Lesson Notes By Weeks and Term v3 - Primary 6
Ratios and Proportions
Download the Lessonotes Mobile Nigeria 2025 app for faster lesson access on Android and iPhone.
Ratios and Proportions
Download the Lessonotes Mobile Nigeria 2025 app for faster lesson access on Android and iPhone.
Subject: General Mathematics
Class: Primary 6
Term: 1st Term
Week: 4
Theme: Number And Numeration
This page supports the lesson note with a companion video and a short classroom-ready summary.
For class groups and homework, share this lesson page so learners also get the summary, objectives, and full lesson context.
Solve problems on ratio; Apply ratio to everyday life; Solve quantitative reasoning problem in volving ratio; Solve problems in direct proportion; Solve problems on quantitative reasoning in volving direct proportion; Solve problems on in verse proportions; Identify some daily life activities that are in versely related; Solve problems on quantitative reasoning in in verse proportions
`3/30 = 7/x` Cross-multiply: `3 x = 7 30` `3x = 210` `x = 210 ÷ 3` `x = ₦70`.
Inverse Proportion: Two quantities are in inverse proportion if an increase in one quantity causes a proportional decrease in the other, and a decrease in one quantity causes a proportional increase in the other.
Identifying Inverse Proportion: If `x` and `y` are inversely proportional, then `xy = k` (a constant value) or `y = k/x`.
Examples: The more workers on a construction site, the less time it takes to complete the building. The faster a car travels, the less time it takes to reach its destination. The more people sharing a given amount of food, the less each person gets.
Solving Inverse Proportion Problems: Unitary Method (Multiplication Logic): If it takes `A` people `B` hours, then 1 person would take `A B` hours. Then divide by the new number of people.
Ratio Method: For inverse proportion `a:b` and `c:d`, the relationship is `ac = bd` (or `a/b = d/c`).
Example 5: If 5 farmers can clear a plot of land in 12 days, how many days will it take 8 farmers to clear the same plot of land? (More farmers means less time, so it's inverse proportion)
Unitary Method:
1. Time taken by 1 farmer = `5 farmers 12 days = 60 days`.
2. Time taken by 8 farmers = `60 days ÷ 8 = 7.5 days`.
Ratio Method: Let `x` be the time taken by 8 farmers. `Farmers: Days` `5 : 12` `8 : x` For inverse proportion, `(Number of farmers) (Time taken) = Constant` `5 12 = 8 x` `60 = 8x` `x = 60 ÷ 8` `x = 7.5 days`. C. Quantitative Reasoning (QR) Quantitative reasoning problems involving ratios and proportions often present a pattern or a set of related values, and students need to deduce the underlying ratio or proportional relationship to find a missing value.
Example 6 (Ratio QR): `[2 : 4]` `[3 : ? ]` `[5 : 10]` Observe the pattern in the given pairs. In `[2:4]`, the second number is twice the first (`4 = 2 2`). In `[5:10]`, the second number is twice the first (`10 = 5 2`).
Apply the same rule to the missing pair: `3 2 = 6`. So, `[3 : 6]`.
Example 7 (Direct Proportion QR): `If (₦50, 2 loaves of bread), then (₦150, ? loaves of bread)` Identify the relationship: As money increases, loaves of bread increase (direct proportion). ₦50 gives 2 loaves. ₦150 is 3 times ₦50 (150/50 = 3). So, the number of loaves will also be 3 times: `2 loaves 3 = 6 loaves`.
Example 8 (Inverse Proportion QR): `If (4 workers, 6 days), then (8 workers, ? days)` Identify the relationship: As workers increase, days decrease (inverse proportion). `4 workers 6 days = 24 (constant work units)`.
For 8 workers: `8 workers X days = 24`. * `X = 24 / 8 = 3 days`. --- A. Ratio A ratio is a comparison of two or more quantities of the same kind, expressed in the simplest form. It shows how much of one quantity there is compared to another.
Notation: A ratio can be written as `a:b` or `a/b`. For example, if there are 3 boys and 2 girls in a group, the ratio of boys to girls is `3:2`.
Terms of a Ratio: In the ratio `a:b`, `a` and `b` are called the terms of the ratio. The first term is the antecedent, and the second term is the consequent.
Simplifying Ratios: Ratios must always be expressed in their simplest form by dividing all terms by their highest common factor (HCF).
Example 1: Simplify the ratio `15:25`. The HCF of 15 and 25 is
5. Divide both terms by 5: `15 ÷ 5 = 3` and `25 ÷ 5 = 5`. The simplified ratio is `3:5`.
Ratios with Different Units: Quantities must be in the same unit before forming a ratio. Convert larger units to smaller units for easier calculation.
