Direct and Inverse Proportions
Class 8 · Mathematics
4 ideas in Class 8 Mathematics, explained in plain words, with games and quizzes to practise.
Recognising Direct Proportion
Two quantities are in direct proportion if they rise or fall together and their ratio stays the same. If 1 kg of rice costs ₹ 50, then 2 kg cost ₹ 100 and 3 kg cost ₹ 150. The ratio of kilograms to cost is always 1 to 50. Quantities that just increase together, like ages, need not be in direct proportion.
Why it matters. Shopping, recipes and travel all use proportion. Children who check only 'both go up' without checking the ratio can choose the wrong method for a problem.
Try it in two minutes
- Ask your child the cost of 1 pen and 3 pens at the same price. Write both in a small table.
- Divide the number of pens by the cost for each row and see that the answer does not change.
- Now compare your age and your child's age now and 5 years later. Ask if the ratio stays the same.
- Ask: do both going up always mean direct proportion?
Try at home. Look at a price list in a shop and ask your child to check whether the price is in direct proportion to the weight.
Solving Problems on Direct Proportion
When two quantities are in direct proportion, the ratio of one pair of values equals the ratio of another pair. If 4 litres of petrol take a car 60 km, then 12 litres take it 180 km. We can write the equal ratios and solve for the missing number, or find the value for one unit first. A map scale is also direct proportion.
Why it matters. Cooking for more people, finding real distances from maps and finding heights from shadows all need this skill.
Try it in two minutes
- Ask your child: if 3 mangoes cost ₹ 60, what do 5 mangoes cost? Let them find the cost of one first.
- Write the same problem as equal ratios and solve for the cost of 5 mangoes.
- Look at a map or a toy car. If 1 cm on paper stands for 10 cm in real life, ask the real length of a 6 cm line.
- Ask them to explain why the cost of one is a good first step.
Try at home. Ask your child to double a recipe for a family gathering and write all the new quantities.
Recognising Inverse Proportion
Two quantities are in inverse proportion when one rises as the other falls, in such a way that their product stays the same. If 4 workers finish a job in 12 days, then 8 workers need 6 days, because 4 × 12 and 8 × 6 are both 48. Doubling the workers halves the time.
Why it matters. Planning travel, work and sharing needs this idea. A common mistake is thinking that any pair where one rises and the other falls is inverse proportion.
Try it in two minutes
- Take 12 coins and arrange them in 2 rows of 6. Ask how many columns there are with 3 rows.
- Write rows and columns in a small table and multiply them in each row.
- Ask: what stays the same? (The product is always 12.)
- Ask: if you walk faster, does the time to school go up or down?
Try at home. Ask your child to find three pairs of numbers whose product is 24 and write them in a table.
Solving Problems on Inverse Proportion
When two quantities are in inverse proportion, x1 × y1 equals x2 × y2. If 6 pipes fill a tank in 80 minutes, then 5 pipes take 6 × 80 ÷ 5, which is 96 minutes. First work out the constant product, then divide by the new value.
Why it matters. Questions about workers, food, speed and time are common in daily life and in exams. Children often use direct proportion by mistake, so the answer goes the wrong way.
Try it in two minutes
- Say: 4 people can eat a pizza box of food in 6 days. Ask how long it lasts for 8 people and whether it goes up or down.
- Ask your child to multiply people and days to find the 'total' of 24.
- Divide 24 by the new number of people and compare.
- Ask how they know the answer should be smaller, before they calculate.
Try at home. Make up a problem about friends sharing a bag of marbles and solve it using the constant product.
What your child will practise
- Recognising Direct Proportion: Mission Control Quest (Collect the Answer)
- Recognising Inverse Proportion: Space Quick Fire (Quick Fire)
- Solving Problems on Direct Proportion: Lab Explorer Quest (Collect the Answer)
- Solving Problems on Direct Proportion: Ocean Puzzle (Clue Puzzle)
- Solving Problems on Inverse Proportion: Jungle Quick Fire (Quick Fire)
- Quiz: Recognising Direct Proportion (8 questions)
- Quiz: Recognising Inverse Proportion (8 questions)
- Quiz: Solving Problems on Direct Proportion (8 questions)
- Quiz: Solving Problems on Inverse Proportion (8 questions)