Ratios and proportions

Overview

A ratio compares two or more quantities. A proportion states that two ratios are equal. These ideas help learners share money, mix materials, compare ages, calculate percentages, read scales, and solve many daily-life problems.

Ratios are not only written numbers; they describe relationships. If cement, sand, and gravel are mixed in the ratio $1:2:4$, the relationship between the parts must stay the same even when the total amount changes.

A ratio should always be read with its context. The ratio $2:3$ might mean $2$ boys for every $3$ girls, $2$ cups of rice for every $3$ cups of water, or $2$ cm on a map for every $3$ km in real life. The numbers alone are not enough; the quantities being compared must also be known.

Prerequisites

• Multiplication and division of whole numbers.
• Simplifying fractions.
• Equivalent fractions.
• Basic percentages.
• Understanding units such as shillings, kilograms, metres, and litres.

Learning Scope

This page covers writing and simplifying ratios, sharing quantities in a given ratio, using proportion to find unknown values, and applying ratios in everyday contexts.

It does not fully cover rates and variation, exchange rates, compound proportionality, or business accounts. Those may connect to this topic but belong to pages such as Rates and variations or accounting-related contexts.

Subtopics

Meaning of Ratio

A ratio compares quantities of the same kind. If a class has $12$ boys and $18$ girls, the ratio of boys to girls is:

$$ 12:18 $$

Simplify by dividing both parts by $6$:

$$ 12:18=2:3 $$

Key insight: ratio order matters. Boys to girls, $2:3$, is different from girls to boys, $3:2$.

A ratio can compare two quantities or more than two quantities. For example, $1:2:4$ may compare cement:sand:gravel. In a three-part ratio, the order of all three words matters.

Ratios and Fractions

A ratio can be connected to fractions. If boys:girls is $2:3$, the total number of parts is:

$$ 2+3=5 $$

The boys are $\frac{2}{5}$ of the class and the girls are $\frac{3}{5}$ of the class.

Key insight: a part-to-part ratio can be converted into part-to-whole fractions by adding all parts.

This is a common place for errors. In the ratio boys:girls $=2:3$, boys are not $\frac{2}{3}$ of the class. They are $\frac{2}{5}$ of the class because the whole class has $2+3=5$ parts.

Simplifying Ratios

Ratios are simplified by dividing all terms by their highest common factor, after making units consistent.

For example:

$$ 500\ \text{g}:1\ \text{kg}=500\ \text{g}:1000\ \text{g}=1:2 $$

Key insight: never simplify a ratio with mixed units before converting to the same unit.

For decimal quantities, it may help to multiply all terms by the same power of $10$ before simplifying. For example:

$$ 1.5:2.5=15:25=3:5 $$

Multiplying both terms by $10$ keeps the relationship unchanged.

Sharing in a Given Ratio

To share an amount in a ratio:

1. Add the ratio parts.
2. Divide the total amount by the sum of parts.
3. Multiply each ratio part by the value of one part.

If Tsh $40,000$ is shared in the ratio $3:5$, the total parts are:

$$ 3+5=8 $$

One part is:

$$ \frac{40,000}{8}=5,000 $$

The shares are:

$$ 3 \times 5,000=15,000,\quad 5 \times 5,000=25,000 $$

Always check:

$$ 15,000+25,000=40,000 $$

The check confirms that the shares have used the whole amount.

Proportion

A proportion is an equality of two ratios.

$$ \frac{a}{b}=\frac{c}{d} $$

If the quantities are proportional, cross-products are equal:

$$ a \times d=b \times c $$

For example:

$$ \frac{2}{5}=\frac{6}{15} $$

because:

$$ 2 \times 15=5 \times 6=30 $$

Proportion problems can often be solved by finding a scale factor. Since $2$ has been multiplied by $3$ to get $6$, $5$ must also be multiplied by $3$ to get $15$. Cross-products are useful, but learners should also understand the scaling idea.

Direct Proportion in Simple Contexts

Two quantities are in direct proportion if they increase or decrease by the same factor. If $3$ exercise books cost Tsh $6,000$, then $6$ exercise books cost twice as much:

$$ 6,000 \times 2=12,000 $$

This page introduces proportional thinking. More formal variation statements are handled in Rates and variations.

The unitary method is another slow approach: first find the value of $1$ item, then multiply. If $3$ books cost Tsh $6,000$, then $1$ book costs Tsh $2,000$, so $6$ books cost Tsh $12,000$.

Percentages as Proportional Comparisons

A percentage is a comparison out of $100$. For example, $20\%$ means:

$$ 20\%=\frac{20}{100}=\frac{1}{5} $$

Percentages often appear together with ratios in profit, loss, discount, and class-composition problems.

