfrac{1}{5} = 0.2$ . Decreasing: $1 : 4, 2 : 9, 1 : 5$ .

14. Total parts $= 5$ . Each part $= 25/5 = 5$ litres. Palm wine: $4 \times 5 = 20$ litres. Water: $5$ litres.

What's next

In the next lesson we cover sharing in a given ratio (the most common WAEC ratio application), direct and inverse proportion, and rates such as speed and exchange rates. We'll deepen the techniques from this lesson with more complex sharing problems, including problems where one share is given and you must find the rest, or where the total is to be determined.