Proportional Reasoning

There are two honest ways to compare two numbers. You can subtract, which tells you how much more one is than the other, or you can divide, which tells you how many times one is of the other. The second kind of comparison is called a ratio, and everything in this chapter grows out of it.

Why subtraction is sometimes the wrong tool. A shop sells a pen for ₹ 10 and a bag for ₹ 500. Next month the pen costs ₹ 12 and the bag costs ₹ 502. Both went up by ₹ 2, so by subtraction the two increases look identical. But the pen now costs one and a fifth of what it used to, while the bag has barely moved. Dividing catches what subtracting misses: 12 ÷ 10 = 1.2, but 502 ÷ 500 = 1.004. Whenever you want to judge value for money, crowding or fairness, divide.

How a ratio is written. The ratio of a to b is written a : b and read as a to b. It means exactly the fraction a/b, so 3 : 4 and 3/4 are two ways of writing the same comparison. In 3 : 4 the first number 3 is called the first term or antecedent, and 4 is the second term or consequent.

Same units first, always. A ratio compares like with like, so before you write anything down, put both quantities into the same unit. The ratio of 750 g to 3 kg is not 750 : 3. Change 3 kg into 3000 g, and the ratio becomes 750 : 3000, which is 1 : 4. Once the units match, they cancel when you divide, and that is why a ratio itself carries no unit — it is simply a number in disguise.

Order matters. The ratios 3 : 5 and 5 : 3 are different comparisons, in the same way that 3/5 and 5/3 are different numbers. If a question asks for the ratio of girls to boys, the number of girls must be written first, every time.

Worked example: which school is more crowded? School A has 300 students and 12 teachers. School B has 500 students and 25 teachers. School B has more students, but that alone settles nothing. Compare by ratio instead. For A, 300 : 12 = 25 : 1, so there are 25 students for every teacher. For B, 500 : 25 = 20 : 1, so 20 students per teacher. School A is the more crowded one, even though it is the smaller school. This is the real point of a ratio: it strips size out of the comparison so that two very different-sized things can be judged side by side.

Part to part, and part to whole. In a class of 40 children there are 24 girls and 16 boys. The ratio of girls to boys is 24 : 16 = 3 : 2, dividing both terms by 8. That is a part-to-part ratio. The ratio of girls to the whole class is 24 : 40 = 3 : 5, which is a part-to-whole ratio. These are different answers to different questions, and it is an easy slip to give one when the other was asked for. One quick safety check: in a part-to-whole ratio the second term is the total, so it can never be smaller than the first.

One more from daily life. Ravi earns ₹ 15,000 a month and spends ₹ 12,000 of it. His savings are 15000 - 12000 = ₹ 3,000. So savings : income = 3000 : 15000 = 1 : 5, and expenditure : income = 12000 : 15000 = 4 : 5. Notice that these two part-to-whole fractions, 1/5 and 4/5, add up to 1, exactly as the two parts add up to the whole income. That is a useful way to check any pair of part-to-whole ratios you write down.

Two ratios are equivalent when they are the same comparison written with different numbers. Because a ratio is a fraction, you build an equivalent ratio exactly as you build an equivalent fraction: multiply or divide both terms by the same non-zero number.

Building a chain. Start with 2 : 3. Multiplying both terms by 2, by 3 and by 10 gives 4 : 6, 6 : 9 and 20 : 30. Every one of these is the same comparison, because 2/3 = 4/6 = 6/9 = 20/30. If a recipe needs sugar and flour in the ratio 2 : 3, then 2 cups to 3 cups, 4 cups to 6 cups and 20 cups to 30 cups all taste the same. What ruins a ratio is adding: going from 2 : 3 to 3 : 4 by adding 1 to each term changes the comparison completely, since 2/3 is not 3/4.

Simplest form. A ratio is in its simplest, or lowest, form when its two terms share no common factor except 1. To get there, divide both terms by their HCF. For 45 : 60 the HCF is 15, so 45 : 60 = 3 : 4. For a harder pair such as 84 : 126, factorise first: 84 = 2 × 2 × 3 × 7 and 126 = 2 × 3 × 3 × 7, so the HCF is 2 × 3 × 7 = 42. Dividing both terms by 42 gives 84 : 126 = 2 : 3.

