Third Proportional MCQ Quiz - Objective Question with Answer for Third Proportional - Download Free PDF

Given:

First number = 28

Second number = 112

Formula used:

If a, b, and c are in proportion, then:

\(\dfrac{a}{b} = \dfrac{b}{c}\)

Third proportional (c) = \(\dfrac{b^2}{a}\)

Calculation:

Third proportional = \(\dfrac{b^2}{a}\)

⇒ c = \(\dfrac{112^2}{28}\)

⇒ c = \(\dfrac{12544}{28}\)

⇒ c = 448

∴ The third proportional is 448, and the correct answer is option (1).

Given:

First number = 36

Second number = 42

Concept:

If a : b = b : c, then c is called the third proportional to a and b.

Formula Used:

Third Proportional = (b × b) / a

Calculation:

Third Proportional = (42 × 42) / 36

= 1764 / 36

= 49

∴ The third proportional of 36 and 42 is 49.

Given:

First proportional = x

Second proportional = x + 100

Third proportional = 405

Condition: x > 100

Formula Used:

If a, b, and c are in continued proportion, then b² = ac.

Here, a = x, b = x + 100, and c = 405.

Calculation:

Using the formula for continued proportion:

(x + 100)² = x × 405

⇒ x² + 2 × x × 100 + 100² = 405x

⇒ x² + 200x + 10000 = 405x

⇒ x² + 200x - 405x + 10000 = 0

⇒ x² - 205x + 10000 = 0

Solve the quadratic equation using the quadratic formula: x = [-b ± √(b² - 4ac)] / 2a

Here, a = 1, b = -205, c = 10000.

Discriminant (D) = b² - 4ac = (-205)² - 4 × 1 × 10000

D = 42025 - 40000

D = 2025

√D = √2025 = 45

Now, find the values of x:

x₁ = [-(-205) + 45] / (2 × 1) = (205 + 45) / 2 = 250 / 2 = 125

x₂ = [-(-205) - 45] / (2 × 1) = (205 - 45) / 2 = 160 / 2 = 80

We are given that x > 100.

Therefore, the value of x is 125.

∴ The value of x is 125.

Given:

First proportional = 3x²

Second proportional = 4xy

Third proportional = 48

Formula Used:

If a, b, and c are in continued proportion, then b² = ac.

Here, a = 3x², b = 4xy, and c = 48.

Calculation:

(4xy)² = (3x²) × 48

⇒ 16x²y² = 144x²

⇒ y² = 144x² / 16x²

⇒ y² = 9

⇒ y = ±√9

⇒ y = ±3

We need the positive value of y.

∴ The positive value of y is 3.

Given:

First term (a) = (3 + √2)

Second term (b) = 2√7

Formula used:

If a, b, and c are in continued proportion, then b² = a × c

Calculation:

Let the third proportional be c.

\(2√7)^2 = (3 + √2) \times c\)

⇒ 4 × 7 = (3 + √2) × c

⇒ 28 = (3 + √2) × c

⇒ c = \( \dfrac{28}{3 + √2} \)

Rationalize the denominator:

⇒ c = \( \dfrac{28}{3 + √2} \times \dfrac{3 - √2}{3 - √2} \)

⇒ c = \( \dfrac{28(3 - √2)}{9 - 2} \)

⇒ c = \( \dfrac{28(3 - √2)}{7} \)

⇒ c = 4(3 - √2)

∴ The correct answer is option (4).

Given:

We have to obtain the third proportional to 9 and 15

Concept Used:

Concept of ratio and proportion

Calculation:

Let, the third proportional be x

Then,

9 : 15 : : 15 : x

⇒ 9/15 = 15/x

⇒ x = (15 × 15) / 9

⇒ x = 25

∴ The required third proportional to 9 and 15 is 25.

