Ratio and proportion
Indices and Logarithm
A ratio is a comparison of the sizes of two or more quantities of the same kind by division.
If a and b are two quantities of the same kind (in same units), then the fraction a/b is called the ratio of a to b. It is written as a : b. Thus, the ratio of a to b = a/b or a : b. The quantities a and b are called the terms of the ratio, a is called the first term or antecedent and b is called the second term or consequent.
- Both terms of a ratio can be multiplied or divided by the same (non-zero) number. Usually a ratio is expressed is lowest terms (or simplest form).
The order of the terms in a ratio is important.
Ratio exists only between quantities of the same kind.
- Quantities to be compared (by division) must be in the same units.
- To compare two ratios, convert them into equivalent like fractions.
One ratio is the inverse of another if their product is 1. Thus a: b is the inverse of b: a and vice – versa.
- A ratio a: b is said to be of greater inequality if a>b and of less inequality if a<b.
- The ratio compound of the two ratios a: b and c: d is ac: bd.
- A ratio compounded of itself is called its duplicate ratio.
Thus a2 : b2 is the duplicate ratio of a : b.
Similarly, the triplicate ratio of a : b is a3 : b3.
- The sub-duplicate ratio of a : b is √a : √b and
the sub triplicate ratio of a : b is 3√a : 3√b.
- If the ratio of two similar quantities can be expressed as a ratio of two integers, the quantities are said to be commensurable; otherwise, they are said to be incommensurable. √3 : √2 cannot be expressed as the ratio of two integers and therefore, √3 and √2 are incommensurable quantities.
- Continued Ratio is the relation (or compassion) between the magnitudes of three or more quantities of the same kind. The continued ratio of three similar quantities a, b, c n
- An equality of two ratios is called a proportion. Four quantities a, b, c, d are said to be in proportion if a : b = c : d (also written as a : b :: c : d) i.e. if a/b = c/d i.e. if ad = bc.
- The quantities a, b, c, d are called terms of the proportion; a, b, c and d are called its first, second, third and fourth terms respectively. First and fourth terms are called extremes (for extreme terms). Second and third terms are called means (or middle terms).a
If a : b = c : d then d is called fourth proportional.
If a : b = c : d are in proportion then a/b = c/d i.e. ad = bc
this is called cross product rule.
Three quantities a,b,c of the same kind (in same units) are said to be in continuous proportion
If a : b = b : c i.e. a/b = b/c i.e. b2 = ac
If a,b,c are in continuous proportion, then the middle term b is called the mean proportional between a and c, a is the first proportional and c is the third proportional between a and c, a is the first proportional and c is the third proportional.
- Note: In a ratio at a : b, both quantities must be of the same kind while in a proportion a : b = c : d, all the four quantities need not be of the same type. The first two quantities should be of the same kind and last two quantities should be of the same kind.
- PROPERTIES OF PROPORTION
- If a : b = c : d, then ad :: bc (By cross-multiplication)
- If a : b = c : d, then b : a = d : c (Invertendo)
- If a : b = c : d, then a : c = b : d (Alternendo)
- If a : b = c : d, then a + b : b = c + d : d (Componendo)
- If a : b = c : d, then a – b : b = c – d : d (Dividendo)
- If a : b = c : d, then a + b : a – b = c + d : c – d (Componendo and Dividendo)
- If a : b = c : d = e : f = ………….., then each of these ratios (Addendo) is equal
(a + c + e + …….) : (b + d + f + …….)
- a : b = c : d = a + c : b + d (Addendo)
(a / b = c / d = a + c / b + d)
- a : b = c : d = a – c : b – d (Subtrahendo)
(a/b = c/d = a – c / b – d)
- If a : b = c : d = e : f = …… Then each of these ratios =
(a – c – e – …) : (b – d – f – ……..)
LAWS OF INDICESG
- (i) am × an = am+n (base must be same)
- (ii) am ÷an = am-n
- (am)n = amn
- (ab)m= am bm
- (v) (a/b)m = am /bm
- ao = 1
- a-m = 1 / am and 1 / a-m = am
- m√a = a1/m
- (ix) if ax = ay, then x = y
- (x) if xa = ya, then x = y
MULTIPLE CHOICE QUESTIONS (M.C.Q)
Note : Pick up the correct answer from the following.