For example, if $30\%$ of a class are boys, then boys:whole class is $30:100=3:10$. This also means boys:girls is $30:70=3:7$.

Key Terms

• Ratio: a comparison between two or more quantities.
• Equivalent ratios: ratios that express the same relationship.
• Simplest form: a ratio with no common factor greater than $1$.
• Proportion: a statement that two ratios are equal.
• Part-to-part ratio: a comparison between separate parts.
• Part-to-whole fraction: a comparison between one part and the total.
• Direct proportion: a relationship where quantities change by the same factor.
• Percentage: a ratio or fraction expressed out of $100$.

Worked Examples

Example 1: Simplify a ratio with units

Simplify $2\ \text{kg}:750\ \text{g}$.

Convert $2\ \text{kg}$ to grams:

$$ 2\ \text{kg}=2000\ \text{g} $$

So:

$$ 2000:750 $$

Divide by $250$:

$$ 2000:750=8:3 $$

Therefore, $2\ \text{kg}:750\ \text{g}=8:3$.

Check the meaning: the first quantity is larger than the second, so the simplified first part should also be larger than the second part. $8:3$ passes this check.

Example 2: Share money in a ratio

Ally and Jane share Tsh $64,000$ in the ratio $3:5$. Find each share and the difference.

Total parts:

$$ 3+5=8 $$

Value of one part:

$$ \frac{64,000}{8}=8,000 $$

Shares:

$$ \begin{aligned} \text{Ally} &= 3 \times 8,000=24,000 \\ \text{Jane} &= 5 \times 8,000=40,000 \end{aligned} $$

Difference:

$$ 40,000-24,000=16,000 $$

The difference is Tsh $16,000$.

Check:

$$ 24,000+40,000=64,000 $$

The shares add back to the original amount.

Example 3: Find an unknown in a proportion

Find $x$ if:

$$ \frac{x}{12}=\frac{5}{8} $$

Cross-multiply:

$$ \begin{aligned} 8x &= 12 \times 5 \\ 8x &= 60 \\ x &= \frac{60}{8} \\ x &= 7.5 \end{aligned} $$

Therefore, $x=7.5$.

Alternative using scale factor: $8$ becomes $12$ by multiplying by $1.5$, so $5$ also becomes $5 \times 1.5=7.5$.

Example 4: Use ratio in a mixture

A builder mixes cement, sand, and gravel in the ratio $1:3:6$. If the total mass is $500\ \text{kg}$, find the mass of cement.

Total parts:

$$ 1+3+6=10 $$

One part:

$$ \frac{500}{10}=50\ \text{kg} $$

Cement is $1$ part, so:

$$ 1 \times 50=50\ \text{kg} $$

The cement needed is $50\ \text{kg}$.

Example 5: Convert a part-to-part ratio to class numbers

In a class, boys:girls $=3:2$. There are $45$ learners altogether. How many boys and girls are there?

Total ratio parts:

$$ 3+2=5 $$

One part:

$$ \frac{45}{5}=9 $$

Boys:

$$ 3 \times 9=27 $$

Girls:

$$ 2 \times 9=18 $$

Check:

$$ 27+18=45 $$

So there are $27$ boys and $18$ girls.

Example 6: Use direct proportion with the unitary method

If $6$ mangoes cost Tsh $3,000$, find the cost of $10$ mangoes.

First find the cost of $1$ mango:

$$ \frac{3,000}{6}=500 $$

Then find the cost of $10$ mangoes:

$$ 10 \times 500=5,000 $$

Therefore, $10$ mangoes cost Tsh $5,000$.

Example 7: Compare two ratios

Are the ratios $4:10$ and $6:15$ equivalent?

Simplify each ratio:

$$ 4:10=2:5 $$

and:

$$ 6:15=2:5 $$

Since both simplify to $2:5$, the ratios are equivalent.

Using cross-products:

$$ 4 \times 15=60,\quad 10 \times 6=60 $$

The equal cross-products confirm the same result.

Common Mistakes

• Reversing the order of a ratio. Correction: read the words carefully; boys:girls is not girls:boys.
• Adding ratio terms when a part-to-part comparison is required. Correction: add terms only when finding the whole number of parts.
• Simplifying before converting units. Correction: convert all quantities to the same unit first.
• Treating $2:3$ as meaning the total is $3$ parts. Correction: the total is $2+3=5$ parts.
• Forgetting to check that shared parts add back to the total.
• Confusing direct proportion with any increase. Correction: in direct proportion, both quantities change by the same factor.
• Treating a ratio as a single fraction without reading the words. Correction: decide whether the question asks for part-to-part or part-to-whole.
• Using cross-multiplication correctly but losing units. Correction: state whether the answer is shillings, kilograms, litres, metres, or learners.
• Assuming a proportional relationship without evidence. Correction: direct proportion means a constant multiplier or constant cost per item.

