Formulae
1. Ratio:
The ratio of two quantities a and b in the same units, is the fraction \( a \over b \)and we write it as a : b.
In the ratio a : b, we call a as the first term or antecedent and b, the second term or consequent.
Eg. The ratio 5 : 9 represents \( 5 \over 9 \) with antecedent = 5, consequent = 9.
Rule: The multiplication or division of each term of a ratio by the same non-zero number does not affect the ratio.
Eg. 4 : 5 = 8 : 10 = 12 : 15. Also, 4 : 6 = 2 : 3.
2. Proportion:
The equality of two ratios is called proportion.
If a : b = c : d, we write a : b :: c : d and we say that a, b, c, d are in proportion.
Here a and d are called extremes, while b and c are called mean terms.
Product of means = Product of extremes.
Thus, a : b :: c : d ⇔ (b x c) = (a x d).
3. Fourth Proportional:
If a : b = c : d, then d is called the fourth proportional to a, b, c.
Third Proportional:
a : b = c : d, then c is called the third proportion to a and b.
Mean Proportional:
Mean proportional between a and b is \( \sqrt {ab} \).
4. Comparison of Ratios:
We say that (a : b) > (c : d) ⇔ \( a \over b \) > \( c \over d \)
Compounded Ratio:
The compounded ratio of the ratios: (a : b), (c : d), (e : f) is (ace : bdf).
5. Duplicate Ratios:
Duplicate ratio of (a : b) is (a2 : b2).
Sub-duplicate ratio of (a : b) is (√a : √b).
Triplicate ratio of (a : b) is (a3 : b3).
Sub-triplicate ratio of (a : b) is (a1/3 : b1/3).
If \( a \over b \) = \( c \over d \) , then \( {a + b} \over {a - b} \) = \( {c + d} \over {c - d} \). [componendo and dividendo]
6. Variations:
We say that x is directly proportional to y, if x = ky for some constant k and we write, x ∝ y.
We say that x is inversely proportional to y, if xy = k for some constant k and
we write, x ∝ \( 1 \over y \)
Exercise : Ratio and Proportion MCQ Questions and Answers
Question 1
Rs.1170 is divided so that 4 times the first share, thrice the 2nd share and twice the third share amount to the same. What is the value of the third share?
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Correct Answer : D. Rs.540
Description :
A+B+C = 1170
4A = 3B = 2C = x
A:B:C = 1/4:1/3:1/2 = 3:4:6
6/13 * 1170 = Rs.540
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Question 2
In a mixture of 65 litres; milk and water are in the ratio of 3:2 . What are the quantities of milk and water in the mixture?
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Correct Answer : C. 39, 26
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Question 3
There is food for 760 men for 22 days. How many more men should join after two days so that the same food may last for 19 days more?
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Correct Answer : D. 40 men
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Question 4
The ages of three persons A, B and C are in the ratio of 4:7:14 and the ages of B, C and D are in the ratio of 2:4:5. What is the ratio of the ages of A, B, C and D ?
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Correct Answer : B. 8:14:28:35
Description :
A:B:C = 4:7:14 ......(i)
Multiplying (i) by 2,
A:B:C = 8:14:28
B:C:D = 2:4:5
Multiply (ii) by 7: B:C:D = 14:28:35
∴ From these two equations,
A:B:C:D = 8:14:28:35
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Question 5
The weights of three boys are in the ratio 4:5:6. If the sum of the weights of the heaviest and the lightest boy is 45 kg more than the weight of the third boy, what is the weight of the lightest boy?
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Correct Answer : C. 36 kg
Description :
Let the weights of the three boys be 4k, 5k and 6k respectively.
4k + 6k = 5k + 45
=> 5k = 45 => k = 9
Therefore the weight of the lightest boy = 4k = 4 * 9 = 36 kg.
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Question 6
Amar started a business, investing Rs 64000. Six months later, Ajay joined with Rs 48000. In what ratio should the profit they earned be distributed at the end of year ?
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Correct Answer : B. 8 : 3
Description :
Ratio = 64000 x 12 : 48000 x 6
= 768000 : 288000 = 8 : 3
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Question 7
A fraction bears the same ratio to 1/27 as 3/7 does to 5/9. The fraction is?
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Correct Answer : C. 1/35
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Question 8
Seats for Mathematics, Physics and Biology in a school are in the ratio 5:7:8. There is a proposal to increase these seats by 40%, 50% and 75% respectively. What will be the ratio of increased seats ?
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Correct Answer : D. 2:3:4
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Question 9
A and B together have Rs. 1210. If 4/15 of A's amount is equal to 2/5 of B's amount, how much amount does B have?
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Correct Answer : C. 484 Rs.
Description :
4/15 A = 2/5 B => A = (2/5 * 15/4)B
A =3/2 B => A/B = 3/2 => A:B = 3:2
B's share = 1210 * 2/5 = Rs. 484.
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Question 10
A garrison of 2000 men has provisions for 54 days. At the end of 15 days, a reinforcement arrives, and it is now found that the provisions will last only for 20 days more. What is the reinforcement?
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Correct Answer : C. 1900 men
Description :
2000 ---- 54
2000 ---- 39
x ----- 20
x*20 = 2000*39
x = 3900
2000
-------
1900
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