Q.1. If a : b = 2 : 3 and b : c = 5 : 7, find a : c & a : b : c.
(a) (i) 10:21 (ii) 10 : 15 : 21 (b) (i) 10 : 15 (ii) 10 : 15 : 21
(c) (i) 10 : 12 (ii) 10 : 12 : 21 (d) None
(i) Fourth proportional to 3, 7, 15. (ii) Third proportional to 16 and 36.
(iii) Mean proportion between 8 and 32.
(a) (i) 35 (ii) 81 (iii) 16 (b) (i) 25 (ii) 81 (iii) 16
(c) (i) 16 (ii) 81 (iii) 16 (d) None
(i) The duplicate ratio of 3 : 5.
(ii) The triplicate ratio of 2 : 3.
(iii) The sub-duplicate ratio of 16 : 25.
(iv) The sub-triplicate ratio of 27 : 8.
(v) The compounded ratio of (2 : 5) and (3 : 4).
(a) (i) 9 : 25 (ii) 8 : 27 (iii) 4:5 (iv) 3 : 2 (v) 3 : 10
(b) (i) 8 : 27 (ii) 9 : 25 (iii) 4 : 5 (iv) 2 : 3 (v) 3 : 10
(c) (i) 3 : 10 (ii) 9 : 25 (iii) 8 : 27 (iv) 4: 5 (v) 3: 2
Q.4. If a : b = 2 : 5, find (3a + 4b) : (4a + 5b).
(a) 26 : 33 (b) 26 : 30 (c) 25 : 35 (d) None
Q.5. Divide Rs.1024 in the ratio 9 : 7.
(a) Rs.576, Rs.448 (b) Rs.550, Rs.660
(c) Rs.570, Rs.450 (d) None
Q.6. Divide 581 among A, B, C so that 4A = 5B = 7C.
(a) Rs.245, 196, 140 (b) Rs.140, 160, 240 (c) 200, 250, 280 (d) None
Q.7. A bag contains rupee, 50 paise and 25 paise coins in the ratio 5 : 6 : 7. If the total amount is Rs.390, find the number of coins of each kind.
(a) 200, 240, 280 (b) 200, 220, 260 (c) 200, 250, 280 (d) None
Q.8. In a mixture of 28 litres, the ratio of milk and water is 5 : 2. If 2 litres of water is added to the mixture, find the ratio of milk and water in the new mixture.
(a) 3 : 1 (b) 2 : 1 (c) 3 : 2 (d) None
Q.9. A and B are two alloys of gold and copper prepared by mixing metals in the ratio 7 : 2 and 7 : 11 respectively. If equal quantities of alloys are melted to form a third alloy C, find the ratio of gold and copper in C.
(a) 7 : 4 (b) 4 : 7 (c) 7 : 5 (d) 5 : 7
Q.10. Find three numbers in the ratio 2 : 3 : 5, the sum of whose squares is 608.
(a) 8, 12 and 20 (b) 8, 20, 60 (c) 12, 8, 22 (d) None
Q.11. A mixture contains alcohol and water in the ratio 4 : 3. If 7 litres of water is added to it, the ratio of alcohol and water becomes 3 : 4. Find the quantity of alcohol in the mixture.
(a) 12 litres. (b) 13 litres (c) 18 litres (d) None
Q.12. If A : B = 3 : 4, B : C = 5 : 6 and C : D = 11 : 9, then A : D is:
(a) 50 : 60 (b) 55 : 72 (c) 60 : 70 (d) 65 : 75
Q.13. If a : b = 3 : 4, then (6a + b) : (4a + 5b) is:
(a) 1 : 2 (b) 3 : 5 (c) 7 : 8 (d) 11 : 16
Q.14. If : : (1 + ) : : : x , then x is equal to :
(a) –3 (b) 1 + (c) 1 – (d) + 3
Q.15. If X and Y shared Rs.1100 in the ratio 1 : 10, how much did X get ?
