RATIO AND PROPORTION Ratio Ratio of two quantities represents how many times a quantity contains another quantity of the same kind. Two quantities are in the ratio a : b => if the first quantity is ax, then the second quantity is bx. a / b is the ratio of a to b; b is not equal to 0. Here the first term “a” or numerator is called the “antecedent” and the second term “b” or denominator is called the “consequent”. If two quantities are in the ratio of a : b, then the first quantity is a/ (a + b) times the sum of the two quantities and the second quantity is b/ (a + b) times the sum of the two quantities. Compound Ratio: Ratios are compounded by multiplying together the antecedents for a new antecedent, and the consequents for a new consequent. For example, compounded ratio of 3 : 4, 2 : 3, 2 : 5 can be calculated by (3/4) * (2/3) * (2/5)=1:5 Inverse Ratio: If a : b is the given ratio, then (1/a) : (1/b) or b : a is called its inverse or reciprocal ratio. Proportion Equality of ratios is called the proportion. The numbers a, b, c and d are said to be in proportion if a : b = c : d. Here, d is called the fourth proportional. ad = bc Direct Proportion: The proportion by which the value of one variable varies directly as the value of another variable. Inverse Proportion: The proportion by which the value of one variable varies inversely as another variable.  Product of means = Product of Extremes.  i.e., a : b : : c : d  (b * c) = (a * d)  (a : b) > (c : d)  (a/b) > (c/d)  Mean proportional: Mean proportional between a and b is sqrt (ab).  Third proportional: If a:b = b:c, then c is called the Third Proportional to a, b.  Fourth Proportional: If a:b = c:d, then d is called the Fourth Proportional to a, b, c.  Componendo and Dividendo: If (a/b) = (c/d), then [(a+b)/ (a-b)] = [(c+d)/(c-d)]  Some other tricks: (a+b)/b = (c+d)/d, (a-b)/b = (c-d)/d, a/c = b/d Formulae 1. When two ratios are equal, they are said to be in proportion. If (a / b) = (c / d), then a/b is in proportion with c/d and can be written as a: b:: c: d. where “a” and “d”

are called “extremes” and “c” and “b” the means. For a, b, c, d to be in proportion the product of the extremes = the product of the means. i.e. ad = bc 2. Compounded ratio of the ratios (a : b), (c : d), (e : f) is (ace : bdf) 3. Duplicate ratio of (a : b) is (a² : b²) 4. Sub Duplicate ratio of (a : b) is (√a : √b) 5. Triplicate ratio of (a : b) is (a³ : b³) 6. Sub Triplicate ratio of (a : b) is (a1/3 : b1/3) 7. If (a/b) = (c/d), then [(a + b)/(a - b)] = [(c + d)/(c - d)] 8. If A & B are in the ratio a : b, as a proportion of the total, A is a/(a+b) & B is b/(a+b) 9. Direct proportion: When a/b = k or a = kb then “a” is directly proportional to “b”, where k is a constant. 10. Inverse proportion: When “a” and “b” are so related that ab = k, a constant, then “a” and “b” are said to be inversely proportional to each other. 11. If a sum of money S is divided in the ratio a : b : c then the three parts are a b c (i) (ii) (iii) S S S a+b+c a+b+c a+b+c 12. If a : b = m : n and b : c = p : q then a : b : c = mp:np:nq 13. If A and B are two partners investing in the ratio of m:n for the same period of time,then the ratio of profits is m : n 14. If the investment is in the ratio m : n and the period in the ratio p : q then the ratio of profits is mp : nq. 15. If m kg of one kind costing “a” rupees/kg is mixed with “n” kg of another kind ma+nb costing Rs. b/kg, then the price of the mixture is m+n 16. If “a” varies as “b”, then a = kb, where „k is called the constant of proportionality. (Direct variation). 17. If “a” varies as “b” and “b” varies as “c”, there a = kb and b = k’c. Where k, k’ are constants.  a = (kk’)c = λc, where λ = kk’, is another constant.  “a” varies as “c”. 18. If “a” varies directly as “b” and “b” varies inversely as “c” then a = kb and b = k'/c  a = (kk'/c) – (λ/c), where λ = kk’. Hence “a” varies inversely as “c” (mixed variation).

