Ganita Prakash I · Chapter 7 — Proportional Reasoning-1

You have surely resized a photo on a phone. Sometimes the picture still looks right; sometimes faces look stretched or squashed. What decides this?

An experiment with five pictures. Suppose one photo is printed in five sizes:

Compare A and C. C's width is half of A's (60 → 30) and C's height is also half of A's (40 → 20). Both measurements were multiplied by the same factor, ½. So C looks like a smaller copy of A.

Compare A and B. B's width is 20 mm less than A's, and its height is also 20 mm less. The same amount was subtracted from both. But B looks wrong! Its height became half (40 → 20) while its width became only two-thirds (60 → 40). The factors are different, so the shape changes.

Compare A and D. 60 → 90 and 40 → 60. Both are multiplied by 1½ (that is, 3/2). So D looks like A, only bigger.

Picture E is a square. Its height grew by a factor of 1½ while its width did not change at all, so it looks squashed.

The big idea. When two quantities change by the same multiplying factor, we say the changes are proportional, and the shape is kept. Adding or subtracting the same amount from both does not keep the shape (unless both started equal).

Where we meet this. Enlarging a map or a drawing, scaling a recipe, making a model of a building, mixing paint to get the same shade, and printing a flag of a different size all need proportional change. In the Indian national flag, for example, the width and height must always be in the same fixed relationship, whatever size it is made.

In the rest of this chapter we learn a precise language for this idea: ratios and proportions, and a powerful method for solving problems with them.

A ratio compares two quantities of the same kind by division. The width-to-height ratio of picture A is written 60 : 40 and read ‘60 to 40’. The numbers 60 and 40 are the terms of the ratio.

Meaning. The ratio a : b tells us that for every a units of the first quantity there are b units of the second. In picture A, for every 60 mm of width there are 40 mm of height. A ratio a : b has the same value as the fraction a/b.

Order matters. Width to height is 60 : 40; height to width is 40 : 60. These are different ratios. Always write the terms in the order the question asks.

Simplest form. Just as we reduce a fraction, we can reduce a ratio by dividing both terms by their HCF (highest common factor). The ratio does not change its value, just like 60/40 = 3/2.

Pictures A, C and D all reduce to 3 : 2. That is exactly why they look alike. B (2 : 1) and E (1 : 1) are different, and they look distorted.

Same units. The two quantities must be measured in the same unit before you write the ratio. The ratio of 50 cm to 2 m is not 50 : 2. Convert first: 2 m = 200 cm, so the ratio is 50 : 200 = 1 : 4. A ratio of two quantities of the same kind has no unit.

More examples.

• A school has 5 teachers and 170 students. Teachers to students = 5 : 170 = 1 : 34 (HCF 5).
• Orange juice 600 mL and apple juice 900 mL: 600 : 900 = 2 : 3 (HCF 300).
• A class has 18 girls and 24 boys: girls to boys = 3 : 4, and girls to the whole class = 18 : 42 = 3 : 7.

Ratio or difference? If Raju has ₹30 and Mona has ₹10, we can say Raju has ₹20 more (difference) or that Raju has 3 times as much (ratio 3 : 1). Ratios answer the question ‘how many times?’, which is what matters for shape, taste and strength.

Two ratios are proportional (or in proportion) if they are the same in their simplest forms. We write this with the symbol ::. So a : b :: c : d means the ratios a : b and c : d are equal, and we read it as ‘a is to b as c is to d’.

From the pictures: 60 : 40 :: 30 : 20 and 60 : 40 :: 90 : 60, since all three reduce to 3 : 2.

Two ways to check.

• Simplest form. Reduce both ratios. 3 : 4 and 72 : 96: HCF of 72 and 96 is 24, so 72 : 96 = 3 : 4. Proportional.
• Same factor. See if one ratio is the other with both terms multiplied by the same number. 4 : 7 and 12 : 21: 4 × 3 = 12 and 7 × 3 = 21. Proportional.

Some tests.

Finding a missing term. Find ratios proportional to 14 : 21 with one term given.

• ___ : 42. Since 42 = 21 × 2, the first term is 14 × 2 = 28. So 28 : 42.
• 6 : ___. The factor taking 14 to 6 is 6/14 = 3/7 (a fraction is fine). So the second term is 21 × 3/7 = 9. So 6 : 9.
• 2 : ___. Divide 14 : 21 by the HCF 7 to get 2 : 3.

Making equivalent ratios. Multiply both terms of 4 : 9 by 2, 3 and 4 to get 8 : 18, 12 : 27 and 16 : 36. All are proportional to 4 : 9. There are infinitely many.

Adding does not work. When Neelima was 3, her mother was 30: ratio 3 : 30 = 1 : 10. Nine years later they are 12 and 39: ratio 12 : 39 = 4 : 13. Adding the same 9 years to both ages changed the ratio. Only multiplying (or dividing) both terms by the same number keeps a ratio the same.

