3.4 Ratio, Proportion & Percentage Problem Solving

3.4 Ratio, Proportion & Percentage Problem Solving

Ratio and percentage are core Grade 6 Number objectives in the National Standards Curriculum (Term 2, Unit 1). In the Ministry's 2026 Grade 6 Mathematics test-preparation publication, many of the Number items ask students to identify equivalent ratios, share an amount using a ratio, find a percentage of a set, or work out a whole number when only a percentage of it is known. This section builds each of those skills step by step.

What a Ratio Is

A ratio compares two (or more) quantities of the same kind. It is written with a colon, $a : b$, and read "$a$ to $b$".

• A class has 12 boys and 18 girls. The ratio of boys to girls is $12 : 18$.
• Order matters: the ratio of girls to boys is $18 : 12$, which is a different statement.
• A part-to-part ratio compares one part with another part (boys : girls). A part-to-whole ratio compares one part with the total (boys : whole class $= 12 : 30$).
• The quantities must be in the same units before you compare them. To compare $50\text{ cm}$ with $2\text{ m}$, change $2\text{ m}$ to $200\text{ cm}$: the ratio is $50 : 200$.

Simplest Form and Equivalent Ratios

Simplify a ratio by dividing both terms by their highest common factor (HCF), just as you simplify a fraction.

$12 : 18 = (12 \div 6) : (18 \div 6) = 2 : 3$

$50 : 200 = 1 : 4$

Equivalent ratios describe the same relationship. Multiply or divide both terms by the same number:

To check whether two ratios are equivalent, simplify both. $8 : 12$ and $10 : 15$ both simplify to $2 : 3$, so they are equivalent.

Solving Problems with Equivalent Ratios

Missing term: $2 : 5 = \square : 35$. The second term was multiplied by 7 ($5 \times 7 = 35$), so the first term must also be multiplied by 7: $2 \times 7 = 14$.

Recipe scaling: a fruit punch uses 3 cups of pineapple juice for every 2 cups of soursop juice. For 9 cups of pineapple juice (three times as much), use $2 \times 3 = 6$ cups of soursop juice.

Using one known part: the ratio of boys to girls in a club is $3 : 4$ and there are 24 girls. Four parts are 24 girls, so one part is $24 \div 4 = 6$. Boys $= 3 \times 6 = 18$, and the club has $18 + 24 = 42$ members.

Sharing an Amount in a Given Ratio

Follow three steps:

1. Add the parts to find the total number of parts.
2. Divide the amount by the total parts to find the value of one part.
3. Multiply each person's number of parts by the value of one part.

Worked example: Tameka and Andre share J$4,500 in the ratio $4 : 5$.

• Total parts: $4 + 5 = 9$.
• One part: J$4,500 $\div 9$ = J$500.
• Tameka: $4 \times$ J$500 = J$2,000; Andre: $5 \times$ J$500 = J$2,500.
• Check: J$2,000 + J$2,500 = J$4,500.

From Ratio to Fraction to Percentage

A part-to-part ratio tells you the fractions of the whole. If boys : girls $= 2 : 3$, there are 5 parts altogether, so boys are $\frac{2}{5}$ of the group and girls are $\frac{3}{5}$.

A common trap is to read $2 : 3$ as "two-thirds". Two-thirds would be the ratio of boys to girls, not boys to the whole group.

The Percentage Problems You Must Be Able to Solve

1. A percentage of a quantity: find $35\%$ of 240. Use 10% chunks: $10\% = 24$, so $30\% = 72$ and $5\% = 12$; $35\% = 72 + 12 = 84$.
2. One number as a percentage of another: 18 students out of 60 walk to school. $\frac{18}{60} = \frac{30}{100} = 30\%$.
3. Finding the whole when a percentage is known: 15% of a number is 45. Then $1\% = 45 \div 15 = 3$ and $100\% = 300$. With a multiple of 5, you can also use $5\% = 15$, so $100\% = 15 \times 20 = 300$.
4. The percentage of a set or object in a diagram: 7 of 20 equal squares are shaded, so $\frac{7}{20} = \frac{35}{100} = 35\%$ is shaded.
5. Percentage of money or measures: 25% of 2 kg of flour is $\frac{1}{4} \times 2000\text{ g} = 500\text{ g}$.