Ratio and Proportion – Basic Concepts, Formulas, Aptitude Questions and Solved Examples with Explanation (Video Lessons)

Formulas and Quick Tricks for Ratio and Proportion

1. If four quantities are in proportion, then Product of Means = Product of Extremes.
   In the proportion a:b::c:d, we have bc = ad
2. If a:b::c:x, x is called the fourth proportional of a, b, c.
   a/b = c/x or, x = bc/a
3. If two numbers are in a:b ratio and the sum of these numbers is x, then
   numbers will be ax/(a+b) and bx/(a+b) respectively
4. If three numbers are in the ratio a:b:c and the sum of these numbers is x, then
   these numbers will be ax/(a+b+c), bx/(a+b+c) and cx/(a+b+c) respectively
5. The ratio of two numbers is a : b. If n is added to each of these numbers, the ratio becomes c : d.
   The two numbers will be given as an(c-d)/(ad-bc) and bn(c-d)/(ad-bc) respectively.
6. The ratio of two numbers is a : b. If n is subtracted from each of these numbers, the ratio becomes c : d.
   The two numbers are given as an(d-c)/(ad-bc) and bn(d-c)/(ad-bc) respectively.
7. If the ratio of two numbers is a: b, then the numbers that should be added to each of the numbers in order to make this ratio c:d is given by (ad-bc)/(c-d)
8. If the ratio of two numbers is a:b, then the number that should be subtracted from each of the numbers in order to make this ratio c:d is given by (bc-ad)/(c-d)
9. The CP of the item that is cheaper is CP_cheaper and the CP of the item that is costlier (dearer) is CP_dearer. The CP of unit quantity of the final mixture is called the Mean Price and is given by:
   

   CPmp =

   CPcheaper − CPmean price CPmean price − CPcheaper

Questions and Solved Examples on Ratio and Proportion

Q.1: The 1st 3 terms of a proportion are 3, 9 and 12. The 4th term is:
A. 18
B. 24
C. 30
D. 36
Answer: D. 36
Explanation: (9 × 12) / 3 = 36

Q.2: A store owner is packing small radios into larger boxes that measure 25 × 42 × 60 inches. If the measurement of each radio is 7 × 6 × 5 inches, then how many radios can be placed in the box?
A. 260
B. 300
C. 340
D. 380
Answer: B. 300
Explanation: Total no's of radios can be placed in the box (25 × 42 × 60) / (7 × 6 × 5) = 6300 / 210 = 300

Q.3: 6 years ago, the ratio of the ages of Kunal and Sagar was 6:5. 4 years hence, the ratio of their ages will be 11:10. What is Sagar's age at present?
A. 12 years
B. 14 years
C. 16 years
D. 18 years
Answer: C. 16 years
Explanation: Let the ages of Kunal and sagar be x and y respectively.
According to the equation, (x - 6) / (y - 6) = 6 / 5
=> (x + 4) / (y + 4) = 11 / 10
After solving we get y = 16

Q.4: If 40% of a number is equal to 2/3rd of another number, what is the ratio of 1st number to the 2nd number?
A. 2 : 5
B. 3 : 7
C. 5 : 3
D. 7 : 3
Answer: C. 5 : 3
Explanation: Let 40% of A = 2 / 3 B. Then,
40A / 100 = 2B / 3 => 2A / 5 = 2B / 3
A / B = (2 / 3 × 5 / 2) = 5 / 3
Hence, A : B = 5 : 3


Q.5: A gardener wants to plant trees in his garden in rows in such a way that the number of trees in each row to be the same. If there are 24 rows the number of trees in each row is 42. If there are 12 more rows find the number of trees in each row?
A. 24 trees
B. 28 trees
C. 32 trees
D. 36 trees
Answer: B. 28 trees
Explanation: Required number of trees = 24 / 36 × 42 = 28

Q.6: The inverse ratio of 3 : 2 : 1 is?
A. 1 : 2 : 3
B. 2 : 3 : 1
C. 3 : 1 : 2
D. 2 : 3 : 6
Answer: D. 2 : 3 : 6
Explanation: 1 / 3 : 1 / 2 : 1 / 1 = 2 : 3 : 6

Q.7: If a:b = 4:1, then find (a – 3b) / (2a – b)?
A. 1 / 7
B. 2 / 7
C. 3 / 7
D. 5 / 7
Answer: A. 1 / 7
Explanation: a / b = 4 / 1 => a = 4b
(a – 3b) / (2a – b) = (4b – 3b) / (8b – b)
= b / 7b => 1 / 7

