Ratios and Proportions short tricks and golden formula

Ratio and proportion is an important topic of aptitude and all exams have questions from this topic.

What is Ratio and Proportion ?

The relative size of two quantities expressed as the quotient of one divided by the other. The ratio can be expressed in two ways such as a/b or a:b. Equality between two ratios is called as proportions. These questions are easy to solve if you understand the formulae and concepts better. Ratios and proportions are frequently asked questions in the aptitude exams of government and interview papers.

Example 1:
The height of the building A is 100 metres.
The height of the building B is 1000 metres.
What is the ratio of these two buildings?

Solution: Let’s divide between A&B.

$\frac{10}{100} = \frac{1}{10}$

The ratio between A&B is 1:10
This means B is ten times of A.

If two ratios are equal to each other then it is called that the two ratios are proportionate to each other.
If A:B = C:D. Then it is called that the A:B is proportionate to C:D.
It is symbolized as A:B :: C:D
It is also written as

$\frac{A}{B} = \frac{C}{D}$

Important note:
If we have two ratios such as A:B and B:C and we are supposed to find A:B:C?
Then make B same in both the ratios (Since B is only common between both the ratios) and give the answer as A:B:C. The following example will make the above note crystal clear.

Question 1: If A:B is 7:3 then B:C is 6:5. What is the ratio of A:B:C ?

Solution: Given,
A:B = 7:3 — (1)
B:C = 6:5 — (2)
Only B is common between (1)&(2)
The value of B is 3 in (1) and 6 in (2).
We are supposed to take LCM of 3 and 6.
LCM of 3 & 6 is 6.
We need to make B as 6 in both the equation.
So, the first equation has to multiplied by 2 to make B as 6.
The second equation already has B as 6. We need not to multiply anything.
So, (1) * 2
New A:B = 7:3 *2 = 14:6

New A:B = 14:6
B:C = 6:5
B has become equal.
The ratio of A:B:C = 14:6:5

Answer: 14:6:5

Important Note –
Ratio can be reduced/increased directly wrt to percentages.

Question 2: The ratio of number of cans of orange, pineapple and mixed fruits kept in store is 8:9:15. If the store sells 25%, 33.33%, and 20% of orange, pineapple and mixed fruits juices cans respectively, then what is the ratio of number of cans of these juices in the remaining stock?

Solution: Given
Orange = 25% sold = 100 – 25% = 75% remaining
Pineapple = 33.33% sold = 100 – 33.33% = 66.66% remaining
Mixed fruit = 20% sold = 100 – 20% = 80% remaining

The cans present were 8:9:15 for orange, pineapple and mixed fruites respectively

$Required ratio = 8*\frac{75}{100} : 9*\frac{66.66}{100} : 15*\frac{80}{100}$

Required ratio = 6:6:12
If we take six as common, then
the ratio becomes 1:1:2

Answer: 1:1:2

Question 3: The salaries of A, B and C are in the ratio of 2:3:5. If the increments of 15%, 10% and 20% are allowed respectively in the salaries, then what will be the new ratio of the salaries?

Solution: Given
A = 15% increment = 100 + 15 = 115%
B = 10% increment = 100 + 10 = 110%
C = 20% increment = 100 + 20 = 120%

The ratios are A = 2; B = 3; C = 5.

$Required ratio = 2*\frac{115}{100} : 3*\frac{110}{100} : 5*\frac{120}{100}$

$Required ratio = 2.3: 3.3: 6$

Answer: 2.3:3.3:6

Question 4: Rs. 3200 is divided among A, B & C in the ratio of 3:5:8. What is the difference (in Rs.) between in the share of B & C?

Solution: For these type of questions add all the ratio and divide it with its own ratio multiplied along with amount.
The ratio of A,B&C is 3:5:8
Add all the ratio = 3+5+8 = 16

$Share-of-A = \frac{3}{16}*3200 = 600$

$Share-of-B = \frac{5}{16}*3200 = 1000$

$Share-of-C = \frac{8}{16}*3200 = 1600$

The difference of share between B and C = 1600 – 1000 = 600

OR

We can also do this sum through
Marking all the ratios as x and finding x
A = 3x; B = 5x; C = 8x
Adding all these will give 16x
Total of all these is 3200; which is 16x = 3200
x = 200
The difference between B&C = 8x – 5x = 3x
We know that X = 200
3x = 3*200 = 600
Difference between B&C = 600

Answer: 600

Question 5: The flight fare between two cities is increased in the ratio 11:13. What is the increase in the fare, if the original fare was Rs. 12,100?

Solution: The ratio from old fare to new fare = 11:13
We are going to solve by marking these both as x
Ratio = 11x:13x
Original fare = 12,100

$11x = 12,100$

$x = \frac{12,100}{11} = 1100$

Now, we know x = 1100
New fare = 13x = 13*1100 = 14300

The increase in the fare = New fare – Original fare = 14300 – 12100 = 2,200

Answer: 2,200

Question 6: A company at the time of inflation reduced the staff in the ratio 5:3 and average salary per employee is increased in the ratio 7:8. By doing so, the company saved 55,000. What was the initial expenditure of the company?

Solution: Given
The ratio from old staff to new staff = 5:3
The ratio of old average salary to new average salary = 7:8
Let’s mark the ratio of staff as x and average salary as y
The ratio from old staff to new staff = 5x:3x
The ratio of old average salary to new salary = 7y:8y

When there were old staff (5x) the old salary was (7y)
Now, for the new staff (3x) the new salary is (8y).

So
The original or previous expenditure = 5x*7y = 35xy
The new or current expenditure = 3x*8y = 24xy

It is said that the company saved 55,000 by doing so. This means that the difference between the old and new expenditure is 55,000

$35xy - 24xy = 55,000$

$11xy = 55,000$

$xy = \frac{55,000}{11} = 5,000$

The question demands the old expenditure

$35xy = 35*5,000 = 1,75,000$

Answer: 1,75,000

Important
Find Third proportional
C is the third proportional to A and B if
If there is a:b:c then this means that a/b = b/c
If we cross multiply then we will get

$b^2 = a*c$

Find Fourth proportional
If there is a:b::c:d = a/b = c/d

Question 7: What is the third proportional to 10 & 25?

Solution: It is the given in the ratio of a:b:c

$\frac{a}{b} = \frac{b}{c}$

If we cross multiply

$b^2 = a*c$

We know that b = 25 and a = 10
We already know the formula which is

$b^2 = a*c$

$25^2 = 10 * c$

$625 = 10c$

$c = \frac{625}{10} = 62.5$

Answer: 62.5

Question 8: What is the fourth proportional to 72, 168, & 150?

Solution: It is given in the ratio a:b = c:d

$\frac{a}{b} = \frac{c}{d}$

Given; a=72, b=168 & c=150

$\frac{72}{168} = \frac{150}{d}$

After reducing

$d = 350$

Answer: 350

Question 9: If 3/5p = 7/2q = 7/5r, then what is the ratio of p, q and r respectively?

Solution: Let us consider k as constant for all the three ratios as shown below

$\frac{3}{5}p =k$