Ratio and Proportion

A ratio compares two quantities of the same kind; a proportion says that two ratios are equal. These two ideas are behind almost every practical calculation — sharing a profit, mixing mortar, reading a map, or working out how much rice a family of six needs for a month. This chapter works through the definitions, then the two kinds of variation that make the difference between a sum that takes one line and a sum that is done backwards.

4.1 Ratio

A ratio compares two quantities of the same kind by division. The ratio of a to b is written a:b and read "a is to b", where

The two quantities must be in the same unit. A ratio has no unit, because it is one quantity divided by another of the same kind.

Statement	Ratio
15 kg to 25 kg	15:25=3:5
500 m to 2 km	500:2000=1:4 (convert to the same unit first)
45 minutes to 2 hours	45:120=3:8

Simplification. Divide every term by their highest common factor. A ratio is usually written with whole numbers and in its lowest terms.

4.1.1 Dividing a quantity in a given ratio

Add the terms of the ratio to find the number of equal parts, then give each share as its term out of that total.

4.1.2 Word problems on ratio

The usual method: write the quantities as ax and bx (the ratio times a common multiplier x), form an equation from the condition, and solve for x.

4.2 Proportion

Four quantities a, b, c, d are in proportion if

In the statement a:b=c:d, the outer terms a and d are the extremes and the inner terms b and c are the means. The fundamental property of a proportion is

Three quantities a, b, c are in continued proportion if a:b=b:c, so that b^2=ac. Then b is the mean proportional between a and c, and

4.3 Direct and inverse variation

Two quantities are in direct variation when they increase or decrease together in the same ratio, written x\propto y and x=ky.

Direct variation	Inverse variation
x\propto y, so x=ky	x\propto\frac1y, so x=\frac{k}{y}
\frac{x_1}{y_1}=\frac{x_2}{y_2}	x_1y_1=x_2y_2
more goods, more money	more workers, less time
more days, more wages	faster speed, less time
more rice, more people fed	more taps, less time to fill a tank
constant is the unit rate	constant is the total work

The constant k is found from the pair of values given in the problem; once it is known, the second pair can be calculated.

4.4 Map scale and similar figures

A map scale is a ratio of a length on the map to the real length, with both in the same unit. A scale of 1:50\,000 means 1 cm on the map represents 50 000 cm, that is 500 m or 0.5 km, on the ground.

4.5 Sharing in a given ratio

Task	Method
Simplify a ratio	divide both terms by their HCF; make units the same first
Share x in m:n	shares are \frac{m}{m+n}x and \frac{n}{m+n}x
Compound ratio of a:b and c:d	ac : bd
Mean proportional between a and b	\sqrt{ab}
Check a proportion a:b = c:d	ad = bc (product of extremes = product of means)

Chapter summary

• A ratio compares two quantities of the same kind and has no unit; write both in the same unit and simplify by the HCF.
• To divide a quantity in the ratio a:b:c, add the terms, find one part, then multiply.
• Four quantities are in proportion if a:b=c:d, which holds exactly when ad=bc.
• Three quantities are in continued proportion if a:b=b:c, and then the mean proportional is b=\sqrt{ac}.
• Direct variation: x\propto y, x=ky, \frac{x_1}{y_1}=\frac{x_2}{y_2}; both quantities rise together.
• Inverse variation: x\propto\frac1y, xy=k, x_1y_1=x_2y_2; one rises as the other falls.
• A map scale is a ratio, so both lengths must be converted to the same unit before multiplying.

Practice questions

Group A — Objective (1 mark each)

1. The ratio 250 g to 1 kg in its lowest terms is [1]a250:1b1:4c4:1d1:250
2. If a:b=c:d, then [1]aab=cdbad=bccac=bdda+b=c+d
3. The mean proportional between 9 and 16 is [1]a12b25c144d7
4. More workers, less time is an example of [1]adirect variationbinverse variationcproportiondcontinued proportion
5. If 3 pens cost Rs 45, then 7 pens cost [1]aRs 90bRs 105cRs 135dRs 63
6. On a map of scale 1:100\,000, 3 cm represents [1]a3 kmb30 kmc0.3 kmd300 km

Answers to Group A

1. (b) — 250:1000=1:4.

