Ratio:
The ratio of two quantities a and b in the same units, is the fraction a/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 5/9 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.
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).
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 ab.
Comparison of Ratios:
We say that (a : b) > (c : d) <=> a/b > c/d
Compounded Ratio:The compounded ratio of the ratios: (a : b), (c : d), (e : f) is (ace : bdf).
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/b = c/d, then a + b/a-b = c + d/c-d [componendo and dividendo]
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/y
MCQS
A and B together have Rs. 1210. If of A’s amount is equal to of B’s amount, how much
amount does B have?
A. Rs. 460
B. Rs. 484
C. Rs. 550
D. Rs. 664
Answer: Option B
Explanation:
4/15 A = 2/5 B
A =(2/5 x 15/4)B
A = 3/2B
A/B = 3:23
A : B = 3 : 2.
B’s share = Rs. 1210 x 2/5 = Rs. 484.
Two numbers are respectively 20% and 50% more than a third number. The ratio of the two
numbers is:
A. 2 : 5
B. 3 : 5
C. 4 : 5
D. 6 : 7
Answer: Option C
Explanation:
Let the third number be x.
Then, first number = 120% of x = 120x/100 = 6x/5
Second number = 150% of x =150x/100 = 3x/2
Ratio of first two numbers = (6x/5 : 3x/2) = 12x : 15x = 4 : 5.
A sum of money is to be distributed among A, B, C, D in the proportion of 5 : 2 : 4 : 3. If C
gets Rs. 1000 more than D, what is B’s share?
A. Rs. 500
B. Rs. 1500
C. Rs. 2000
D. None of these
Answer: Option C
Explanation:
Let the shares of A, B, C and D be Rs. 5x, Rs. 2x, Rs. 4x and Rs. 3x respectively.
Then, 4x – 3x = 1000
x = 1000.
B’s share = Rs. 2x = Rs. (2 x 1000) = Rs. 2000.
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?
A. 2 : 3 : 4
B. 6 : 7 : 8
C. 6 : 8 : 9
D. None of these
Answer: Option A
Explanation:
Originally, let the number of seats for Mathematics, Physics and Biology be 5x, 7x and 8x respectively.
Number of increased seats are (140% of 5x), (150% of 7x) and (175% of 8x).
(140/100 x 5x) ,(150/100 x 7x) and (175/100 x 8x)
7x, 21x/2 and 14x.
The required ratio = 7x : 21x/2 : 14x
14x : 21x : 28x
2 : 3 : 4.
In a mixture 60 litres, the ratio of milk and water 2 : 1. If this ratio is to be 1 : 2, then the quanity
of water to be further added is:
A. 20 litres
B. 30 litres
C. 40 litres
D. 60 litres
Answer: Option D
Explanation:
Quantity of milk =(60 x 2/3) litres = 40 litres.
Quantity of water in it = (60- 40) litres = 20 litres.
New ratio = 1 : 2
Let quantity of water to be added further be x litres.
Then, milk : water =(40/20 +x)
Now(40/20 + x) = 1/2
20 + x = 80
x = 60.
Quantity of water to be added = 60 litres.
The ratio of the number of boys and girls in a college is 7 : 8. If the percentage increase in the
number of boys and girls be 20% and 10% respectively, what will be the new ratio?
A. 8 : 9
B. 17 : 18
C. 21 : 22
D. Cannot be determined
Answer: Option C
Explanation:
Originally, let the number of boys and girls in the college be 7x and 8x respectively.
Their increased number is (120% of 7x) and (110% of 8x).
=> (120/100x 7x) and (110/100 x 8x)
=> 42x/5 and 44x/5
The required ratio = 42x/5 : 44x/5 = 21 : 22.
Salaries of Raiz and Salman are in the ratio 2 : 3. If the salary of each is increased by Rs.
4000, the new ratio becomes 40 : 57. What is Salman’s salary?
A. Rs. 17,000
B. Rs. 20,000
C. Rs. 25,500
D. Rs. 38,000
Answer: Option D
Explanation:
Let the original salaries of Raiz and Salman be Rs. 2x and Rs. 3x respectively.
Then,2x + 4000/3x + 4000 = 40/57
57(2x + 4000) = 40(3x + 4000)
6x = 68,000
3x = 34,000
Salman’s present salary = (3x + 4000) = Rs.(34000 + 4000) = Rs. 38,000.
If 0.75 : x :: 5 : 8, then x is equal to:
A. 1.12
B. 1.2
C. 1.25
D. 1.30
Answer: Option B
Explanation:
(x x 5) = (0.75 x 8) x = (6/5) = 1.20
The sum of three numbers is 98. If the ratio of the first to second is 2 :3 and that of the second to
the third is 5 : 8, then the second number is:
A. 20
B. 30
C. 48
D. 58
Answer: Option B
Explanation:
Let the three parts be A, B, C. Then,
A : B = 2 : 3 and B : C = 5 : 8 = (5 x 3/5) : (8 x 3/5) = 3 : 24/5
A : B : C = 2 : 3 : 24/5 = 10 : 15 : 24
B = (98 x 15/49) = 30.
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