Finding a missing term. If 3 : 5 = 12 : x, ask what turned 3 into 12. It was multiplication by 4, so the same must happen to 5, giving x = 20. The same answer comes from cross-multiplying: 3x = 5 × 12 = 60, so x = 20.

Comparing two ratios. Which is greater, 5 : 8 or 7 : 11? Turn both into fractions over a common denominator. The LCM of 8 and 11 is 88, so 5/8 = 55/88 and 7/11 = 56/88. Since 56/88 is the larger, 7 : 11 is the greater ratio. The shortcut is cross-multiplication: compare 5 × 11 = 55 against 7 × 8 = 56, and the bigger product sits on the side of the bigger ratio. Never try to compare 5 : 8 with 7 : 11 by looking at the first terms alone; 7 is bigger than 5, but that on its own proves nothing.

Dividing a quantity in a given ratio. This is the most useful skill in the section. To divide ₹ 3,500 between two people in the ratio 3 : 4, first add the parts: 3 + 4 = 7 equal parts altogether. One part is therefore 3500 ÷ 7 = ₹ 500. The shares are 3 × 500 = ₹ 1,500 and 4 × 500 = ₹ 2,000. Check twice, as you should in every sharing question: the shares add back to 1500 + 2000 = ₹ 3,500, and 1500 : 2000 does simplify to 3 : 4.

Three or more parts. The method does not change at all. Share 60 sweets among three children in the ratio 2 : 3 : 5. The total number of parts is 2 + 3 + 5 = 10, so one part is 60 ÷ 10 = 6 sweets. The three shares are 12, 18 and 30, and 12 + 18 + 30 = 60, as it must. A faster version of the same idea uses fractions directly: the middle child gets 3/10 of 60 = 18 sweets, because that child holds 3 of the 10 parts.

A ratio compares two quantities. A proportion is a statement that two ratios are equal. When a : b and c : d are the same comparison, the four numbers are said to be in proportion, and this is written a : b :: c : d, read as a is to b as c is to d.

Extremes and means. In a : b :: c : d the outer two numbers, a and d, are the extremes, and the inner two, b and c, are the means. Because the statement really says a/b = c/d, cross-multiplying gives the rule that does all the work here: the product of the extremes equals the product of the means, a × d = b × c.

Testing four numbers. Are 4, 6, 10, 15 in proportion, in that order? Extremes: 4 × 15 = 60. Means: 6 × 10 = 60. The two products match, so yes, 4 : 6 :: 10 : 15. You can see the same thing by simplifying each ratio separately: 4 : 6 = 2 : 3 and 10 : 15 = 2 : 3. Now try 2, 5, 8, 18. Extremes: 2 × 18 = 36. Means: 5 × 8 = 40. The products differ, so these four numbers are not in proportion.

Order matters here too. The numbers 4, 6, 10, 15 are in proportion in that order, but 4, 6, 15, 10 are not: the extremes now give 4 × 10 = 40 while the means give 6 × 15 = 90. Swapping the last two numbers destroyed the proportion, so never shuffle the given numbers to make an answer come out.

Finding a missing term. Suppose 8 : 12 :: 14 : x. The product of the extremes is 8 × x, and the product of the means is 12 × 14 = 168. Setting them equal gives 8x = 168, so x = 168 ÷ 8 = 21. Check by simplifying: 8 : 12 = 2 : 3 and 14 : 21 = 2 : 3, so the proportion does hold.

Proportion at the shop. If 6 notebooks cost ₹ 210, what will 11 identical notebooks cost? The comparison between notebooks and rupees does not change, so 6 : 210 :: 11 : x. Then 6x = 210 × 11 = 2310, giving x = 2310 ÷ 6 = ₹ 385. Set the statement up carefully: notebooks on the left of both colons, rupees on the right of both. Writing 6 : 210 :: x : 11 would compare notebooks against rupees in one ratio and rupees against notebooks in the other, and the answer would be meaningless.

