Ganita Prakash II · Chapter 3 — Proportional Reasoning-2

Idli batter is often made with 2 cups of rice for every 1 cup of urad dal, a ratio of 2 : 1. Viswanath uses 6 cups of rice and 3 cups of dal; Puneet uses 4 cups of rice and 2 cups of dal. Will their idlis taste the same?

Proportional relationship. When two related quantities change by the same factor, we say they are in a proportional relationship. Viswanath multiplied both amounts by 3, and Puneet multiplied both by 2, so both batters keep the ratio 2 : 1. We write 6 : 3 :: 4 : 2 and read it as ‘6 is to 3 as 4 is to 2’.

The cross-multiplication test. Two ratios a : b and c : d are proportional exactly when

a × d = b × c, which is the same as a/b = c/d.

For 6 : 3 and 4 : 2, we get 6 × 2 = 12 and 3 × 4 = 12. The products are equal, so the ratios are proportional.

• Lemonade. 3 spoons of sugar for 2 glasses of water against 9 spoons for 6 glasses: 3 × 6 = 18 and 2 × 9 = 18, so they taste equally sweet.
• Not proportional. 5 : 4 and 6 : 5: 5 × 5 = 25 but 4 × 6 = 24. The products differ, so the ratios are not equal (5 : 4 is slightly larger).

Why does the test work? a : b :: c : d means a/b = c/d. Multiply both sides by b × d and you get a × d = c × b.

Finding a missing term (the rule of three). If a : b :: c : d and three of the numbers are known, cross-multiplying finds the fourth: d = (b × c)/a. For example, if 5 workers move 4500 bricks in a day, then the number of workers for 18,000 bricks satisfies 4500 : 18000 :: 5 : x, so x = 18000 × 5/4500 = 20 workers. This is a direct proportion: more bricks need more workers, in the same ratio.

Direct proportion in symbols. If x and y are directly proportional, then x/y stays the same for every pair of values: x₁/y₁ = x₂/y₂ = k, a constant. Doubling one doubles the other; dividing one by 3 divides the other by 3.

Simplest form. Dividing both terms by their HCF gives the simplest form of a ratio: 6 : 3 becomes 2 : 1 and 12 : 18 becomes 2 : 3. Two ratios are proportional exactly when their simplest forms are the same.

This chapter extends these ideas in three directions: ratios with more than two terms, ratios in maps and pie charts, and a new kind of proportion where one quantity goes up as the other goes down.

Look at the corner of a printed map and you will often see something like 1 : 60,00,000. This is the map’s scale, written as a Representative Fraction (RF). It is the ratio of a distance on the map to the matching distance on the ground, in the same units.

Reading the RF. 1 : 60,00,000 means 1 cm on the map stands for 60,00,000 cm on the ground. Change units: 60,00,000 cm = 60,000 m = 60 km. So every centimetre on this map is 60 km of real distance.

Using it. If two cities are 4.5 cm apart on this map, the real distance is 4.5 × 60 = 270 km. To go the other way, a real distance of 150 km appears as 150 ÷ 60 = 2.5 cm on the map.

Straight-line distance. A map scale gives the geographical distance, measured in a straight line ‘as the crow flies’. The road distance is usually longer, because roads bend around hills, rivers and towns.

Different maps, same answer. A large atlas page might have RF 1 : 25,00,000 (1 cm = 25 km), while a small map of India might have 1 : 1,00,00,000 (1 cm = 100 km). The cities will be different distances apart on paper, but measuring and multiplying should give about the same real distance. Small differences come from measuring errors and thick printed dots.

Large and small scales. A town map with RF 1 : 25,000 shows much more detail: 1 cm = 25,000 cm = 250 m. A street 6 cm long on it is 6 × 250 = 1500 m = 1.5 km long.

Making your own scale drawing. To draw your classroom at a scale of 1 : 50, divide every real length by 50. A wall 8 m = 800 cm long becomes 800 ÷ 50 = 16 cm, and a desk 1 m long becomes 2 cm. Mark the board, desks, fans and lights with simple symbols in their scaled positions.

Why the RF has no units. Because both parts of the ratio are in the same unit, the units cancel. That is why the same RF works whether you measure in centimetres or inches: 1 inch on the map is 60,00,000 inches on the ground.

Viswanath grinds a spice powder from 8 spoons of coriander seeds, 4 red chillies, 2 spoons of toor dal and 1 spoon of fenugreek (methi). The ratio coriander : chillies : dal : methi is 8 : 4 : 2 : 1. A ratio can have three, four or more terms, as long as all the quantities change by the same factor.

Scaling a many-term ratio. Puneet has only 2 red chillies, half of Viswanath’s 4. To get the same taste he halves everything: 4 spoons coriander, 2 chillies, 1 spoon dal and half a spoon of methi. So

8 : 4 : 2 : 1 :: 4 : 2 : 1 : 0.5.

In general, a : b : c : d :: p : q : r : s means a/p = b/q = c/r = d/s. Every term is multiplied (or divided) by the same number.

Simplest form. Divide all terms by their HCF. 12 : 10 : 8 : 6 : 4 has HCF 2, so it becomes 6 : 5 : 4 : 3 : 2.

