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Cambridge O Level · Mathematics 4024

Ratio and proportion

Learn to simplify ratios, share a total, recover missing quantities and use proportional reasoning in recipes, map scales and best-value comparisons. This original practice pack targets section 1.11; it is not a complete course or a formal treatment of algebraic direct and inverse proportion.

2025 / 2026 / 2027 · Academic review not recorded · Published 2026-09-08

AI-assisted practice — not independently academically reviewed. Answers may contain errors; check important results against your course materials.

Guided tutorial

Learn the topic

16 lesson sections

What you will learn

  • Interpret ordered ratios and distinguish part-to-part from part-to-whole comparisons.
  • Simplify two-part and three-part ratios, including ratios involving decimals or fractions.
  • Convert matching quantities to the same units before forming a ratio.
  • Divide a total in a given ratio and check that the shares add to the total.
  • Find a missing share or total when one share or a difference is known.
  • Scale recipes and solve contextual problems using equal multiplicative factors.
  • Use map scales in both directions with appropriate unit conversions.
  • Compare value using a common quantity and explain when proportional reasoning is justified.

Explanation

Read the ratio in the stated order

A ratio compares quantities in an order. Red:blue = 2:3 means two equal-sized parts of red for every three parts of blue. It does not mean there are only five objects: 4 red and 6 blue also fit the ratio.

Label each part before calculating. Reversing the order gives blue:red = 3:2. The colon is not a decimal point, and the entries are relative amounts rather than necessarily actual counts.

Explanation

Part-to-part is not part-to-whole

If red:blue = 2:3 and those are the only colours, there are 2 + 3 = 5 parts altogether. The red fraction of the total is 2/5, not 2/3. The comparison of red to blue alone is 2/3.

For positive quantities in ratio a:b, their fractions of the combined total are a/(a+b) and b/(a+b). With three categories, add all three entries before finding a fraction of the whole.

Worked example

Worked example: ratio and fraction of a whole

A box contains only green and yellow counters in ratio 3:7. There are 10 equal parts, so 3/10 of all counters are green and 7/10 are yellow.

If the box holds 80 counters, one part is 80 ÷ 10 = 8 counters. Green = 3 × 8 = 24 and yellow = 7 × 8 = 56. Check: 24 + 56 = 80 and 24:56 simplifies to 3:7.

Explanation

Equivalent ratios and simplest form

Multiply or divide every entry by the same positive number to preserve a positive ratio. For example, 4:6 = 8:12 = 2:3. Adding the same number to each entry usually changes the ratio.

To give a ratio in simplest whole-number form, remove any fractions or decimals and divide all entries by their highest common factor. For three entries, use a common factor of all three, not just the first two.

Worked example

Worked example: a three-part ratio

Simplify 18:30:42. The highest common factor of all three entries is 6, so divide each entry by 6 to obtain 3:5:7.

Although 18 and 30 are both divisible by 3, stopping at 6:10:14 leaves a common factor of 2. Check that the final entries have no common factor greater than 1.

Explanation

Match units before comparing

When comparing two lengths, masses or volumes as a dimensionless ratio, first express them in the same units. For instance, 2 m:50 cm becomes 200 cm:50 cm = 4:1.

Writing 2:50 without converting would compare the numerical labels rather than the actual lengths. A comparison of unlike quantities, such as cost per kilogram, is a unit rate; keep its units when interpreting the answer.

Worked example

Worked example: decimal and fractional ratios

Simplify 0.45:1.2. Multiply both entries by 100 to get 45:120, then divide both by 15 to obtain 3:8.

Simplify (2/3):(5/6). Multiply both entries by 6 to remove the denominators: 4:5. The multiplier must act on every entry, not just on the fractions you find awkward.

Explanation

Share a total using equal parts

To share a total in ratio a:b, add the entries to find the number of equal parts. Divide the total by that sum to find one part, then multiply by each entry.

Attach units and keep the order of the recipients. Finally, add the shares and simplify their ratio. If a problem involves indivisible objects, check that each share is a whole number before accepting the split.

Worked example

Worked example: share a budget three ways

A club shares MVR 960 between books, games and equipment in ratio 3:2:7. There are 12 parts, so one part is MVR 960 ÷ 12 = MVR 80.

The shares are MVR 240, MVR 160 and MVR 560 respectively. Their sum is MVR 960, and 240:160:560 simplifies to 3:2:7.

Worked example

Worked example: one share is known

Flour:sugar = 5:2 in a recipe. If sugar is 140 g, those 140 g represent two parts, so one part is 70 g. Flour is five parts: 5 × 70 = 350 g.

The combined flour and sugar mass is 350 + 140 = 490 g. Do not divide 140 by seven: the stated 140 g is one ingredient, not the combined total.

Worked example

Worked example: the difference is known

Two ribbons have lengths in ratio 7:4. The longer ribbon is 18 cm longer. The difference is 7 − 4 = 3 parts, so one part is 18 ÷ 3 = 6 cm.

The lengths are 42 cm and 24 cm; their difference is 18 cm and their total is 66 cm. A known difference uses the difference of the ratio entries, not their sum.

Explanation

Scale recipes multiplicatively

If every serving is the same size, increasing the number of servings multiplies every ingredient by the same scale factor. For a recipe changed from 4 servings to 10, the factor is 10/4 = 2.5.

Do not add six grams to each ingredient merely because six more servings are needed. Ratios preserve multiplication, not addition. Real cooking times or container sizes do not necessarily scale in the same way as ingredients.

Worked example

Worked example: adjust a recipe

A recipe uses 300 g of flour for 4 equal servings. For 10 equal servings, multiply by 10/4: 300 × 2.5 = 750 g.

For 3 servings, the factor is 3/4 and the flour is 225 g. The unitary method gives the same results: one serving needs 300 ÷ 4 = 75 g.

Worked example

Worked example: map distance and real distance

On a map with scale 1:50 000, each map length represents 50 000 times that length in reality. A map distance of 3.6 cm represents 180 000 cm = 1 800 m = 1.8 km.

In the reverse direction, 4 km is 400 000 cm. Its map length is 400 000 ÷ 50 000 = 8 cm. Convert to matching units before applying the scale and state whether your answer is on the map or in reality.

Worked example

Worked example: compare best value

Pack A contains 600 g of the same rice for MVR 48. Pack B contains 900 g for MVR 67.50. Assuming equal quality and no extra charges, compare cost for a common amount: A costs MVR 8 per 100 g and B costs MVR 7.50 per 100 g.

Pack B has the lower unit cost, even though it has the higher total price. If the question instead asks which purchase is affordable or avoids waste, the cheapest unit price may not answer that different question.

Explanation

Check whether a proportional model fits

A constant unit price with no fixed charge gives proportional cost: doubling the quantity doubles the cost. A fixed delivery charge breaks that rule for total cost. Always read the conditions before scaling.

A model should match the situation. Equal portions in a recipe or a stated map scale justify one common multiplier; unrelated measurements do not. Formal equations and graphs for direct and inverse proportion belong to a later algebra topic.

Original practice, not an official examination paper. Readnary is not affiliated with the awarding body. Prepared with AI assistance.