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Ratio and proportion — KCSE Mathematics

KCSE Mathematics · 109 practice questions · 4 syllabus objectives · 4 revision lessons

36 easy36 medium37 hard

Last updated · Aligned to the KNEC KCSE syllabus

What You'll Learn

Key learning outcomes for this topic, aligned to the KNEC KCSE syllabus.

Express and simplify ratios; divide a quantity in a given ratio

Solve problems involving direct and inverse proportion using the unitary method and algebraic approach

Apply ratio and proportion to map reading (scales), mixing solutions and sharing resources

Ratio and proportion

Revision Notes

Concise lesson notes for Ratio and proportion, written to the KCSE Mathematics marking standard. Read the first lesson free below.

Understanding Ratios and Proportions

Ratios compare two or more quantities. To express a ratio, write it in the form a:b, where a and b are the quantities being compared. To simplify a ratio, divide both terms by their greatest common divisor (GCD).

Example of Simplifying Ratios:

  • For the ratio 8:12, the GCD is 4.
  • Simplifying gives 8 ÷ 4 : 12 ÷ 4 = 2:3.

When dividing a quantity in a given ratio, follow these steps:

  1. Add the parts of the ratio together.
  2. Divide the total quantity by the sum of the parts.
  3. Multiply the result by each part of the ratio.

Example of Dividing a Quantity:

  • Divide 60 in the ratio 2:3.
  • Total parts = 2 + 3 = 5.
  • Each part = 60 ÷ 5 = 12.
  • First part = 2 × 12 = 24; Second part = 3 × 12 = 36.
  • Thus, 60 is divided into 24:36.

Key points to remember

  • Ratios compare two or more quantities in a specific format.
  • To simplify, divide by the greatest common divisor.
  • When dividing, sum the ratio parts before dividing the total.
  • Multiply the result by each part of the ratio for distribution.
  • Express ratios in the simplest form for clarity.

Worked example

Divide 100 in the ratio 1:4. Total parts = 1 + 4 = 5. Each part = 100 ÷ 5 = 20. First part = 1 × 20 = 20; Second part = 4 × 20 = 80.

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More lessons in this topic

Lesson 2: Understanding Ratio and Proportion

Objective: Solve problems involving direct and inverse proportion using the unitary method and algebraic approach

In mathematics, ratio and proportion are essential concepts used to compare quantities. Direct proportion occurs when two quantities increase or decrease together, while inverse proportion occurs when one quantity increases as the other decreases. To solve problems involving these concepts, we can use the unitary method or algebraic approach.

Unitary Method:

  1. Identify the relationship between the quantities.
  2. Find the value of one unit.
  3. Multiply to find the required quantity.

Algebraic Approach:

  1. Set up an equation based on the relationship.
  2. Solve for the unknown.

Example of Direct Proportion: If 5 apples cost 100 shillings, how much do 8 apples cost?

  • Cost of 1 apple = 100 / 5 = 20 shillings.
  • Cost of 8 apples = 8 * 20 = 160 shillings.

Example of Inverse Proportion: If 4 workers can complete a task in 10 days, how long will 2 workers take?

  • Total work = 4 workers * 10 days = 40 worker-days.
  • Days taken by 2 workers = 40 / 2 = 20 days.
  • Direct proportion: quantities increase or decrease together.
  • Inverse proportion: one quantity increases while the other decreases.
  • Use unitary method for simple calculations.
  • Use algebra to set up equations for complex problems.
  • Always check the relationship between quantities before solving.

If 6 kg of rice costs 480 shillings, how much does 10 kg cost?

  • Cost of 1 kg = 480 / 6 = 80 shillings.
  • Cost of 10 kg = 10 * 80 = 800 shillings.
Lesson 3: Understanding Ratio and Proportion in Real Life

Objective: Apply ratio and proportion to map reading (scales), mixing solutions and sharing resources

Ratio and proportion are essential concepts in mathematics that help us solve real-world problems. Ratio is a comparison of two quantities, while proportion states that two ratios are equal. Here are key applications:

  • Map Reading: When using a map, the scale indicates the ratio of the distance on the map to the actual distance. For instance, a scale of 1:50,000 means 1 cm on the map equals 50,000 cm in reality.
  • Mixing Solutions: When mixing solutions, the ratio helps determine the concentration. For example, if you need to mix salt and water in a 1:4 ratio, for every 1 liter of salt, use 4 liters of water.
  • Sharing Resources: When sharing resources, use ratios to ensure fairness. If three friends share 12 apples in a ratio of 2:3:7, you can find each person’s share using the total parts (2+3+7=12). Each part is 1 apple, so:
    • First friend: 2 apples
    • Second friend: 3 apples
    • Third friend: 7 apples.
  • Ratio compares two quantities directly.
  • Proportion states two ratios are equal.
  • Use ratios for map scales and distances.
  • Apply ratios in mixing solutions accurately.
  • Ratios ensure fair sharing of resources.