Example 1: paint. A purple shade is made with red : blue : white = 2 : 3 : 5. Yasmin has 10 litres of white. White is 5 parts, so 1 part = 10 ÷ 5 = 2 litres. Red = 2 × 2 = 4 litres and blue = 3 × 2 = 6 litres. Total purple = 4 + 6 + 10 = 20 litres.

Example 2: concrete. Strong concrete for pillars and roofs uses cement : sand : gravel = 1 : 1.5 : 3. With 3 bags of cement, multiply every term by 3: 3 : 4.5 : 9. That is 3 + 4.5 + 9 = 16.5 bags of dry mix.

Example 3: a library. Odia : Hindi : English books = 3 : 2 : 1, and there are 288 Odia books. One part = 288 ÷ 3 = 96. So there are 2 × 96 = 192 Hindi books and 96 English books.

Method: find one part.

• Find the term whose real amount is known.
• Divide the amount by that term to get the size of one part.
• Multiply each other term by one part.
• Check that the new amounts are in the same ratio.

Decimals in ratios. A ratio like 1 : 1.5 : 3 is fine, but you can clear decimals by multiplying all terms by 2 to get 2 : 3 : 6. It is the same ratio.

Sometimes we know the total and must split it in a ratio. To divide 12 in the ratio 2 : 1, add the terms (2 + 1 = 3), find one part (12 ÷ 3 = 4) and multiply: 2 × 4 = 8 and 1 × 4 = 4. The same idea works for any number of terms.

Rule. To divide a quantity x in the ratio a : b : c : ..., the parts are

x × a/(a + b + c + ...), x × b/(a + b + c + ...), x × c/(a + b + c + ...), and so on.

Example 1: concrete. 110 units of concrete are needed with cement : sand : gravel = 1 : 1.5 : 3. The terms add to 5.5, so one part is 110 ÷ 5.5 = 20. Cement = 20 units, sand = 30 units, gravel = 60 units. Check: 20 + 30 + 60 = 110.

Example 2: paint. 50 ml of purple paint with red : blue : white = 2 : 3 : 5. The terms add to 10, so red = 50 × 2/10 = 10 ml, blue = 15 ml and white = 25 ml.

Example 3: a cricket session. A 150-minute practice is split as warm-up : batting : bowling : fielding = 3 : 4 : 3 : 5. The sum is 15, so one part is 10 minutes: 30, 40, 30 and 50 minutes.

Example 4: angles of a triangle. Construct a triangle with angles in the ratio 1 : 3 : 5. The angles add to 180° and the terms add to 9, so one part is 20°. The angles are 20°, 60° and 100°. Draw a base AB, make 20° at A and 60° at B; the rays meet at C, and the angle there is 100°.

Example 5: coins. 100 coins are ₹10, ₹5, ₹2 and ₹1 coins in the ratio 4 : 3 : 2 : 1. One part = 100 ÷ 10 = 10 coins, so there are 40, 30, 20 and 10 coins. The money is 400 + 150 + 40 + 10 = ₹600. Notice: we divide the number of coins in the ratio, not the money.

Side lengths. A triangle with sides in the ratio 3 : 4 : 5 can be drawn in many sizes (3, 4, 5 cm or 6, 8, 10 cm). They all have the same shape but not the same size, so they are not all congruent. A triangle with sides in the ratio 1 : 3 : 5 is impossible at any size, because 1 + 3 is less than 5: the two shorter sides cannot reach each other.

Always check that the parts add back to the whole.

A pie chart shows how a whole is shared out among parts, as slices of a circle. Its slices are an example of dividing a whole in a ratio: here the whole is the full angle of 360°.

The data. In a test, 40 students got these grades.

Step 1: the ratio. The angle of each slice must be proportional to the number of students. So divide 360° in the ratio 12 : 10 : 8 : 6 : 4, which simplifies (dividing by the HCF 2) to 6 : 5 : 4 : 3 : 2.

Step 2: the angles. The terms add to 20, so one part = 360° ÷ 20 = 18°.

Check: 108 + 90 + 72 + 54 + 36 = 360. Another way to get each angle: (students ÷ total) × 360°. For grade A, 12/40 × 360° = 108°.

Step 3: draw. Draw a circle and one radius. With a protractor at the centre, measure 108° from that radius and draw the second radius: that is slice A. From the new radius measure 90° for B, then 72° for C and 54° for D. What is left should be 36° for E. Colour and label each slice.

From percentages. If data is given in percent, each 1% is 3.6°. A TV survey says Entertainment 50%, Sports 25%, News 15%, Information 10%; the angles are 180°, 90°, 54° and 36°.

Reading a pie chart. A slice of 90° is a quarter of the whole, and 120° is a third. If a survey pie shows a 60° slice for ‘car’ and 18 children travel by car, then 60° stands for 18 children, so 360° stands for 18 × 6 = 108 children.

When to use a pie chart. Use it when the parts make up one whole and you want to show each part’s share. It is less useful for comparing parts that are close in size, or for data that changes over time.