Continued proportion. Three quantities a, b, c are in continued proportion when a : b = b : c, which by the same cross-multiplication means b × b = a × c. The numbers 4, 8, 16 are in continued proportion, because 4 : 8 = 1 : 2 and 8 : 16 = 1 : 2; checking by products, 8 × 8 = 64 and 4 × 16 = 64. The repeated middle term b is called the mean proportional between a and c. To find the mean proportional between 9 and 25, solve b × b = 9 × 25 = 225, so b = 15.

Two safe rearrangements. Since a proportion is really the equation a/b = c/d, you may turn both ratios upside down to get b : a :: d : c, or swap the two middle quantities to get a : c :: b : d, and the statement stays true. Starting from 4 : 6 :: 10 : 15, those give 6 : 4 :: 15 : 10 and 4 : 10 :: 6 : 15. Both survive the product test: 6 × 10 = 60 = 4 × 15, and 4 × 15 = 60 = 10 × 6.

The unitary method is the plainest problem-solving tool in arithmetic: first find the value of one unit, then find the value of as many units as the question asks for. Two steps, in that order, and nearly every rate question at this level falls to it.

Step 1 divides, step 2 multiplies. If 9 pens cost ₹ 144, then one pen costs 144 ÷ 9 = ₹ 16. That is step 1. Now 13 pens cost 13 × 16 = ₹ 208. That is step 2. Notice the shape of it: going from many to one is a division, and going from one to many is a multiplication. Students who hold on to that shape rarely go wrong.

Working the other way round. Sometimes you are given an amount of money and asked for a quantity. If ₹ 84 buys 7 apples, then one apple costs 84 ÷ 7 = ₹ 12, so ₹ 132 buys 132 ÷ 12 = 11 apples. Step 1 is unchanged, but step 2 is now a division, because the question is really asking how many twelves fit into 132.

Write the units down. The best protection against dividing the wrong way round is to carry the units through the working. In the apple question, ₹ 84 ÷ 7 apples gives ₹ 12 per apple, and then ₹ 132 ÷ ₹ 12 per apple gives 11 apples. If the units of your final answer are not the units the question asked for — rupees when it wanted a number of apples, say — you know at once that a step has gone the wrong way.

Worked example: fuel. A car travels 240 km on 15 litres of petrol. How far will it go on 22 litres at the same mileage? One litre takes it 240 ÷ 15 = 16 km. Then 22 litres take it 22 × 16 = 352 km.

Worked example: shopping. If 15 kg of rice costs ₹ 1,050, what will 8 kg cost? One kilogram costs 1050 ÷ 15 = ₹ 70, so 8 kg cost 8 × 70 = ₹ 560. Then sanity-check the size of the answer: 8 kg is a little more than half of 15 kg, and ₹ 560 is a little more than half of ₹ 1,050, so the answer sits where it should.

Two routes, one answer. The unitary method and the proportion method are the same calculation wearing different clothes. For the rice, proportion gives 15 : 1050 :: 8 : x, so 15x = 1050 × 8 = 8400 and x = 8400 ÷ 15 = ₹ 560, the same as before. Use whichever you find clearer. The unitary method has one quiet advantage: its middle step, ₹ 70 per kg, is a number you can weigh against real life, so a silly answer announces itself early.

When the unitary method must not be used. It assumes every unit costs the same. If an auto charges a fixed ₹ 30 the moment you sit down and then ₹ 12 for each kilometre, a 1 km ride costs 30 + 12 = ₹ 42 and a 2 km ride costs 30 + 24 = ₹ 54, not 2 × 42 = ₹ 84. Treating ₹ 42 as the rate per kilometre is wrong, because part of that money was never charged per kilometre at all. The same warning covers offers such as buy two get one free, and bulk rates where a larger packet works out cheaper per kilogram. Before dividing, ask yourself whether every unit really does cost the same.

Two quantities are in direct proportion when increasing one multiplies the other by the same factor: double the first and the second doubles, treble the first and the second trebles. The precise test is that the ratio of the two never changes. If x and y are in direct proportion, then y ÷ x gives the same number for every matching pair of values, and that fixed number is called the constant of proportionality, usually written k.

