CBSE Class 8 Maths Chapter 3 Proportional Reasoning 2 Notes 2025-26

Proportional reasoning is used to compare quantities in different situations, and it is one of the most important concepts in mathematics for daily life. A proportional relationship means two or more quantities change in the same ratio or by the same factor. For example, if we mix 2 cups of rice with 1 cup of urad dal for idli batter, the ratio is 2 : 1. If another mixture uses 6 cups rice to 3 cups dal or 4 cups rice to 2 cups dal, all these maintain the same proportionality since the ratios are equal when simplified.

To check if two ratios are proportional, we can use cross-multiplication. If 6 : 3 and 4 : 2 are compared, then 6 × 2 = 3 × 4, both resulting in 12, so the mixtures have the same taste if prepared in proportion. In general, two ratios a : b and c : d are proportional if a × d = b × c or if a/c = b/d.

Ratios in Maps Maps often show distances using a ratio called the Representative Fraction (RF), such as 1 : 60,00,000. This means that 1 cm measured on the map equals 60,00,000 cm (or 60 km) on the ground. If you want to find the real distance between two cities, you should measure the distance on the map and multiply it by the scale value. Different maps may use different scales, but the method remains the same, helping us find the actual distance between places.

Map-making activities often use ratios to help students practice scaling. For example, you might sketch your classroom using a scale of 1 : 50 and mark places like the teacher’s desk or blackboard with correct proportions.

Ratios with More than Two Terms Often, we see ratios that involve more than two terms. Let's say a spice mix recipe uses 8 spoons coriander seeds, 4 red chillies, 2 spoons toor dal, and 1 spoon fenugreek seeds, resulting in a ratio of 8 : 4 : 2 : 1. If someone wants to make a smaller batch, they can use a proportional amount, such as 4 : 2 : 1 : 0.5. As long as the ratios of each ingredient stay the same compared to the original, the taste will also be similar.

For ratios with three or more terms, they are proportional if each corresponding part, when divided, gives the same value; for example, a : b : c : d :: p : q : r : s are proportional if a/p = b/q = c/r = d/s.

In practical life, suppose paint has to be mixed in ratio Red : Blue : White :: 2 : 3 : 5, and you already have 10 litres of white paint. Since white is 5 parts, 1 part = 2 litres. Thus, you need 4 litres of red and 6 litres of blue to keep the colour shade consistent. Similarly, in construction, if you have the cement : sand : gravel ratio as 1 : 1.5 : 3, and you use 3 bags of cement, you must use 4.5 bags of sand and 9 bags of gravel.

Dividing a Whole in a Given Ratio Frequently, problems ask you to split a whole quantity into parts using a specific ratio. For example, splitting 12 in the ratio 2 : 1 means you first add the ratio numbers (2 + 1 = 3), then divide 12 by 3 to get 4, and finally multiply to get 8 and 4 as parts.

This rule works for any number of parts. If you have a sum x to split among a : b : c, the formula is x × (a/total), x × (b/total), x × (c/total). For example, if 110 units of concrete are to be split in the ratio 1 : 1.5 : 3, the total is 5.5. So, cement receives 20 units, sand gets 30 units, and gravel is assigned 60 units. If you are making 50 ml of purple paint in a 2 : 3 : 5 ratio, red will be 10 ml, blue is 15 ml, and white is 25 ml.

This method can also be used in geometry. To construct a triangle whose angles are in the ratio 1 : 3 : 5, sum the parts (1 + 3 + 5 = 9), divide 180° by 9 to get 20°, then multiply each part: the triangle will have angles of 20°, 60°, and 100°.

• A cricket coach divides 150 minutes of practice among activities in the ratio 3 : 4 : 3 : 5.
• Library contains books in Hindi, English, and Odiya in ratio 3 : 2 : 1. If there are 288 Odiya books, the numbers of Hindi and English books can be calculated similarly.
• 100 coins are divided in the ratio 4 : 3 : 2 : 1 for ₹10, ₹5, ₹2, ₹1 coins respectively; you can find the total value using ratios.
• Triangles with side lengths in ratio 3 : 4 : 5 will always be similar but not necessarily congruent.
• It is impossible to construct a triangle with side ratios of 1 : 3 : 5 due to triangle inequality.

Pie Charts and Proportion Pie charts visually show how a whole is split into proportional sectors. For example, if student grades are distributed as A: 12, B: 10, C: 8, D: 6, E: 4, you first simplify the ratios (dividing by 2 gives 6 : 5 : 4 : 3 : 2) with a total of 20 parts. Each part gets 18°, so A = 108°, B = 90°, C = 72°, D = 54°, and E = 36°. Drawing a pie chart helps you understand data organisation by measuring angles for each sector.

Pie charts are common for survey data, such as distributing votes among favourite seasons or student subject preferences. To draw, make a circle, mark a starting line (radius), then use a protractor to measure each angle based on the calculated degrees for the given ratios.

Inverse Proportions In many situations, two quantities are in inverse proportion: as one increases, the other decreases, and their product stays constant. If 5 workers shift 4,500 bricks in a day, then to move 18,000 bricks, we set up the ratio 4500 : 18000 = 5 : x. Using the rule of three, x = (18000 × 5)/4500 = 20 workers.

Another common example is speed and time: higher speed means less time for the same distance. Suppose travelling from Lucknow to Kanpur takes 3 hours at 30 km/h, but only 1.5 hours at 60 km/h. The formula x₁y₁ = x₂y₂ applies: if you double the speed, time halves. The same principle works for work assignments (workers and days), pumps and time, or students and days for available food.

When several people work together at different speeds, you can add their rates to find the time needed for one unit of work. For example, if Ram does a job in 1 hour (1 unit/hr) and Shyam in 1.5 hours (2/3 unit/hr), together their rate is 5/3 units/hr, so they finish 1 unit of work in 3/5 hour.

1. Given values of x and y, check if they are in inverse proportion by multiplying each pair. If the product stays the same, they are inversely related.
2. If x: 16, 12, 36; y: 9, __, 48, fill in the missing value so the products remain constant.
3. 20 workers can finish laying a road in 4 days. With 10 workers, the work will take 8 days.
4. 2 pumps fill a tank in 18 hours; 4 pumps will fill it in 9 hours.
5. 80 students have food for 15 days. If 20 more join, the food lasts for 12 days.
6. Ram and Shyam, working at different rates, finish a vegetable-cutting job in less time when together.

Summary

• Ratios compare quantities in forms like a : b : c : d.
• To divide a quantity among any number of parts, multiply the total by each ratio term divided by the sum of all terms.
• In direct proportion, quantities increase or decrease together such that (x/y) = constant.
• In inverse proportion, when one value multiplies by n, the other divides by n so their product remains constant (xy = constant).

Class 8 Maths Chapter 3 Notes – Proportional Reasoning 2: Key Revision Points

These Class 8 Maths Chapter 3 notes on Proportional Reasoning 2 summarise key ideas like direct and inverse proportions, ratios with multiple terms, and pie charts. Quickly revise concepts and important formulas for faster exam preparation. All sections use easy examples and stepwise explanations to boost your confidence.

With these CBSE Class 8 Proportional Reasoning 2 notes, you can understand map scales, solve ratio word problems, and practise real-life applications—essential for scoring well. Clear tables and methods help you quickly recall and apply important proportion rules during your Maths exams.