Use in drawing. If your blackboard is 360 cm by 120 cm (3 : 1), a drawing of it in your notebook that is 9 cm by 3 cm keeps the same look.

Many everyday questions are really questions about proportional ratios.

Sweet enough lemonade. Kesang made 6 glasses of lemonade with 10 spoons of sugar. She now has to make 18 glasses with the same sweetness. Glasses to spoons must stay proportional: 6 : 10 :: 18 : ?. The first term went from 6 to 18, a factor of 3. So the spoons go from 10 to 30. She needs 30 spoons.

Is the wall weaker? Nitin built a 60 ft wall with 3 bags of cement; Hari built a 40 ft wall with 2 bags. Nitin worries that Hari used less cement. Compare the ratios: 60 : 3 = 20 : 1 and 40 : 2 = 20 : 1. Both walls used one bag for every 20 ft, so they are equally strong. A smaller amount does not mean a weaker mix; the ratio is what counts.

Stronger or lighter coffee? A shop makes regular filter coffee with 15 mL of decoction and 35 mL of milk (3 : 7). For each mix below, compare the decoction-to-milk ratio with 3 : 7. One easy way is to find the decoction per 1 mL of milk, the value of the ratio.

A bigger cup is not a stronger cup. The 300 : 600 mix is stronger than regular because it has more decoction for each mL of milk, not because it is bigger.

Patterns in a wall. In a brick wall where each repeating block has 9 grey bricks and 6 coloured bricks, the ratio of grey to coloured is 9 : 6 = 3 : 2. Because the pattern repeats, the whole wall has the same ratio.

How to approach such problems.

• Decide which two quantities must stay in the same ratio.
• Write the ratios in the same order and units.
• Compare their simplest forms, or find the factor of change and apply it to the other term.
• Check that your answer makes sense: more glasses should need more sugar.

Proportional thinking is about relationships, not single numbers. Always ask: compared to what?

A school cook makes 15 kg of rice for 120 students. One rainy day only 80 students come. How much rice should be cooked so that none is wasted?

Students to rice must stay proportional: 120 : 15 :: 80 : ?. The number of students changed by a factor of 80/120 = 2/3. So the rice must be 15 × 2/3 = 10 kg.

Here three quantities are known and a fourth, linked to them proportionally, is unknown. Such problems are very common. Let us find a general method.

Cross multiplication. Suppose a : b :: c : d. Then c = fa and d = fb for the same factor f. So f = c/a and f = d/b, which gives c/a = d/b. Multiplying both sides by ab:

The product of the outer terms equals the product of the inner terms. This also gives a quick test: 4 : 7 :: 12 : 21 is true because 4 × 21 = 84 = 7 × 12.

The Indian Rule of Three. Indian mathematicians called this trairāśika (three quantities). Āryabhaṭa, in the 5th century CE, named the three known numbers pramāṇa (the measure, a), phala (the fruit or yield of the measure, b) and icchā (the requisition, c). The answer is the icchā-phala (d). His rule: multiply the phala by the icchā and divide by the pramāṇa, that is, d = (b × c) ÷ a. This method was used for trade, astronomy and taxes for centuries.

Example: the car. A car goes 90 km in 150 minutes. At the same speed, how far does it go in 4 hours? First make the units match: 4 hours = 240 minutes. Then 150 : 90 :: 240 : x, so 150 × x = 90 × 240 and x = 21600 ÷ 150 = 144 km.

Example: bricks. A 10 ft wall needs about 1450 bricks. A house needs 108 ft of wall of the same height and thickness. 10 : 1450 :: 108 : x gives x = 1450 × 108 ÷ 10 = 15660 bricks.

Other ways to think. You do not have to cross multiply every time. In the car problem you can find the distance in 30 minutes (90 ÷ 5 = 18 km) and multiply by 8 (there are eight 30-minute blocks in 240 minutes) to get 144 km. This ‘unit’ way of thinking and cross multiplication give the same answer.

Proportional reasoning goes wrong if we are careless with units or if we assume proportion where there is none.

Same units, again. A farmer in Himachal Pradesh sells a 200 g packet of tea for ₹200. A big estate in Meghalaya sells a 1 kg packet for ₹800. Is the weight-to-price ratio the same? Writing 1 : 800 for Meghalaya would mix kilograms with grams. In grams, the ratios are 200 : 200 = 1 : 1 and 1000 : 800 = 5 : 4. They are not proportional.

Which tea costs more? Compare the price of the same weight. 200 g is 1/5 of a kilogram, so 1 kg of Himachal tea costs 5 × ₹200 = ₹1000. Meghalaya tea costs ₹800 per kg. The Himachal tea is more expensive.

Bigger packs are often not proportional. Here are sample prices for one shampoo:

Price is not proportional to volume: the large bottle is much cheaper per mL. Companies save on packing and transport for big packs. Comparing the unit price (price per mL, per kg or per item) is the fair way to compare. Small sachets also create far more plastic waste for the same amount of shampoo.