Example 2: Express the ratio of 50 kobo to ₦2 in its simplest form. First, convert ₦2 to kobo: ₦1 = 100 kobo, so ₦2 = 2 100 kobo = 200 kobo. The ratio is `50 kobo : 200 kobo`.
Simplify: Divide both by 50 (HCF). `50 ÷ 50 = 1`, `200 ÷ 50 = 4`. The simplified ratio is `1:4`.
Sharing in a Given Ratio: To share a quantity in a given ratio, follow these steps:
1. Add the parts of the ratio to find the total number of parts.
2. Divide the total quantity by the total number of parts to find the value of one part.
3. Multiply the value of one part by each individual ratio part to find the share of each.
Example 3: Share ₦1,200 between Aminu and Bola in the ratio `3:5`.
1. Total parts = `3 + 5 = 8` parts.
2. Value of one part = `₦1,200 ÷ 8 = ₦150`.
3. Aminu's share = `3 parts ₦150/part = ₦450`.
4. Bola's share = `5 parts ₦150/part = ₦750`.
5. Check: `₦450 + ₦750 = ₦1,200`.
B. Proportion Proportion is a statement that two ratios are equal. If `a:b` and `c:d` are two equal ratios, then we can write `a:b = c:d` or `a/b = c/d`.
Direct Proportion: Two quantities are in direct proportion if an increase in one quantity causes a proportional increase in the other, and a decrease in one quantity causes a proportional decrease in the other.
Identifying Direct Proportion: If `x` and `y` are directly proportional, then `y/x = k` (a constant value) or `y = kx`.
Examples: The more notebooks you buy, the more money you spend. The more fuel you put in a car, the further it can travel. The more garri a farmer harvests, the more sacks are needed.
Solving Direct Proportion Problems: Unitary Method: Find the value for one unit, then multiply by the required number of units.
Ratio Method: Set up an equivalence of ratios.
Example 4: If 3 sachets of "Pure Water" cost ₦30, how much will 7 sachets cost?
Unitary Method:
1. Cost of 1 sachet = `₦30 ÷ 3 = ₦10`.
2. Cost of 7 sachets = `7 ₦10 = ₦70`.
Ratio Method: Let `x` be the cost of 7 sachets. `3 sachets : ₦30 = 7 sachets : x` `3/30 = 7/x` Cross-multiply: `3 x = 7 30` `3x = 210` `x = 210 ÷ 3` `x = ₦70`.
Inverse Proportion: Two quantities are in inverse proportion if an increase in one quantity causes a proportional decrease in the other, and a decrease in one quantity causes a proportional increase in the other.
Identifying Inverse Proportion: If `x` and `y` are inversely proportional, then `xy = k` (a constant value) or `y = k/x`.
Examples: The more workers on a construction site, the
A. Introduction (10 minutes)
Teacher Activity: Begins by posing a real-life scenario: "Imagine we have ₦1,000 to share between two friends, Emeka and Tunde. If Emeka is to get twice as much as Tunde, how much will each person get?" This hooks students into the idea of sharing unequally, leading to the concept of ratio.
Student Activity: Students brainstorm possible ways to share the money. Some may suggest equal sharing, while others might attempt to guess. The teacher guides them to understand that a systematic approach is needed.
B. Concept Development (30 minutes)
Teacher Activity (Ratio): Defines ratio using examples from the classroom (e.g., number of boys to girls, number of tables to chairs). Demonstrates how to write ratios (e.g., `3:5`). Explains simplifying ratios using examples relevant to Nigerian context (e.g., `10 mangoes to 15 oranges`, `25 kobo to ₦1`). Demonstrates sharing a quantity in a given ratio (e.g., sharing ₦2,000 profit from selling groundnuts in a `2:3` ratio).
Student Activity (Ratio): Students write down ratios for various items in the classroom. Students practice simplifying given ratios on their individual whiteboards or jotters. Students attempt a guided problem on sharing money or objects in a ratio within small groups.
Teacher Activity (Direct Proportion): Introduces proportion, then specifically direct proportion. Explains that as one quantity increases, the other increases proportionally (and vice versa), using everyday examples: `cost of bread and number of loaves`, `amount of garri bought and its weight`. Demonstrates solving problems using both the unitary method and the ratio method with clear, step-by-step calculations.
Student Activity (Direct Proportion): Students identify more examples of direct proportion in their daily lives. Students work in pairs to solve direct proportion problems, discussing which method (unitary or ratio) they find easier.
Teacher Activity (Inverse Proportion): Introduces inverse proportion. Explains that as one quantity increases, the other decreases proportionally (and vice versa), using examples: `number of bricklayers and time to build a wall`, `speed of a vehicle and time to reach a village`. Demonstrates solving problems using both the unitary method (adjusted for inverse) and the ratio method.
