Ratio, Proportion and Sharing

Simplifying ratios, dividing quantities in given ratios, direct proportion, and inverse proportion.

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Curated Video Lesson

Visual explanation and practical step-by-step walk-through

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Comprehensive Study Notes

Aligned with Ghana NaCCA & WAEC BECE syllabus standards

Topic Introduction & Real-World Context:

A ratio is a mathematical comparison of two or more quantities of the same kind by division. Ratios and proportions are used constantly in commerce, sharing business capital and dividends, mixing concrete in masonry (cement, sand, and stone in ratio 1:2:4), scaling maps, and calculating recipe portions. In the BECE exam, questions on sharing quantities, equivalent ratios, and direct vs inverse proportion are regular favorites.

What You Will Master in This Lesson (NaCCA Objectives):

Express comparisons as ratios in simplest form.

Convert ratios with fractions or decimals into whole-number ratios.

Divide or share a quantity into a given ratio.

Solve direct proportion problems using the unitary method.

Distinguish between direct proportion and inverse (indirect) proportion.

1. What is a Ratio and How is it Expressed?

A ratio compares quantities of the SAME kind and unit. Notation: a : b or a/b. Important Rules: 1. Units must be identical before forming a ratio! (e.g. 50 pesewas to 2 Cedis: 50 : 200 = 1 : 4). 2. A ratio has NO units. It is a pure dimensionless number. 3. Order matters: 3 : 5 is not the same as 5 : 3.

2. Sharing a Quantity in a Given Ratio

Standard 3-Step Method: Step 1: Calculate Total Ratio Parts by adding the ratio terms together. Step 2: Find the value of 1 part = Total Quantity / Total Parts. Step 3: Multiply the value of 1 part by each ratio term to determine individual shares.

3. Direct vs Inverse Proportion

• Direct Proportion: An increase in one quantity leads to a proportional increase in the other (e.g. books vs cost). Use the unitary method: find for 1, then multiply. • Inverse Proportion: An increase in one quantity leads to a proportional DECREASE in the other (e.g. workers vs days). Product remains constant: Workers × Days = Constant.

Common Mistakes Students Make in BECE Examinations:

⚠️Comparing quantities with different units without converting them first.

⚠️Applying direct proportion to inverse proportion problems (workers vs days).

⚠️Dividing by an individual ratio term instead of the sum of parts.

Teacher's BECE Exam Pro-Tips:

⭐Always add the ratio numbers together first to find total parts.

⭐Check that individual shares add up to the original total quantity.

Quick Revision Summary Checklist:

Ratio has no units; units must match before simplifying.

Share = (Individual ratio part / Total parts) × Total Amount.

Direct proportion: more produces more. Inverse proportion: more produces less.

Step-by-Step Worked Examples (1)

Real BECE exam-standard problems with complete solution steps

Example 1: Dividing Money in a Ratio

Problem StatementShare GHS 600 between Kwame and Ama in the ratio 3 : 2.

Step-by-Step Solution:

Step 1: Total parts = 3 + 2 = 5 parts.

Step 2: Value of 1 part = 600 / 5 = GHS 120.

Step 3: Kwame (3 parts) = 3 × 120 = GHS 360.

Step 4: Ama (2 parts) = 2 × 120 = GHS 240.

Step 5: Check: 360 + 240 = 600.

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Key Takeaway / Exam Rule: Always verify that individual shares sum up to the total original quantity.

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