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
There are some pigeons and hares in a zoo. If heads are counted, there are 200. If legs are counted, there are 580. The number of hares in the zoo is?
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Correct Answer : A. 1:1
Description :
5/6 * 3/2 * 4/5 = 1/1
1:1
View Answer
50/50
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Question 2
Some ballons are divided between two persons in the ratio of 3:5. If one person gets 20 less ballons than that of the other, find the total number of ballons.
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Correct Answer : A. 80
Description :
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Question 3
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 : C. 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
Question 4
If a:b=2:5, b:c=3:7 and c:d=1:3 find a:b:c:d?
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Correct Answer : A. 6:15:35:105
Description :
a:b = 2:5, b:c = 3:7, c:d = 1:3
2:5
3:7
( a = 2 ? 3 = 6, b = 5 ? 3 = 15 and c = 5 ? 7 = 35)
(a = a ? b, b= b ? b and c= b ? c)
a:b:c = 6 : 15 : 35
a:b:c = 6 : 15 : 35 and c:d = 1 : 3
(Note: First a,b,c multiplication with c means 1 and last c means 35
multiplication with d means 3
a:b:c:d = 6:15:35:105
Question 5
If 40% of a number is equal to two-third of another number, what is the ratio of first number to the second number.
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Correct Answer : A. 5:3
Description :
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Question 6
A bag contains 50p,Rs.1/- and Rs2/- coins in the ratio of 4:2:1 respectively. If the total money in the bag is Rs.60/-. Find the number of 50p coins in that bag?
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Correct Answer : C. 40 coins
Description :
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Question 7
Narendra started a business, investing Rs 30,000. Six months later, Chandan joined him with Rs 15,000. If they make a profit of Rs 10,000 at the end of the year, what should be the share of Narendra ?
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Correct Answer : D. Rs 8000
Description :
30000 x 12 : 15000 x 6
= 4 : 1
Because, Narendra's share = 44/5 x 10000 = Rs. 8000
Question 8
15 binders can bind 1400 books in 21 days. How many binders will be required to bind 1600 books in 20 days?
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Correct Answer : B. 18
Description :
Binders Books Days
15 1400 21
x 1600 20
x/15 = (1600/1400) * (21/20) => x = 18
Question 9
The least whole number which when subtracted from both the terms of the ratio 6 : 7 to give a ratio less than 16 : 21, is
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Correct Answer : B. 3
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Question 10
Find the mean proportional of the following pairs of numbers 6 and 24.
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Correct Answer : C. 12
Description :
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