Student Activity (Inverse Proportion): Students identify real-life situations that exhibit inverse proportionality. Students solve inverse proportion problems individually, comparing their answers with a partner.
Teacher Activity (Quantitative Reasoning): Explains that QR tests logical thinking using patterns and relationships, often involving ratios and proportions. Presents simple QR examples on the board, guiding students to identify the underlying ratio/proportional rule.
Student Activity (Quantitative Reasoning): Students attempt to identify the pattern in QR problems related to ratios and proportions. Students solve simple QR exercises in groups, explaining their reasoning.
C. Consolidation and Wrap-up (5 minutes)
Teacher Activity: Summarizes the key concepts of ratio, direct proportion, and inverse proportion. Asks probing questions to check understanding and address any lingering misconceptions.
Student Activity: Students articulate what they have learned, ask clarifying questions, and participate in a quick Q&A session. --- The teacher should guide students through these problems, encouraging them to try each step before revealing the solution.
Question 1 (Ratio Sharing): Mr. Okoro shared ₦4,500 between his two children, Ada and Obi, in the ratio of their ages. If Ada is 8 years old and Obi is 7 years old, how much did each child receive?
Solution: Determine the ratio of their ages: Ada:Obi = 8:
7. Calculate the total number of parts: `8 + 7 = 15 parts`.
Find the value of one part: `₦4,500 ÷ 15 parts = ₦300 per part`.
Calculate Ada's share: `8 parts * ₦300/part = ₦2,400`.
Calculate Obi's share: `7 parts * ₦300/part = ₦2,100`.
Commentary: This problem assesses the ability to share a quantity in a given ratio, directly addressing Performance Objective 1 and
2. Question 2 (Direct Proportion): A trader buys 5 bags of yam for ₦12,
5
0
0. How much would 8 bags of yam cost at the same rate?
Solution: Identify relationship: As the number of bags increases, the cost increases. This is direct proportion.
Using Unitary Method: Cost of 1 bag of yam = `₦12,500 ÷ 5 = ₦2,500`. Cost of 8 bags of yam = `8 ₦2,500 = ₦20,000`.
Commentary: This problem tests the understanding and application of direct proportion in a real-life market scenario, aligning with Performance Objective
4. Question 3 (Inverse Proportion): If 6 painters can paint a school block in 10 days, how many days will it take 4 painters to paint the same school block?
Solution: Identify relationship: Fewer painters mean more days. This is inverse proportion.
Using Inverse Unitary Method: Time taken for 1 painter = `6 painters 10 days = 60 days`. (One painter takes longer) Time taken for 4 painters = `60 days ÷ 4 = 15 days`.
Commentary: This problem assesses the ability to solve problems involving inverse proportion, aligning with Performance Objective
6. Question 4 (Quantitative Reasoning - Ratio): Observe the pattern: `[6 : 3]`, `[10 : 5]`, `[14 : ? ]`. Find the missing number.
Solution: Analyze the given pairs: In `[6 : 3]`, the second number is half of the first (`3 = 6 ÷ 2`). In `[10 : 5]`, the second number is half of the first (`5 = 10 ÷ 2`).
Apply the pattern to the incomplete pair: `14 ÷ 2 = 7`. The missing number is
7. Commentary: This problem evaluates quantitative reasoning skills based on ratio relationships, aligning with Performance Objective 3. ---
Cooking and Recipes (Cultural/Community): Ratios are fundamental in Nigerian cuisine. When preparing local dishes like Jollof rice, Akara, or Egusi soup, cooks constantly adjust ingredient ratios (e.g., rice to water, beans to pepper, melon seeds to palm oil) based on the number of people to be served. If a recipe for 4 people calls for 2 cups of rice, making it for 10 people requires calculating the new proportional amount of rice and other ingredients. This highlights direct proportion. Resource Allocation and Business (Economic/Community): Ratios are used to fairly distribute resources or profits. For example, if a group of farmers pools resources to buy fertilizer, the cost or profit can be shared in the ratio of land size each farmer owns or the amount of their initial contribution. Small business owners in Nigeria use ratios to share profits based on capital invested by partners. This is a direct application of sharing in a given ratio. Construction and Manual Labour (Community/Environmental): In building houses or community projects, ratios are critical for mixing materials like cement, sand, and gravel for concrete. A common mix ratio like `1:2:4` (cement:sand:gravel) ensures structural integrity. Inverse proportion is seen when a community mobilizes more volunteers (e.g., for clearing a path or building a local market stall); the more hands available, the less time it takes to complete the task. ---