Ratio & Proportion

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Q. Find First , Mean , Third , Fourth Proportional.

1. First Proportional

Q. What is the first proportional of 12 and 8 ?

Formula: a²/b

Trick : : Square the 1st number and divide by 2nd

Solⁿ

C = 12²/8 = 144/8 = 18

Ans: 18

2. Mean Proportional

Q.What is the mean proportional of 18 and 50 ?

Formula: Mean = sqrt a×b​

Trick : Multiply the numbers then take square root

Solⁿ

sqrt 18×50 = 900 = 30

Ans: 30

3. Third Proportional

Q. What is the third proportional of 8 and 20 ?

Formula : If a : b = b : c

                                   c = b²/a

Trick: Square the 2^nd number, divide by 1st

Solⁿ

C = 20²/8 = 400/8 = 50

Ans: 50

 4. Fourth Proportional

Q. What is the fourth proportional of 6, 16 and 9 ?

Formula: If a : b = c : d

                             D = b×c/a

Trick: Multiply 2nd × 3rd, divide by 1st

Solⁿ

d = 16×9/6 = 144/6 = 24

Ans: 24

Q.The ratio of two quantities is 8 : 15. If the first quantity is 40, what is the second quantity ?

Options: (a) 120 (b) 75 (c) 45 (d) 15

Solⁿ

Given ratio: First quantity : Second quantity = 8 : 15
Given: First quantity = 40

Write the ratio as: 40 : Second quantity = 8 : 15

This means: Second quantity = 40×15/8

Calculate: Second quantity = 600/8=75

Ans: (b) 75

Explanation: When the first quantity (8 parts) becomes 40, each part becomes 40 ÷ 8 = 5
So the second quantity (15 parts) = 15×5=75.

Q. The ratio of salary of A and B is 5:3 the difference between them is ₹40 so what is the salary of A ?

Given:

Ratio of salaries (A : B) = 5 : 3
Difference = ₹40

Assume each ratio value = Y

5Y − 3Y = 40

2Y = 40

Y = 40 ÷ 2 = 20

Now calculate salaries:

Salary of A = 5Y = 5 × 20 = ₹100

Salary of B = 3Y = 3 × 20 = ₹60

Ans: A’s salary = ₹100, B’s salary = ₹60 Difference = 100 − 60 = ₹40

Q. If a² + b² = c² prove that a⁶ + b⁶ + 3a²b²c² = c⁶

Solⁿ

1^st : Start with the left-hand side (LHS) = a⁶+b⁶+3a²b²c²

Write powers in cube form: = (a²)³ + (b²)3 + 3a²b²c²

2^nd : Use the identity: x³ + y³ + 3xy(x+y) = (x+y)³

Here, x = a², y = b²

So, (a²)³ + (b²)³ + 3(a²)(b²)(a²+b²) = (a²+b²)³

3^rd : Substitute the given value

We know^: a² + b² = c²

So, (a²+b²)³ = (c²)³ = c⁶

Ans : a⁶ + b⁶ + 3a²b²c² = c⁶