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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

The ratio of first and second class train fares between two stations is 40:1 and that of the number of passengers travelling between these stations by first and second class is 1:20. If on a particula

A.  
B.  
C.  
D.  

Correct Answer : B. Rs.1800/-

Description :
No answer description available for this question. Let us discuss.

Question 2

A box contains rupee, fifty paise and twenty five paise coins whose values are in the proportion of 2:3:4. if the total number of coins are 480 find the value of each coins and total amount in rupees

A.  
B.  
C.  
D.  

Correct Answer : C. 40, 60, 80 and 180

Description :
No answer description available for this question. Let us discuss.

Question 3

In 4 : 7 : : 16 : 28, 7 and 16 are called

A.  
B.  
C.  
D.  

Correct Answer : D. middle terms

Description :
No answer description available for this question. Let us discuss.

Question 4

If a:b = 4:1, then find (a - 3b)/(2a - b)?

A.  
B.  
C.  
D.  

Correct Answer : C. 1/7

Description :
a/b = 4/1 => a = 4b
(a - 3b)/(2a - b) = (4b - 3b)/(8b - b)
= b/7b => 1/7

Question 5

Divide Rs.118000 among three persons A, B and C such that the ratio of the shares of A and B is 3:4 and that of B:C is 5:6?

A.  
B.  
C.  
D.  

Correct Answer : B. A = 30000 B = 40000 C = 48000

Description :
Compound ratio of A:B:C
A:B = 3:4
B:C = 5:6
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A:B:C = 15:20:24
We can divide Rs.118000 in this ratio.
Share of A = 15/59 * 118000 = 30000
Share of B = 20/59 * 118000 = 40000
Share of C = 24/59 * 118000 = 48000

Question 6

Two numbers are in the ratio 7 : 9. If the sum of the numbers is 112, then the larger number is:

A.  
B.  
C.  
D.  

Correct Answer : B. 63

Description :
No answer description available for this question. Let us discuss.

Question 7

If an amount of Rs 168480 is distributed equally among 72 persons. How much amount would each person get ?

A.  
B.  
C.  
D.  

Correct Answer : C. Rs. 2340

Description :
Total amount = 168480
No. of persons = 72
each person get amount = 168480/72
= 2340

Question 8

The milk & water in two vessels A & B are in 4:3 and 2:3. In what ratio, the liquids be mixed in both the vessels so that the new liquid contains half milk and half water?

A.  
B.  
C.  
D.  

Correct Answer : D. 7 : 5

Description :
No answer description available for this question. Let us discuss.

Question 9

The average age of two persons A and B is 32 years. After four years, the ratio of the ages of A and B will be 5:13. What is the present age of B ?

A.  
B.  
C.  
D.  

Correct Answer : C. 48 years

Description :
Let the ages of A and B be x and y respectively.
x + y/2 = 32 ∴ x + y = 64 ....(1)
x + 4/y + 4 = 5/13
∴ 13x + 52 = 5y + 20or, 13x - 5y = -32 .....(2)
From equation (1) and (2)
x = 16, y = 48

Question 10

In a colony of 70 residents, the ratio of the number of men and women is 4:3. Among the women, the ratio of the educated to the uneducated is 1:4. If the ratio of the number of education to uneducated

A.  
B.  
C.  
D.  

Correct Answer : A. 2:3

Description :
Number of men in the colony = 4/7 (70) = 40
Number of women in the colony = 3/7 (70) = 30
Number of educated women in the colony = 1/5 (30) = 6
Number of uneducated women in the colony = 4/5 (30) = 24
Number of educated persons in the colony = 8/35 (70) = 16
As 6 females are educated, remaining 10 educated persons must be men.
Number of uneducated men in the colony = 40 - 10 = 30
Number of educated men and uneducated men are in the ratio 10:30 => 1:3

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