Q.8: The ratio of the incomes of Chetan and Dinesh is 3:4. The ratio of their expenditures is 5:7. If each of them saves Rs.200, find the incomes of both?
A. Rs.600, Rs.800
B. Rs.1200, Rs.1600
C. Rs.1500, Rs.2000
D. Rs.1800, Rs.2400
Answer: B. Rs.1200, Rs.1600
Explanation: The savings of Chetan and Dinesh are 3x – 5y and 4x – 7y respectively.
3x – 5y = 200 --- (1)
4x – 7y = 200 --- (2)
Multiplying (1) by 7 and (2) by 5 and subtracting the resultant equation (2) from resultant equation (1), we get x = 400.
The incomes of Chetan and Dinesh are 3x = Rs.1200 and 4x = Rs.1600 respectively.


Q.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?
A. Rs. 460
B. Rs. 484
C. Rs. 550
D. Rs. 664
Answer: B. Rs. 484
Explanation: 4 / 15 A = 2 / 5 B
=>A = (2 / 5 × 15 / 4) B
=>A = 3 / 2 B
=>A / B = 3 / 2
=>A : B = 3 : 2
Therefore B's share = Rs. (1210 × 2 / 5) = Rs. 484

Q.10: 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: C. 4 : 5
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

Q.11: Three Rabbits A, B and C move in such a way that when A takes 7 steps, B takes 8 steps and C takes 9 steps. But 4 steps of A are equal to 5 steps of B and 6 steps of C. What is the ratio of their speeds?
A. 28 : 40 : 54
B. 42 : 40 : 36
C. 35 : 32 : 30
D. 30 : 32 : 35
Answer: C. 35 : 32 : 30
Explanation: A : B : C = 7 : 8 : 9
Size of step, 4A = 5B = 6C
The ratio of speeds = 7 / 4 : 8 / 5 : 9 / 6
Hence, 35 : 32 : 30

Q.12: In the year 2019, in a company the ratio of employees in three different departments HR, MT and TA are 2:3:4 respectively. In the year 2020, in each department if 'n' number of employees left then the ratio becomes 3:5:7. In the year 2021 in each department 5 employees left then the ratio became 1:2:3. Then what is the total number of employees in the year 2020?
A. 70
B. 72
C. 75
D. 78
Answer: C. 75
Explanation: In the year 2019 = 2x : 3x : 4x
In the year 2020 = 3y : 5y : 7y
3y - 5 / 5y - 5 = ½ => y = 5
Employees in 2020 = 15, 25, 35
Hence, total number of employees in 2020 = 75.

Q.13: In a horse racing, there are three horses, A, B and C. The Payoffs at A is 3:7, at B, is 4:9, at C, is 5:11. Sharif bets Rs. 693 on a horse which would fetch maximum amount. Luckily his horse has won. Then what is the total amount won by him?
A. Rs. 990/-
B. Rs. 1001/-
C. Rs. 1008/-
D. Rs. 1011/-
Answer: C. Rs. 1008/-
Explanation: If he bets on horse A: 693 × 3 / 7 = 297
If he bets on horse B: 693 × 4 / 9 = 308
If he bets on horse C: 693 × 5 / 11 = 315
So, he bets on horse C and he wins Rs. 315/-
Total = 693 + 315 = Rs. 1008/-

Q.14: A, B, C and D are four friends living in a room. A bought some sweet and went to the room. He found no one is in the room. He then divided the sweets into four parts and ate one part. After that B came to the room they then divided the remaining sweets into four parts at one part each. After that C came to the room, now together they divided the remaining sweets into four parts and ate their parts. After that D came now together they divided the sweets into four parts and finished the sweets. D ate 6 sweets only. Then how many sweets does A eat?
A. 140
B. 142
C. 144
D. 146
Answer: B. 142
Explanation: D = 3x / 128 = 6
Total sweets = 256
A = x / 4 + 3x / 16 + 3x / 32 + 3x / 128
So, sweets that A ate = 142.

Q.15: An amount of money is to be divided between P, Q and R in the ratio of 3:7:12. If the difference between the shares of P and Q is Rs. x and the difference between Q and R's share is Rs. 3000/- Find the total amount of money?
A. Rs. 11,600/-
B. Rs. 12,400/-
C. Rs. 13,200/-
D. Rs. 14,300/-
Answer: C. Rs. 13,200/-
Explanation: 12a - 7a = 3000
5a = 3000 =>a = 600
7a - 3a = x =>x = 4a =>x = 2400
Total amount of money = 22 × 600 = Rs. 13,200/-