The test on a table. A petrol pump gives these readings: 2 litres for ₹ 220, 3 litres for ₹ 330, 5 litres for ₹ 550 and 8 litres for ₹ 880. Divide each cost by its volume: 220 ÷ 2 = 110, 330 ÷ 3 = 110, 550 ÷ 5 = 110 and 880 ÷ 8 = 110. The ratio is 110 every single time, so cost is in direct proportion to volume, with k = 110 rupees per litre. The relation can be written as cost = 110 × litres.

Using the constant to predict. Once you know k, every other question about that table is one step away. The cost of 14 litres is 14 × 110 = ₹ 1,540, and ₹ 715 buys 715 ÷ 110 = 6.5 litres. You need not find k at all if you prefer the paired form x₁/y₁ = x₂/y₂, which simply says that any two rows of the table give the same ratio.

What the graph looks like. Plot volume across and cost up, and the four points lie on a single straight line — and that line passes through the origin, because zero litres cost zero rupees. Both features have to be there. A straight line that misses the origin is not a direct proportion, and neither is a curve of any shape.

Worked example. Eight identical books weigh 2.4 kg. What do 15 such books weigh? One book weighs 2.4 ÷ 8 = 0.3 kg, so 15 books weigh 15 × 0.3 = 4.5 kg. Written as a proportion instead, 8/2.4 = 15/x gives 8x = 2.4 × 15 = 36, so x = 36 ÷ 8 = 4.5 kg. Two different routes must agree, and they do.

Maps and scales are direct proportion. A map drawn to a scale of 1 cm to 25 km is saying that map distance and real distance are in direct proportion with k = 25 km per cm. Two towns 6.5 cm apart on the map are really 6.5 × 25 = 162.5 km apart, and towns that are 200 km apart on the ground appear 200 ÷ 25 = 8 cm apart on the map.

Pairs that are not in direct proportion. Three traps are worth learning by heart:

• An auto fare of ₹ 30 fixed plus ₹ 12 per km. A 1 km ride costs ₹ 42 and a 2 km ride costs ₹ 54, but 42 ÷ 1 = 42 while 54 ÷ 2 = 27, so the ratio is not constant.
• The side of a square and its area. A side of 3 cm gives 9 cm², while a side of 6 cm gives 36 cm². Doubling the side made the area four times as big, not twice as big.
• The number of workers and the days needed to finish a fixed job. More workers means fewer days, so the two quantities move in opposite directions, and that is a different relationship altogether.

A person’s age and height rise together for some years, but not in a constant ratio, so that pair fails the test as well. The lesson is worth repeating: both quantities getting larger is not enough. The ratio has to stay fixed.

A percentage is a ratio whose second term has been fixed at 100. The word comes from the Latin per centum, meaning for every hundred, and the symbol % is shorthand for out of 100. So 37% means 37 out of every 100, which is the ratio 37 : 100 and the fraction 37/100.

Why fix the second term at 100? Because comparison then becomes simple reading. Suppose Aarti scored 18 out of 25 in one test and 27 out of 40 in another. Which was the better performance? The denominators differ, so the two scores cannot be judged by simply looking at them. Rewrite both out of 100. For the first, 18/25 = (18 × 4)/(25 × 4) = 72/100 = 72%. For the second, multiply top and bottom by 2.5 to get 27/40 = 67.5/100 = 67.5%. The first test was the better one. A percentage is nothing more than a ratio put into a standard uniform so that any two of them can be lined up and compared instantly.

From a fraction to a percentage. Multiply the fraction by 100 and attach the % sign. So 3/8 becomes (3 ÷ 8) × 100 = 300/8 = 37.5%, and 7/20 becomes 700/20 = 35%. When the division does not terminate, either round or keep the fraction: 5/6 gives 500/6 = 83.33% to two decimal places, which is exactly 83 1/3 %.

From a percentage to a fraction. Write the number over 100 and simplify. So 45% = 45/100 = 9/20, and 12% = 12/100 = 3/25. This also shows plainly that every percentage is a ratio in disguise: 45% is the ratio 45 : 100, which in simplest form is 9 : 20.

Finding a percentage of a quantity. Here the word of means multiply. So 15% of ₹ 2,400 = 15/100 × 2400 = ₹ 360, and 8% of 750 = 8/100 × 750 = 60. There is a fast mental route for many of these: 10% of a number is that number with its decimal point moved one place to the left, and 5% is half of the 10%. For ₹ 2,400, then, 10% is ₹ 240, so 5% is ₹ 120, and 15% is 240 + 120 = ₹ 360, which agrees with the working above.