Solved Problems 1. If a : b = 7 : 5 and b : c = 4 : 3, find a : b : c. Answer: a : b = 7 : 5 and b : c = 4 : 3 = (4 * 5/4) : (3 * 5/4) = 5 :15/4 a :b : c = 7 : 5 : 15/4 = 28 : 20 :15 2. Find the fourth proportional to 3, 4, 12 Answer: 3 : 4 : : 12 : x x =16

3. Find the third proportional to 12 and 36. Answer: 12 : 36 : : 36 : x 12x = 36 * 36 x = 108 4. Find the mean proportional between 0.08 & 0.18. Answer: √(0.08 * 0.18) = √(8/100 * 18/100) = √(144/10000) = 0.12 5. Divide Rs. 1248 among A, B, C in the ratio 12: 4: 8. Answer: Sum of ratio = 24 1st part = Rs. (1248 * 12/24) = Rs. 624 2nd part = Rs. (1248 * 4/24) = Rs. 208 3rd part = Rs. (1248 * 8/24) = Rs. 416 6. A box contains 20 p, 50 p and 25 p coins in the ratio 5 : 2 : 3, amounting to Rs.341. Find the no. of coins of each type. Answer: Let the no. of 20p, 50p & 25p be 5x, 2x & 3x respectively. Then 5x/5 + 2x/2 + 3x/4 = 341 11x = 1364 x = 124 No. of 20p coins = 620, No. of 50p coins = 248, No. of 25p coins = 372 7. If 20% of a number is equal to two third of another number, what is the ratio of first number to the second number? Answer: Let 20% of A = (2/3)B Then, 20 A/ 100 = 2 B/3 A/5 = 2B/3 , A : B = 10 : 3 8. A sum of Rs. 36.90 is made up of 180 coins which are either 10 paise coins or 25 p coins. The number of 10 p coins is Step (i) Total number of coins = 180 Let x be number of 10p coins and y be number of 25p coins x+y=180 -------------- (i) Step (ii) Given 10p coins and 25p coins make the sum = Rs. 36.90 10x/100 + 25y/100 = 36.90 ⇒ 10x+25y=3690 --------- (ii) Step (iii) Solving (i) and (ii) 10x+10y=1800 -------------------- [ (i)×10 ] 10x+25y=3690 ---------------------(ii) ⇒ −15y=−1890 ⇒ y=1890/15 =126 Substitute y value in equation (i) , x=180–126=54 Number of 10p coins = 54

9. Rs.432 is divided amongst three workers A, B and C such that 8 times A’s share is equal to 12 times B’s share which is equal to 6 times C’s share. How much did A get? Sol: 8 times A’s share = 12 times B’s share = 6 times C’s share Note that this is not the same as the ratio of their wages being 8:12:6 In this case, find out the L.C.M of 8, 12 and 6 and divide the L.C.M by each of the above numbers to get the ratio of their respective shares. The L.C.M of 8, 12 and 6 is 24. Therefore, the ratio A:B:C::24/8:24/12: 24/6 ⇒ A:B:C::3:2:4 The sum of the total wages=3x+2x+4x=432 ⇒ 9x=432 or x=48. Hence A gets 3×48= Rs. 144 10. The monthly incomes of A and B are in the ratio 4 : 5, their expenses are in the ratio 5 : 6. If 'A' saves Rs.25 per month and 'B' saves Rs.50 per month, what are their respective incomes? SOL: Let A's income be = 4x A's expenses, therefore=4x–25 Let B's income be =5x B's expenses, therefore =5x–50 We know that the ratio of their expenses =5:6 ⇒ 24x−150=25x−250 ⇒ Therefore, x=100. ⇒ A's income =4x= 400 and B's income =5x= 500. 11. The salaries A, B, C are in the ratio 2 : 3 : 5. If the increments of 15%, 10% and 20% are allowed respectively in their salaries, then what will be new ratio of their salaries?

Explanation: Let A = 2k, B = 3k and C = 5k. 115 of 2k =

A's new salary =

23k

115 x 2k

=

100

100

10

110

110

33k

of 3k =

B's new salary =

x 3k

100

100

120

120 of 5k =

C's new salary = 100

10 x 5k

100

=

= 6k

23k New ratio

33k : 6k

: 10

= 23 : 33 : 60

10

12. If 40% of a number is equal to two-third of another number, what is the ratio of first number to the second number? Explanation: 2 Let 40% of A =

B 3

40A Then,

2B =

100 2A

3

2B =

5

3

A

2 =

B

5 x

3

5 =

2

3

A : B = 5 : 3. 13. A sum of Rs. 1300 is divided amongst P, Q, R and S such that P's share Q's share

=

Q's share R's share

=

R's share S's share

=

2 3

. Then, P's share is :

SOL: Let P = 2x and Q = 3x.Then, Q/R= 2/3 , R=3/2Q=> 3/2*3x= 9x/2 Also, R/S= 2/3, S=3/2R=> 3/2*(9x/2)=> 27x/4 Thus, P=2x, Q=3x, R=9x/2 and S=27x/4 Now, P + Q + R + S =2x+3x+(9x/2)+(27x/4)=1300 Solving x=80, P’s Share = Rs. (2*80)= Rs. 160