(a) Rs.99 (b) Rs.100 (c) Rs.101 (d) Rs.110
Q.16. If 2x = 3y = 4z, then x : y : z is :
(a) 2 : 3 : 4 (b) 4 : 3 : 2 (c) 6 : 4 : 3 (d) 3 : 4 : 2
Q.17. If x : y = 7 : 9 and y : z = 5 : 4, then x : y : z is :
(a) 7 : 45 : 36 (b) 35 : 45 : 36 (c) 28 : 36 : 35 (d) None
Q.18. The fourth proportional to 3, 5, and 21 is :
(a) 35 (b) (c) (d) 12.6
Q.19. Mean proportional between 7 and 28 is:
(a) 17.5 (b) 12 (c) 14 (d) 16
Q.20. Third proportional to 9 and 12 is:
(a) 6 (b) 10.5 (c) 16 (d) None
Q.21. What must be added to each term of the ratio 49 : 68 so that it becomes 3 : 4 ?
(a) 3 (b) 5 (c) 8 (d) 9
Q.22. What least number must be added to each one of 6, 14, 18, 38 to make them in proportion?
(a) 1 (b) 2 (c) 3 (d) 4
Q.23. What least number must be subtracted from each of the numbers 14, 17, 34, 42 so that the remainders may be proportional ?
(a) 0 (b) 1 (c) 2 (d) 7
Q.24. A fraction bears the same ratio to as does to. The fraction is:
(a) (b) (c) (d)
Q.25 Rs.1980 are divided among A, B, C so that half of A’s part, one-third of B’s part and one-sixth of C’s part are equal. Then, B’s part is:
(a) Rs.660 (b) Rs.360 (c) Rs.1080 (d) Rs.540
Q.26. If A’s money is to B’s money as 4 : 5 and B’s money is to C’s money as 2 : 3 and A has Rs.800, then C has:
(a) Rs.1000 (b) Rs.1200 (c) Rs.1500 (d) Rs.2000
Q.27. 94 is divided into two parts in such a way that the fifth part of the first and the eighth part of the second are in the ratio 3 : 4. The first part is :
(a) 27 (b) 30 (c) 36 (d) 48
Q.28. Rs.1360 have been divided among A, B, C such that A gets of what B gets and B gets of what C gets. Then, B’s share is:
(a) Rs.120 (b) Rs.160 (c) Rs.240 (d) Rs.320
Q.29. Rs.770 have been divided among A, B, C in such a way that A receives th of what B and C together receive. Then A’s share is:
(a) Rs.140 (b) Rs.154 (c) Rs.165 (d) Rs.170
Q.30. Rs.4850 have been divided among A, B, C such that if their shares be diminished by Rs.15, Rs.10 and Rs.25 respectively, the remainders are in the ratio 3 : 4 : 5. Then, B’s share is:
(a) Rs.1595 (b) Rs.1610 (c) Rs.1626.66 (d) Rs.1600
Q.31. A sum of Rs.7000 is divided among A, B, C in such a way that shares of A and B are in ratio 2 : 3 and those of B and C are in the ratio 4 : 5. The amount received by C is:
(a) Rs.2600 (b) Rs.2800 (c) Rs.Rs.3000 (d) Rs.3900
Q.32. A sum of Rs.53 is divided among A, B, C in such a way that A gets Rs.7 more than what B gets and B gets Rs.8 more than what C gets. The ratio of their shares is :
(a) 16 : 9 : 18 (b) 25 : 18 : 10 (c) 18 : 25 : 10 (d) 15 : 8 : 30
Q.33. A bag contains Rs.600 in the form of one-rupee, 50-paise and 25-paise coins in the ratio 3 : 4 : 12. The number of 25-paise coins is:
(a) 600 (b) 900 (c) 1200 (d) 1376
Q.34. A sum of money is divided among A, B, C such that to each rupee A gets, B gets 65 paise and C gets 35 paise. If C’s share is Rs.560, the sum is:
(a) Rs.2400 (b) Rs.2800 (c) Rs.3200 (d) Rs.3600
Q.35. Rs.5625 are divided among A, B, C so that A may receive one-half as much as B and C together receive and B receives one-fourth of what A and C together receive. The share of A is more than that of B by :
(a) Rs.750 (b) Rs.775 (c) Rs.1500 (d) Rs.1600
Q.36. A certain amount was divided between X and Y in the ratio 4 : 3. If B’s share was Rs.4800, the total amount was :
(a) Rs.11,200 (b) Rs.6,400 (c) Rs.19,200 (d) Rs.39,200
Q.37. The ratio of number of boys and girls in a school of 720 students is 7 : 5. How many more girls should be admitted to make the ratio 1 : 1?
