# Ratio and Proportion

**Compounded Ratio**of two ratios a/b and c/d is ac/bd, i.e., ac : bd.

**Duplicate ratio**of a : b is a² : b²

**Triplicate ratio**of a : b is a³ : b³

Sub-duplicate ratio of a : b is a : b

Sub-triplicate ratio of a : b is³a :³b

**Reciprocal ratio**of a : b is b : a**Proportion.**Four (non-zero) quantities of the same kind a, b, c, d are in proportion,

written as a : b :: c : d iff a/b = c/d- The non-zero quantities of the same kind a, b, c, d, ... are in
**continued proportion**iff

a/b = b/c = c/d = ...

In particular, a, b, c are in continued proportion iff a/b = c/d. In this case b is called the**mean proportion**; b = ac; c is called third proportional. If a, b, c, d are in proportion, then d is called fourth proportional. **Invertendo.**If a : b :: c : d then b : a :: d : c

**Alternendo.**If a : b :: c : d then a : c :: b : d

**Componendo.**If a : b :: c : d then (a +b) : b :: (c +d) : d

**Dividendo.**If a : b :: c : d then (a -b) : b :: (c -d) : d

**Componendo and dividendo.**

If a : b :: c : d then (a +b) : (a -b) :: (c +d) : (c -d)

i.e., a/b = c/d => (a +b)/(a - b) = (c +d)/(c +d)- If a/b = c/d = e/f = ..., then each ratio = (a +c +e +...)/(b +d +f +...)

## Exercise

- Find the ratio of

(i) 45 minutes to 5¾ hours

(ii) 4 months and 2½ years

(iii) 1·2 kg and 60 gm. - Find the compounded ratio of

(i) 5 : 7 and 9 : 10

(ii) (x +y) : (x -y) and (x -y) : (x +y)

(iii) 2a : 3b, 2b : 3a, a² : b² - Find the following

(i) the duplicate ratio of 3 : 7

(ii) the triplicate ratio of 2 : 5

(iii) the sub-duplicate ratio of 36 : 25

(iv) the sub-triplicate ratio of 27 : 1

(v) the reciprocal ratio of 9 : 11. - Find the following

(i) the duplicatae ratio of 2x : 3y

(ii) the sub-duplicatae ratio of 16a² : 25b²

(iii) the sub-triplicate ratio of a^{6}: 8b³ - Which ratio is greater 17 : 21 or 23 : 28?
- Arrange the following ratios in descending order of magnitude

13 : 9, 25 : 23, 16 : 11 and 20 : 17 - Arrange the ratios 2 : 3, 3 : 4, 4 : 5 in ascending order of magnitude.
- A man earns Rs 5000 per month and spends Rs 3500 per month. Find the ratio of his

(i) expenditure to income

(ii) savings to income

(iii) savings to expenditure. - If A : B = 4 : 5, B : C = 6 : 7, find A : C
- If A : B = 6 : 7, B : C = 8 : 9, find A : B : C
- Find the number which bears the same ratio to 7/33 that 8/21 bears to 4/9.
- Two numbers are in ratio 7 : 11. If 7 is added to each of the numbers, the ratio becomes 2 : 3. Find the numbers.
- A ratio is equal to 3 : 7. If the antecedent is 5, find the consequent.
- Find two numbers in the ratio of 8 : 7 such that when each is decreased by 12½, they are in ratio 11 : 9.
- Divide Rs 170 in the ratio 3 : 7.
- Find the value of x in the following proportions

(i) 10 : 35 = x : 42

(ii) 3 : x = 24 : 2

(iii) 2·5 : 11·5 = x : 3

(iv) x : 50 :: 3 : 2. - Find the fourth proportional to

(i) 1/3, 1/4, 1/5

(ii) 1·5, 2·5, 4·5

(iii) 9·6 kg, 7·2 kg, 28·8 kg

(iv) 2xy, x², y² - Find the third proportional to

(i) 5, 10

(ii) 1·3, 3·9

(iii) 21/4 and 7

(iv) x² -y², (x +y)²

(v) 2 x, 4 x². - Find the mean proportion of

(i) 5 and 80

(ii) 1/12 and 1/75

(iii) 8·1 and 2·5

(iv) 5 +2, 5 -2

(v) (a -b)³, (a -b)^{5} - What should be subtracted from 23, 30, 57 and 78 so that the remainders may be in proportion?
- What number must be added to each of the numbers 6, 15, 20 and 43 to make them proportional?
- Find two numbers such that their mean proportional is 9 and the third proportional is 243.
- If three quantities are in continued proportion, prove that first is to third is the duplicate ratio of the first to the second.
- If a b and a : b is the duplicate ratio of (a +c) and (b +c), prove that c is the mean proportional between a and b.
- If b is the mean proportional between a and c, prove that

(a² -b² +c²)/(a^{-2}-b^{-2}+c^{-2}) = b^{4} - If x/a = y/b = z/c, prove that each ratio is equal to
- If y is the mean proportional between x and z, prove that xyz(x +y +z)³ = (xy +yz +zx)³.
- If x/a = y/b = z/c, prove that

(i)

(ii) - If x/(b-c) = y/(c-a) = z/(a-b), prove that

(i) ax +by +cz = 0

(ii) x +y +z = 0 - If ax = by = cz, prove that x²/yz + y²/xz + z²/xy = bc/a² + ca/b² +ab/c²
- Find x from the following equations:

(i) [(a +x) +(a -x)]/[(a + x) -(a - x)] = c/d

(ii) 16

## Answers

**1.**(i) 3 : 23 (ii) 2 : 15 (iii) 20 : 1

**2.**(i) 9 : 14 (ii) 1 : 1 (iii) 4a² : 9b²

**3.**(i) 9 : 49 (ii) 9 : 125 (iii) 6 : 5 iv) 3 : 1 (v) 11 : 9

**4.**(i) 4x : 9y (ii) 4a : 5b (iii) a² : 2b.

**5.**23 : 28

**6.**16 : 11, 13 : 9, 20 : 17, 25 : 23

**7.**2 : 3, 3 : 4, 4 : 5

**8.**(i) 7 : 10 (ii) 3 : 10 (iii) 3 : 7

**9.**24 : 25.

**10.**48 : 56 : 63

**11.**2/11

**12.**49 and 77

**13.**35/3

**14.**40, 35

**15.**Rs 51, Rs 119

**16**(i) 12 (ii) 1/4 (iii) 5 (iv) 75

**17.**(i) 3/20 (ii) 7·5 (iii) 21·6 kg (iv) x y/2

**18.**(i) 20 (ii) 11·7 (iii) 28/3 (iv) (x +y)³/(x-y) (v) 8 x³

**19.**(i) 20 (ii) 1/30 (iii) 4·5 (iv) 3 (v) (a -b)

^{4}

**20.**6

**21.**3

**22.**3,27

**31.**(i) 2acd/(c² +d²) (ii) a/3