Class 8 Maths Part 2 Chapter 3: Proportional Reasoning-2 Notes

What is this chapter about?

Ratios turn up in the kitchen, on maps, on building sites and in survey results. This chapter picks up where Proportional Reasoning-1 stopped. It reads the scale on a map, works with ratios of three, four or more quantities, splits a whole in a given ratio, draws pie charts, and meets a new kind of proportion, inverse proportion, in which one quantity shrinks as the other grows.

Proportion: a quick recap

Two quantities are in a proportional relationship when they change by the same factor. Idli batter can use 2 cups of rice to 1 cup of urad dal. Viswanath uses 6 cups to 3, and Puneet 4 cups to 2. Are 6 : 3 and 4 : 2 proportional? Cross-multiply: 6 × 2 = 12 and 3 × 4 = 12. The products match, so the ratios are proportional, and if everything else is in proportion too, the idlis should taste alike. In general a : b :: c : d when a × d = b × c.

Ratios in maps

Most maps carry a ratio such as 1 : 60,00,000 in a corner. This representative fraction (RF) compares a distance on the map with the real distance on the ground: 1 cm on the map stands for 60,00,000 cm, which is 60 km. Measure two cities with a ruler and multiply: about 4.8 cm between Bengaluru and Chennai means about 290 km. This is the straight-line, geographical distance; the road is longer. A classroom plan drawn at 1 : 50 shows each metre as 2 cm.

Ratios with more than two terms

Viswanath grinds a spice powder from coriander seeds (8 spoons), red chillies (4), toor dal (2 spoons) and fenugreek seeds (1 spoon): the ratio 8 : 4 : 2 : 1. Puneet has only 2 chillies, half as many, so he halves everything: 4 : 2 : 1 : 0.5. The two are proportional, written 8 : 4 : 2 : 1 :: 4 : 2 : 1 : 0.5, and in general a : b : c : d :: p : q : r : s when a ÷ p = b ÷ q = c ÷ r = d ÷ s. Paint mixed red : blue : white = 2 : 3 : 5 with 10 litres of white needs 4 litres of red and 6 of blue. Read ratios with more than two terms.

Dividing a whole in a given ratio

To split 12 in the ratio 2 : 1, add the terms (3), divide (12 ÷ 3 = 4) and multiply: 8 and 4. The same works with more terms: each part of x in the ratio a : b : c is x × a ÷ (a + b + c), and so on. Concrete for pillars and roofs is cement : sand : gravel = 1 : 1.5 : 3, so 110 units need 20 units of cement, 30 of sand and 60 of gravel. If a triangle’s angles are in the ratio 1 : 3 : 5, they measure 20°, 60° and 100°.

Pie charts

A pie chart shows how a whole is shared out. Each slice’s angle is proportional to its share, and the whole circle is 360°. For 40 students with grades A to E scored by 12, 10, 8, 6 and 4 students, the ratio simplifies to 6 : 5 : 4 : 3 : 2, which is 20 parts of 18°. The slices are 108°, 90°, 72°, 54° and 36°. Draw a radius, measure the first angle with a protractor, then each next angle from the last line drawn, and colour and label the slices. Read pie charts.

Inverse proportion

Going from Lucknow to Kanpur takes 18 hours on foot at 5 km/h, 6 hours by bicycle at 15 km/h, 3 hours by motorcycle at 30 km/h and 1.5 hours by car at 60 km/h. When the speed is multiplied by 3, the time is divided by 3. The two change by the same factor but in opposite directions: that is inverse proportion. Their product stays fixed, here 90 km, the distance: x × y = k, so x₁ × y₁ = x₂ × y₂. Fewer workers need more days, more pumps fill a tank faster, and more students make the food run out sooner. Read inverse proportion.

Working together

Ram cuts a heap of vegetables in 1 hour and Shyam in 1.5 hours. In one hour Ram does the whole job and Shyam two-thirds of it, so together they do 5/3 of the job each hour. The work done and the time taken are in direct proportion, so one whole job takes 3/5 of an hour, which is 36 minutes.

In a nutshell

A ratio of many terms means every quantity scales by the same factor; dividing a whole uses each term’s share of the sum; a pie chart gives each share of 360°; and in inverse proportion the product, not the quotient, stays the same. Answers to every question are in the NCERT solutions for Part 2 Chapter 3.