(a) 90 (b) 120 (c) 220 (d) 240
Q.38. A boy 1.4 m tall casts a shadow 1.2 m long at the time when a building casts a shadow 5.4 m long. The height of the building is:
(a) 4.63 m (b) 3.21 m (c) 6.3 m (d) 5.6 m
Q.39. The incomes of A and B are in the ratio 3 : 2 and their expenditures in the ratio 5 : 3. If each saves Rs.1500 then B’s income is:
(a) Rs.6000 (b) Rs.4500 (c) Rs.3000 (d) Rs.7500
Q.40. The prices of scooter and a moped are in the ratio 9 : 5. If a scooter costs Rs.6800 more than a moped, the price of a scooter is:
(a) Rs.17,000 (b) Rs.13,600 (c) Rs.15,300 (d) None
Q.41. The cost of making an article is divided between materials, labour and overheads in the ratio of 5 : 3 : 1. If the materials cost Rs.6.90, the cost of the article is:
(a) Rs.13.80 (b) Rs.12.42 (c) Rs.11.56 (d) Rs.9.83
Q.42. The ratio of zinc and copper in a brass piece is 13 : 7. How much zinc will be there in 100 kg of such a piece?
(a) 20 kg (b) 35 kg (c) 55 kg (d) 65 kg
Q.43. What is the ratio whose terms differ by 40 and the measure of which is ?
(a) 16 : 56 (b) 14 : 56 (c) 15 : 56 (d) 16 : 72
Q.44. Two numbers are in the ratio 3 : 5. If 9 be subtracted from each, then they are in the ratio of 12 : 23. The second number is:
(a) 52 (b) 53 (c) 54 (d) 55
Q.45. In a mixture of 60 litres, the ratio of milk and water is 2 : 1. If the ratio of milk and water is to be 1 : 2, then the amount of water (in litres) to be further added is:
(a) 20 (b) 30 (c) 40 (d) 60
Q.46. A mixture contains milk and water in the ratio 5 : 1. On adding 5 litres of water, the ratio of milk and water becomes 5 : 2. The quantity of milk in the original mixture is:
(a) 16 litres (b) 25 litres (c) 22.75 litres (d) 32.5 litres
Q.47. 20 litres of a mixture contain milk and water in the ratio 5 : 3. If 4 litres of this mixture are replaced by 4 litres of milk, the ratio of milk to water in the new mixture will become:
(a) 2 : 1 (b) 6 : 3 (c) 7 : 3 (d) 8 : 3
Q.48. An alloy contains zinc and copper in the ratio 5 : 8 and another alloy contains zinc and copper in the ratio 5 : 3. If equal amounts of both the alloys are melted together, then the ratio of zinc and cooper in the resulting alloy is:
(a) 25 : 24 (b) 3 : 8 (c) 103 : 105 (d) 105 : 103
Q.49. If ==, then = ?
(a) 7 (b) 2 (c) (d)
Q.50. If (a + b) : (a – b) = 1: 5, then (a2 – b2) : (a2 + b2) equals:
(a) 2 : 3 (b) 3 : 2 (c) 5 : 13 (d) 13 : 5
Q.51. Two whole numbers whose sum is 64 can not be in the ratio :
(a) 5 : 3 (b) 7 : 1 (c) 3 : 4 (d) 9 : 7