If a map scale is 1:100,000, how far is 3 cm on the map in reality?

  • Actual distance = 3 cm × 100,000 = 300,000 cm or 3 km.
Lesson 4: Understanding Ratio and Proportion

Objective: Ratio and proportion

Ratio is a way to compare two quantities, showing how much of one exists relative to the other. It is expressed as 'a:b' or 'a/b'. Proportion, on the other hand, states that two ratios are equal. For example, if 3:2 = 6:4, then the two ratios are in proportion.

Key concepts to remember:

  • Simplifying Ratios: Always simplify ratios to their lowest form. For instance, the ratio 8:4 simplifies to 2:1.
  • Proportional Relationships: If a:b = c:d, then a, b, c, and d can be used to find missing values.
  • Cross Multiplication: To solve proportions, use cross multiplication. For example, if a/b = c/d, then ad = bc.

Practical applications: Ratios and proportions are used in recipes, maps, and financial calculations. Understanding these concepts helps in real-life problem-solving.

  • Ratios compare two quantities using a:b format.
  • Proportions state that two ratios are equal.
  • Simplify ratios to their lowest form for clarity.
  • Use cross multiplication to solve proportion problems.
  • Ratios and proportions are useful in everyday situations.

If the ratio of boys to girls in a class is 3:5, how many girls are there if there are 12 boys?

  • Set up the proportion: 3/5 = 12/x.
  • Cross multiply: 3x = 60.
  • Solve for x: x = 20. Thus, there are 20 girls.

Sample Questions

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1
easySHORT ANSWER4 marks

Identify the ratio in which two solutions of salt water are mixed if solution A has a concentration of 15% and solution B has a concentration of 5%. If the final mixture needs to be 10% concentrated, state the ratio of solution A to B required. (4 marks)

Answer & marking scheme

Part (a) — 4 marks
Let the ratio be x:y, where x = amount of solution A and y = amount of solution B (1 mk)
Set up the equation 10 = (15x + 5y) / (x + y) (1 mk)
Solve to find x:y = 1:1 (1 mk)
State that the required ratio is 1:1 (1 mk)
2
easySHORT ANSWER2 marks

Identify the scale of a map that represents a distance of 50 km as 5 cm. If a distance of 20 km is to be represented on the same map, calculate the length in centimetres. (2 marks)

Answer & marking scheme

Part (a) — 2 marks
Scale = 1 cm represents 10 km (1 mk)
Length for 20 km = 2 cm (1 mk)
3
easySHORT ANSWER2 marks

State the new price of a product if its original price was KES 800 and it is increased in the ratio of 3:5. (2 marks)

Answer & marking scheme

Part (b) — 2 marks
Calculate total parts of the ratio = 3 + 5 = 8 (1 mk)
New price = 800 × (5/3) = KES 1333.33 (1 mk)
4

State the amount of money each student receives if KES 600 is shared among 4 students in the ratio 2:3:1:4. (3 marks)

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Frequently asked questions

What does the KCSE Mathematics topic "Ratio and proportion" cover?

Ratio and proportion covers Express and simplify ratios; divide a quantity in a given ratio; Solve problems involving direct and inverse proportion using the unitary method and algebraic approach; Apply ratio and proportion to map reading (scales), mixing solutions and sharing resources, and more, all aligned to the official KNEC KCSE Mathematics syllabus.

How many practice questions are available for Ratio and proportion?

HighMarks has 109 Ratio and proportion practice questions for KCSE Mathematics, each with a full marking scheme. The first 3 are free; sign up to access the rest, plus all KCSE mock exams and past papers.

Are these aligned with the KNEC KCSE syllabus?

Yes. Every objective on this page is taken directly from the official KNEC KCSE Mathematics syllabus. Practice questions match the KCSE exam format and are graded against the standard KNEC marking scheme.

How should I revise Ratio and proportion for the KCSE exam?

Start with the revision notes on this page to refresh the core concepts, then work through the practice questions in increasing difficulty. Sign up for HighMarks to get a personalised study plan that adapts to the topics you keep getting wrong, plus mock exams, subject-wide practice, and detailed performance tracking. See pricing.

Why Practise Ratio and proportion?

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Related topics in KCSE Mathematics

Continue revising adjacent topics from the Mathematics syllabus.

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Decimals (operations, approximation, significant figures)
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Indices (laws of indices)
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Fractions (operations, mixed numbers)
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Logarithms (laws of logarithms, common logarithms, applications)
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Factors and multiples (LCM, HCF/GCD)
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Standard form (scientific notation)
Previous topic
Percentages (applications: profit, loss, discount, commission)
Next topic
Rates and variation (direct, inverse, joint)

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