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

1. Find the ratio of
   (i) 45 minutes to 5¾ hours
   (ii) 4 months and 2½ years
   (iii) 1·2 kg and 60 gm.
2. 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
3. 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.
4. Find the following
   (i) the duplicatae ratio of 2x : 3y
   (ii) the sub-duplicatae ratio of 16a² : 25b²
   (iii) the sub-triplicate ratio of a⁶ : 8b³
5. Which ratio is greater 17 : 21 or 23 : 28?
6. Arrange the following ratios in descending order of magnitude
      13 : 9, 25 : 23, 16 : 11 and 20 : 17
7. Arrange the ratios 2 : 3, 3 : 4, 4 : 5 in ascending order of magnitude.
8. 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.
9. If A : B = 4 : 5, B : C = 6 : 7, find A : C
10. If A : B = 6 : 7, B : C = 8 : 9, find A : B : C
11. Find the number which bears the same ratio to 7/33 that 8/21 bears to 4/9.
12. Two numbers are in ratio 7 : 11. If 7 is added to each of the numbers, the ratio becomes 2 : 3. Find the numbers.
13. A ratio is equal to 3 : 7. If the antecedent is 5, find the consequent.
14. Find two numbers in the ratio of 8 : 7 such that when each is decreased by 12½, they are in ratio 11 : 9.
15. Divide Rs 170 in the ratio 3 : 7.
16. 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.
17. 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
18. 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.
19. 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
20. What should be subtracted from 23, 30, 57 and 78 so that the remainders may be in proportion?
21. What number must be added to each of the numbers 6, 15, 20 and 43 to make them proportional?
22. Find two numbers such that their mean proportional is 9 and the third proportional is 243.
23. If three quantities are in continued proportion, prove that first is to third is the duplicate ratio of the first to the second.
24. 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.
25. If b is the mean proportional between a and c, prove that
           (a² -b² +c²)/(a^-2 -b^-2 +c^-2) = b⁴
26. If x/a = y/b = z/c, prove that each ratio is equal to 
27. If y is the mean proportional between x and z, prove that xyz(x +y +z)³ = (xy +yz +zx)³.
28. If x/a = y/b = z/c, prove that
    (i) 
    (ii) 
29. If x/(b-c) = y/(c-a) = z/(a-b), prove that
    (i) ax +by +cz = 0
    (ii) x +y +z = 0
30. If ax = by = cz, prove that x²/yz + y²/xz + z²/xy = bc/a² + ca/b² +ab/c²
31. 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