The parts of a whole add up to 100%. In a school of 800 students, 45% chose Hindi and 30% chose English as their second language, and the rest chose some other language. The percentages have to total 100, so the rest make up 100 - 45 - 30 = 25%. In actual numbers: Hindi 45/100 × 800 = 360 students, English 30/100 × 800 = 240 students, and others 25/100 × 800 = 200 students. Check the total: 360 + 240 + 200 = 800, which is the whole school.

Percentages can go past 100. Nothing stops a percentage at 100, so long as it is not describing a part of a whole. If a shirt cost ₹ 200 last year and ₹ 250 this year, the new price is 250/200 × 100 = 125% of the old price. Percentages can also be very small: 0.5% of 4,000 is 0.5/100 × 4000 = 20. Both of these are perfectly ordinary once you remember that the % sign only ever means divide by a hundred.

A fraction, a decimal and a percentage can all describe the same amount. Three quarters of a cake is 3/4, or 0.75, or 75%, and nothing about the cake changes. Being able to slide between the three forms quickly is what makes percentage questions easy, so all six conversions are gathered here in one place.

Fraction to decimal: divide the top by the bottom. So 3/4 = 3 ÷ 4 = 0.75, and 5/8 = 0.625. Decimal to fraction: write the digits over the matching power of ten, then simplify. So 0.75 = 75/100 = 3/4, and 0.625 = 625/1000 = 5/8.

Decimal to percentage: multiply by 100, which just shifts the decimal point two places to the right. So 0.75 becomes 75%, 0.045 becomes 4.5%, and 1.6 becomes 160%. Percentage to decimal: divide by 100, shifting the point two places to the left. So 6% = 0.06, 125% = 1.25, and 0.5% = 0.005. A very easy slip is shifting the point one place instead of two, which turns 0.045 into 0.45% instead of 4.5%.

Fraction to percentage and back. These appeared earlier, but the two-step route is worth having: turn the fraction into a decimal first, then shift the point. So 5/8 = 0.625 = 62.5%. Going backwards, 62.5% = 62.5/100 = 625/1000 = 5/8.

The conversions worth memorising. A handful of these turn up so often that recognising them on sight saves real time:

• 1/2 = 0.5 = 50%, and 1/4 = 0.25 = 25%, and 3/4 = 0.75 = 75%
• 1/5 = 0.2 = 20%, and 2/5 = 0.4 = 40%, and 3/5 = 0.6 = 60%
• 1/8 = 0.125 = 12.5%, and 3/8 = 0.375 = 37.5%, and 5/8 = 0.625 = 62.5%
• 1/10 = 0.1 = 10%, and 1/20 = 0.05 = 5%, and 1/25 = 0.04 = 4%
• 1/3 = 0.333... = 33 1/3 %, and 2/3 = 0.666... = 66 2/3 %

Expressing one quantity as a percentage of another. Make a fraction with the quantity you are describing on top and the quantity you are comparing against on the bottom, then multiply by 100. What percentage of 48 is 36? That is 36/48 × 100 = 75%. A student who scores 63 out of 75 has scored 63/75 × 100 = 84%. Units must be matched first, exactly as with ratios: 250 g as a percentage of 2 kg is 250/2000 × 100 = 12.5%, and writing 250/2 × 100 would be badly wrong.

Working backwards from a percentage. If 30% of a number is 96, what is the number? Write the sentence as an equation: 30/100 × n = 96, so n = 96 × 100 ÷ 30 = 9600 ÷ 30 = 320. Then check the answer against the question: 30% of 320 = 0.3 × 320 = 96, as required.

A shortcut worth knowing. Because p% of A is (p × A)/100 and A% of p is (A × p)/100, and multiplication can be done in either order, p% of A always equals A% of p. So 8% of 25 is the same as 25% of 8, which is 2 — far easier to do in your head. In the same way, 4% of 50 = 50% of 4 = 2, and 16% of 25 = 25% of 16 = 4. Whenever one of the two numbers is a friendly percentage such as 25 or 50, flip them round.