Practice Tasks

1. Simplify $18:24$.
2. Simplify $45\ \text{cm}:1.5\ \text{m}$.
3. Write the ratio $4:7$ as part-to-whole fractions.
4. Share Tsh $36,000$ in the ratio $2:7$.
5. A drink is mixed using syrup and water in the ratio $1:5$. How much water is needed for $2$ litres of syrup?
6. Find $x$ if $\frac{x}{15}=\frac{4}{5}$.
7. If $6$ mangoes cost Tsh $3,000$, find the cost of $10$ mangoes assuming direct proportion.
8. A class has boys:girls in the ratio $3:2$. If there are $45$ learners, how many are girls?
9. A price is reduced by $15\%$. Express $15\%$ as a fraction in simplest form.
10. Explain why units must be made the same before simplifying a ratio.
11. Simplify $1.2:0.8$.
12. Simplify $750\ \text{ml}:2\ \text{litres}$.
13. A rope is cut in the ratio $4:5$. If the shorter piece is $24\ \text{m}$, find the longer piece.
14. Cement, sand, and gravel are mixed in the ratio $1:2:5$. If there are $80\ \text{kg}$ of mixture, how much sand is used?
15. Are $9:12$ and $15:20$ equivalent? Show your method.
16. If $8$ pens cost Tsh $4,800$, find the cost of $13$ pens assuming direct proportion.
17. A class has $40\%$ girls. Write the ratio girls:boys in simplest form.
18. A learner says that in the ratio $5:7$, the first quantity is $\frac{5}{7}$ of the whole. Correct the mistake.

Generated Question Layer

• Ratio-writing questions: convert word descriptions into ratio notation.
• Simplification questions: simplify ratios with and without unit conversion.
• Sharing questions: divide money, mass, time, or learners in a given ratio.
• Proportion questions: solve missing values by equivalent ratios or cross-products.
• Percentage bridge questions: connect percentages to fractions and ratios.
• Application questions: mixtures, ages, class composition, building materials, and shopping.
• Error-analysis questions: identify reversed ratio order or wrong total parts.

Learner Aid Opportunities

• diagram: ratio bar model showing parts and whole.
• chart: ratio, fraction, decimal, and percentage equivalences.
• interactive: share-an-amount tool where learners adjust ratio parts and see values update.
• animation: scaling a ratio up and down while preserving the relationship.
• video: local-context mixture and sharing examples using money, cement, and classroom groups.
• LLM tutor: step-by-step questioning that asks learners to identify the total number of parts first.
• scaffold: ratio table with columns for original parts, scale factor, scaled parts, and total.
• interactive: unit-conversion ratio simplifier for grams, kilograms, millilitres, litres, centimetres, and metres.
• diagnostic: prompts that ask whether a question needs part-to-part, part-to-whole, or unitary-method reasoning.
• worked-example overlay: reveal check step showing that shared amounts add back to the total.

Exam-Derived Signals

• topic_frequency_2021_2025.json reports $8$ primary mapped records for this topic across 2021-2025: $2$ in 2021, $1$ in 2022, $1$ in 2023, $2$ in 2024, and $2$ in 2025.
• Primary unreviewed or needs-manual-review examples include ratio of polygon angles, loss and selling price, sharing Tsh $64,000$ in a ratio, class percentages, age ratios, a fraction-to-percentage context, and net profit wording.
• Several secondary mappings link ratios to variation, percentages, radicals, progressions, and mixtures. These are unreviewed and sometimes flagged as multi-topic candidates or missing marks.
• exam_format_topic_crosswalk_2022.jsonl maps this topic to "Ratios, profit and loss/Accounts" with $1$ item and $7.14\%$ weight, but unmatched terms "Profit and loss" and "Accounts" require manual review.
• These signals should guide practice variety but should not be treated as reviewed past-question coverage.

Source And Review Notes

• Official syllabus alignment comes from data/curriculum_map.json and raw/syllabuses/csee/2023/csee_mathematics_syllabus_2023.pdf.
• The exam-format crosswalk is assessment guidance; it does not expand the syllabus topic into full accounting.
• Exam-derived question mappings are explicitly unreviewed and need checking against original papers.
• Worked examples are original and designed to reflect common ratio reasoning without copying past-paper solutions.