Proportional Reasoning-2
Class 8 · Mathematics
5 ideas in Class 8 Mathematics, explained in plain words, with games and quizzes to practise.
Proportions and Map Scales
Part of: Ratio and proportion
Two ratios a : b and c : d are proportional when a × d is equal to b × c. This is called cross-multiplication. Maps use the idea: a representative fraction (RF) of 1 : 100000 means that 1 cm on the map stands for 100000 cm, which is 1 km, on the ground. The distance found this way is the straight-line geographical distance, not the road distance.
Why it matters. Ratios let children scale recipes, drawings and maps up or down. Reading a map scale helps them plan trips and understand the world around them.
Try it in two minutes
- Ask: 2 cups of rice go with 1 cup of dal. How much rice goes with 4 cups of dal?
- Show a map with a scale. Ask your child to measure the distance between two towns with a ruler and convert it.
- Ask your child to check whether 6 : 9 and 8 : 12 are proportional by cross-multiplying.
Try at home. Draw a plan of your room using a scale of 1 cm for 50 cm and mark the bed and the door.
Ratios with More Than Two Terms
Part of: Ratio and proportion
A ratio can have many terms, such as nuts : raisins : seeds is 6 : 3 : 1. When every quantity is multiplied by the same factor, the mixture keeps its taste and the new ratio is proportional to the old one. To find the other quantities when one is known, find the factor and multiply all terms by it.
Why it matters. Recipes, mixtures, paints and building materials use more than two ingredients. Scaling all parts by the same factor keeps results the same.
Try it in two minutes
- Write a recipe with three ingredients, such as 4 spoons of flour, 2 of sugar and 1 of butter.
- Ask your child what quantities are needed if you use 8 spoons of flour.
- Ask if the mix is still the same if only the flour is doubled.
Try at home. Ask your child to scale a recipe at home to feed twice as many people and write the new quantities.
Dividing a Whole in a Given Ratio
Part of: Ratio and proportion
To divide a quantity in a ratio, add all the terms of the ratio and divide the quantity by that sum to find the value of one part. Then multiply each term by this value. To divide 60 in the ratio 2 : 3, one part is 60 ÷ 5 = 12, so the shares are 24 and 36. The angles of a triangle add up to 180 degrees, so they can be split in a given ratio the same way.
Why it matters. Sharing money, ingredients or space fairly according to a ratio is a daily skill. It also helps with angles and with charts.
Try it in two minutes
- Ask your child to share 60 sweets between two children in the ratio 2 : 3.
- Ask them to check that the two shares add up to 60.
- Ask them to split 180 degrees in the ratio 1 : 2 : 3 and see what kind of triangle the angles make.
Try at home. Share a bag of 50 coins or beads among three family members in a ratio of your choice.
Pie Charts and Angles
Part of: Ratio and proportion
A pie chart is a circle cut into slices to show how a whole is shared. The angle at the centre of each slice is proportional to the amount it stands for. The angles of all slices add up to 360 degrees, so each angle is found by dividing 360 degrees in the ratio of the data. To find the angle for a value, take its share of the total and multiply by 360.
Why it matters. Pie charts are used in newspapers, school reports and surveys. Children need to know how to read the slices, not just how they look.
Try it in two minutes
- Ask your child to count the family members who like tea, coffee or juice.
- Work out the angle for each group by finding its share of the whole and multiplying by 360 degrees.
- Check that all the angles add up to 360 degrees.
Try at home. Make a small survey of favourite fruits among five friends and ask your child to compute the angle for each fruit.
Inverse Proportion
Part of: Ratio and proportion
Two quantities are in inverse proportion when one increases by some factor and the other decreases by the same factor. Their product stays constant, so x1 × y1 is x2 × y2. If 20 workers take 4 days to do a job, then 10 workers take 8 days, because 20 × 4 and 10 × 8 are both 80. More workers means fewer days. Speed and time for a fixed distance behave the same way.
Why it matters. Many everyday situations need this idea: more helpers, faster travel, fewer days. Without it children may wrongly treat everything as direct proportion.
Try it in two minutes
- Ask: 4 people clean a hall in 6 hours. How long would 8 people take if they work at the same speed?
- Show that 4 × 6 and 8 × 3 are equal.
- Ask your child to find one example of a direct and one of an inverse proportion at home.
Try at home. Time yourselves walking a fixed distance at two speeds and check that the speed times the time stays the same.
What your child will practise
- Dividing a Whole in a Given Ratio: Ocean Quick Fire (Quick Fire)
- Inverse Proportion: Mission Control Sort (Sort It Out)
- Pie Charts and Angles: Ocean Puzzle (Clue Puzzle)
- Proportions and Map Scales: Lab Explorer Quest (Collect the Answer)
- Ratios with More Than Two Terms: Space Quick Fire (Quick Fire)
- Quiz: Dividing a Whole in a Given Ratio (8 questions)
- Quiz: Inverse Proportion (8 questions)
- Quiz: Pie Charts and Angles (8 questions)
- Quiz: Proportions and Map Scales (8 questions)