Puneeth’s father rides from Lucknow to Kanpur, 90 km away, at 30 km/h and takes 3 hours. In a car at 60 km/h, does the time double too? No, faster travel means less time. Look at the table.

From walking to cycling, speed is multiplied by 3 and time is divided by 3. From motorcycle to car, speed is multiplied by 2 and time is divided by 2. The two quantities change by the same factor, but in opposite directions. This is an inverse proportion.

The test. Two quantities x and y are inversely proportional when their product stays the same:

x × y = k, a constant.

Here k = 90 km, the fixed distance. So for any two pairs of values, x₁y₁ = x₂y₂. This also means x₁/x₂ = y₂/y₁: the ratio of the x values is the inverse of the ratio of the y values. For walking and car, 5/60 = 0.0833... and 1.5/18 = 0.0833... as well.

In words. If one quantity is multiplied by n, the other is multiplied by 1/n. Double one, halve the other; triple one, divide the other by 3.

Compare with direct proportion.

Checking a table. Is this table an inverse proportion? x: 40, 80, 25, 16 and y: 20, 10, 32, 50. The products are 800, 800, 800, 800, so yes. If one product is different, as with 25 × 12.5 = 312.5 next to 40 × 20 = 800, it is not.

Careful. ‘One goes up, the other goes down’ is not enough. As a candle burns for more minutes its height goes down, but burning twice as long does not halve its height, so the product of time and height is not fixed. For inverse proportion the product must stay the same.

Many everyday problems are inverse proportions: more workers finish a job in fewer days, more pumps fill a tank faster, more people use up food sooner. The method is always the same: the product stays the same.

Step-by-step method.

• Decide whether the relation is direct or inverse: if one quantity doubles, does the other double or halve?
• For inverse proportion, write x₁ × y₁ = x₂ × y₂.
• Put in the three known values and solve for the fourth.
• Check that the answer makes sense (more workers should mean fewer days).

Example 1: workers. 20 workers lay a road in 4 days. How long will 10 workers take? Fewer workers, more days: 20 × 4 = 10 × y, so y = 80 ÷ 10 = 8 days. Halving the workers doubles the time.

Example 2: pumps. 2 pumps fill a tank in 18 hours. With 2 more pumps (4 in all): 2 × 18 = 4 × x, so x = 9 hours.

Example 3: food. A hostel has food for 80 students for 15 days. If 20 more students join: 80 × 15 = 100 × x, so x = 1200 ÷ 100 = 12 days.

Example 4: chairs. Chairs are set in 25 rows of 12. With 20 chairs in a row, the number of rows is 25 × 12 ÷ 20 = 300 ÷ 20 = 15 rows.

Example 5: periods. A school day has 8 periods of 45 minutes. With 9 periods in the same time, each is 8 × 45 ÷ 9 = 360 ÷ 9 = 40 minutes.

Example 6: machines. 42 machines make a batch of toys in 63 days. To make it in 54 days: 42 × 63 = 54 × m, so m = 2646 ÷ 54 = 49 machines.

Assumptions. These answers assume every worker, pump or machine works at the same steady rate, and that adding more does not cause crowding or delays. Real life is not always so neat, which is why the answers are good estimates rather than promises. For the water tank shared by families, we must also assume each family uses the same amount every day.

Do not mix up the two kinds. If 24 pencils cost ₹120, then 20 pencils cost 120 × 20/24 = ₹100. That is direct, not inverse: fewer pencils cost less. And if a pump fills 2 tanks in 6 hours, it fills 5 tanks in 15 hours, also direct, because one pump and more tanks means more time.

Ram cuts a basket of vegetables in 1 hour; Shyam takes 1.5 hours for the same basket. How long will they take working together? Adding the times (2.5 hours) is clearly wrong: together they must be faster than Ram alone.

Think in work per hour. Call the whole basket 1 unit of work.

• Ram does the whole job in 1 hour, so he does 1 unit per hour.
• Shyam does it in 1.5 hours, so he does 1 ÷ 1.5 = 2/3 unit per hour.
• Together they do 1 + 2/3 = 5/3 units per hour.

Work done and time are directly proportional at a fixed rate: 5/3 : 1 :: 1 : x. So (5/3) × x = 1 and x = 3/5 hour = 36 minutes.

Two pumps. A small pump fills a tank in 3 hours and a large one in 2 hours. Per hour they fill 1/3 and 1/2 of the tank, together 1/3 + 1/2 = 5/6. So the tank fills in 6/5 hours = 1 hour 12 minutes.

Why this works. Time to finish and rate of work are inversely proportional: rate × time = 1 job. So we add the rates, never the times, and then take the inverse.

Direct or inverse? A checklist. Ask: if one quantity doubles, what happens to the other?

Not every pair is proportional. Some quantities are related but neither directly nor inversely proportional. How long different animals sleep has nothing to do with their weight in any fixed ratio, and a person’s height does not double when their age doubles. Always test with the quotient (direct) or the product (inverse) before using a rule.

Summary of methods. Direct: x/y is constant, so cross-multiply ratios. Inverse: x × y is constant, so multiply and divide. Working together: add the rates and invert.