Comparing crowding. Delhi has an area of about 1484 km² and a population of about 30 million; Mumbai has about 550 km² and 20 million. People per km²: Delhi about 20,200, Mumbai about 36,400. Mumbai is more crowded, even though it has fewer people, because they live in a smaller area.

When the Rule of Three does not work. Puneeth's father rides from Lucknow to Kanpur in 2 hours at 50 km/h. How long at 75 km/h? Writing 50 : 2 :: 75 : ? would give 3 hours, which is absurd: going faster cannot take longer! When speed goes up, time goes down. Speed and time here are not in direct proportion, so the Rule of Three in this form does not apply. (The distance is 100 km, so at 75 km/h the ride takes 100 ÷ 75 = 1⅓ hours, that is, 1 hour 20 minutes.) Such ‘inverse’ relationships are studied later.

Always ask: If one quantity doubles, should the other double too? If yes, use proportion. If not, think again.

Two friends share 12 marbles in the ratio 3 : 1. One way: the first takes 3 and the second takes 1, then again 3 and 1, and again 3 and 1. The marbles are finished. The first friend has 9 and the second has 3. And 9 : 3 = 3 : 1.

This works, but it is slow for large numbers. Let us find a quicker way.

Groups. In the ratio 3 : 1, each round uses 3 + 1 = 4 marbles. Since 12 is 3 times 4, there are 3 rounds, so the parts are 3 × 3 = 9 and 3 × 1 = 3.

Example. Share 42 counters in the ratio 4 : 3. Total parts = 4 + 3 = 7. Size of one part = 42 ÷ 7 = 6. The shares are 4 × 6 = 24 and 3 × 6 = 18. Check: 24 + 18 = 42 and 24 : 18 = 4 : 3.

Sharing profit. Prashanti invested ₹75,000 and Bhuvan ₹25,000 in a food cart. The month's profit of ₹4,000 is shared in the ratio of their investments, 75000 : 25000 = 3 : 1. One part = 4000 ÷ 4 = ₹1000. Prashanti gets ₹3000 and Bhuvan gets ₹1000.

Mixtures. A lab solution has acid and water in the ratio 1 : 5. A 240 mL bottle has 240 ÷ 6 = 40 mL per part: 40 mL of acid and 200 mL of water.

Changing a mixture. 40 kg of a mixture has sand and cement in the ratio 3 : 1. That is 30 kg of sand and 10 kg of cement. How much cement must be added to make the ratio 5 : 2? The sand stays 30 kg. We need 5 : 2 :: 30 : ?, so the cement must be 30 × 2/5 = 12 kg. Add 12 − 10 = 2 kg of cement.

Adding one ingredient. Green paint is mixed blue to yellow 3 : 5. For 40 mL we use 15 mL blue and 25 mL yellow. If 20 mL more yellow is added, the new ratio is 15 : 45 = 1 : 3, a lighter green.

Always check: the shares should add up to the whole, and their ratio should be the given ratio.

Many real proportion problems mix units: feet and metres, acres and square feet, mL and litres. Here are some useful conversions.

Length, area and volume conversions are proportional: 2 metres is twice as many feet as 1 metre. Temperature is different. 0 °C is 32 °F, not 0 °F, so the scales do not start together. That is why the formula has a ‘+ 32’. Example: 25 °C = (9/5) × 25 + 32 = 45 + 32 = 77 °F. And 212 °F = (5/9) × 180 = 100 °C, the boiling point of water.

Problem 1: the bucket. A tap fills a 500 mL mug in 15 seconds. How long for a 10 litre bucket? Same units: 10 litres = 10,000 mL. 500 : 15 :: 10000 : t gives t = 15 × 10000 ÷ 500 = 300 seconds, that is, 5 minutes.

Problem 2: manure. Good practice is 10 tonnes of manure per acre. A plot is 200 ft by 500 ft. Its area is 100,000 square feet = 100000 ÷ 43560 ≈ 2.30 acres. Manure needed ≈ 10 × 2.30 ≈ 23 tonnes.

Problem 3: buses. Last year 3 full buses carried 162 people, so each bus holds 54. This year 204 people go: 204 ÷ 54 ≈ 3.8, so 4 buses are needed, and they will not all be full (4 × 54 = 216, 12 seats spare).

Problem 4: gold and water. Equal volumes of gold and water have masses in the ratio 37 : 2 (roughly). 1 litre of water has mass 1 kg, so 1 litre of gold has mass 37/2 = 18.5 kg.

Problem 5: the coin. A ₹10 coin of mass 7.74 g is made of copper and nickel in the ratio 3 : 1. Copper = 7.74 × 3/4 ≈ 5.805 g and nickel ≈ 1.935 g. At ₹906 per kg of copper and ₹1341 per kg of nickel, the metal costs about 5.26 + 2.59 ≈ ₹7.85.

Strategy for long problems. Write down what is given with units, convert to one unit, set up the proportion, solve, and then ask whether the answer is sensible.