2. (b) — the product of the extremes equals the product of the means.

3. (a) — \sqrt{9\times16}=12.

4. (b) — one quantity rises as the other falls.

5. (b) — one pen is Rs 15, so seven cost Rs 105.

6. (a) — 3\times100\,000=300\,000 cm =3 km.

Group B — Short answer (2 marks each)

1. Express the ratio 45\text{ min}:2\text{ hr} in its lowest terms. [2]
2. Find x if x:8=9:12. [2]
3. Divide Rs 5,400 in the ratio 2:3:4. [2]
4. If 8 men can do a job in 15 days, how long will 12 men take? [2]

Answers to Group B

1. 45:120=3:8.

2. 12x=72, so x=6.

3. Total parts =9, one part =600; shares Rs 1200, Rs 1800 and Rs 2400.

4. Total work =8\times15=120 man-days; with 12 men the time is 120\div12=10 days.

Group C — Long answer (4 marks each)

1. A sum of money is divided between two brothers in the ratio 5:7. If the elder brother receives Rs 8,400, find the total sum and the younger brother's share. [4]
2. A truck covers a certain distance in 6 hours at 40 km/h. How long will it take at 60 km/h? State what kind of variation this is, find the constant, and verify the answer. [4]

Answers to Group C

1. The elder brother's share is the 7-part share: 7\text{ parts}=\text{Rs }8400, so 1 part =8400\div7=\text{Rs }1200. The younger brother gets 5\times1200=\text{Rs }6000, and the total is 6000+8400=\text{Rs }14\,400. Check. 6000:8400=5:7 ✓ and 6000+8400=14\,400 ✓.

2. Kind. Faster speed means less time, so this is inverse variation (\text{speed}\times\text{time}= constant distance). Constant. The distance is 40\times6=240 km. Time at 60 km/h. 240\div60=4 hours. Verify. 40\times6=240 and 60\times4=240 ✓ — the product is the same, as inverse variation requires, and the time fell from 6 hours to 4 as the speed rose.

Extra practice — textbook-style questions

1. Find the ratio of (a) 3 kg and 80 g (b) 2 hours and 45 minutes (c) Rs 5 and 90 paisa.

2. A map is drawn to a scale 1:4000. Two places are 4 cm apart on the map. Find the real distance.

3. Two numbers are in the ratio 3:4. If 5 is added to each, the ratio becomes 4:5. Find the numbers.

4. What should be subtracted from 24 and 30 to make the ratio 2:3?

5. If 12 notebooks cost Rs 600, how many notebooks can be bought for Rs 900?

6. Find the mean proportional between 8 and 18.

Extra practice — answers

1. (a) 3000:80=75:2 (b) 120:45=8:3 (c) 500:90=50:9.

2. 4\times4000=16\,000 cm =160 m.

3. \frac{3x+5}{4x+5}=\frac45 → 15x+25=16x+20 → x=5; the numbers are 15 and 20. Check: 20:25=4:5 ✓.

4. \frac{24-y}{30-y}=\frac23 → 72-3y=60-2y → y=12. Check: 12:18=2:3 ✓.

5. \frac{600}{900}=\frac{12}{x} → x=\frac{900\times12}{600}=18 notebooks.

6. \sqrt{8 \times 18} = \sqrt{144} = 12.

Objective practice

Choose the correct answer. [5×1=5]

1. If 4:7=x:35, then x is (a) 15 (b) 20 (c) 25 (d) 28

2. The mean proportional between 9 and 16 is (a) 12 (b) 12.5 (c) 25 (d) 144

3. Rs 960 shared in the ratio 5:7 gives the larger share (a) Rs 400 (b) Rs 480 (c) Rs 560 (d) Rs 600

4. The compound ratio of 3:4 and 2:5 is (a) 5:9 (b) 3:10 (c) 6:9 (d) 1:20

5. 40 minutes : 2 hours in simplest form is (a) 40:2 (b) 1:3 (c) 2:5 (d) 1:5

Objective practice — answers

1. (b) — x=\frac{4\times35}{7}=20.

2. (a) — \sqrt{144}=12.

3. (c) — \frac{7}{12}\times960=560.

4. (b) — \frac{6}{20}=3:10.

5. (b) — 40:120=1:3.