Given:

First number (a) = (x2 - y2)

Second number (b) = (x - y)

Formula used:

Third proportional = {2nd number (b)}²/first number (a)

(x2 - y2) = (x - y) × (x + y)

Calculation:

Third proportional = (x - y)²/(x2 - y2)

⇒ {(x - y) × (x - y)}/{(x - y) × (x + y)} 

⇒ \(\rm \frac{x-y}{x+y}\)

∴ The correct answer is \(\rm \frac{x-y}{x+y}\).

Concept used:

The third proportional proportion is the second term of the mean terms.

For example, if we have a ∶ b = c ∶ d, then the term ‘c’ is the third proportional to ‘a’ and ‘b’.

Represented as:

a : b ∷ b : c

Calculation:

Let the third proportion to 16 and 24 be x

⇒ 16/24 = 24/x

⇒ x = (24 × 24)/16

⇒ x = 36

∴ The third proportional to 16 and 24 is 36

Concept used:

Third proportion- a ∶ b ∶ ∶ b ∶ c

Calculation:

Let the number added be x

16 ∶ 28 ∶∶ 28 ∶ (40 + x)

16/28 = 28/(40 + x)

40 + x = (28 × 28)/16

⇒ x = 9

Given data:

First term = b² - a²

Second term = b² - ab

Concept: Third proportional to two given terms x and y is (y² / x).

Step-by-step solution:

Third proportional = (b² - ab)² / (b² - a²) = \(\rm \frac{b^2(b-a)}{(b+a)}\)

Hence, the third proportional of (b² - a²) and (b² - ab) is \(\rm \frac{b^2(b-a)}{(b+a)}\).

Given:

Numbers = 12, 24 and 27

Calculation:

Forth proportions 12, 24 and 27 is n,

⇒ 12 : 24 :: 27 : n 

⇒ 12/24 = 27/n

⇒ n = 54

Then,

Third proportional to A and 36 is 54.

⇒ A : 36 = 36 : 54

⇒ 54A = 36²

⇒ A = 24

∴ The value of A is 24.

Given:

Find the third proportion of x and 30, when 45 : 12 : : 75 : x.

Formula used:

Third Proportional:

Let 'z' be the third proportional for a and b.

then, (a : b :: b : z)

Therefore, 

z = \(\frac{b^2}{a}\)

Calculation:

According to the question,

45 : 12 : : 75 : x.

It can be written as:

⇒ \(\frac{45}{12} = \frac{75}{x}\)

⇒ x = \(\frac{12 \times 75}{45}\) = 20

Now, 

Let y be the third proportional of 20 and 30.

⇒ y = \(\frac{30^2}{20}\)= 45

The third proportional of 20 and 30 = 45

Therefore, '45' is the required answer.

Additional Information

1. First Proportional:

Let 'x' be the first proportional for a, b and c.

then, (x : a :: b : c)

Therefore, 

x = \(\frac{ab}{c}\)

2. Mean Proportional:

Let 'x' be the mean proportional for a and b.

then, (a : x :: x : b)

Therefore, 

x = \(\sqrt{ab}\)

3. Fourth Proportional:

Let 'x' be the first proportional for a, b and c.

then, (a : b :: c : x)

Therefore, 

x = \(\frac{bc}{a}\)

Given:

a = 2

b = 3

Concept:

We need to find the third proportional to a³ + b³ and a² + ab + b².

Solution:

⇒ Substitute a and b into the two expressions to get the first and second numbers.

⇒ First number = a³ + b³ = 2³ + 3³ = 8 + 27 = 35

⇒ Second number = a2 + ab + b2 = 22 + 2*3 + 32 = 4 + 6 + 9 = 19

⇒ The third proportional (T) to two numbers (x and y) is given by the formula T = (y²)/x

So, substituting the first and second numbers:

⇒ T = (19²)/35 = 10.31

Therefore, the third proportional to a³ + b³ and a² + ab + b², when a = 2 and b = 3, is approximately 10.31 (correct to two decimal places).

Given:

The expression = 3x2, 4xy and 48

Concept:

If a, b, and c are in proportion.

\({a\over b}={b\over c}\)

Formula used:

If a, b, and c are in proportion then third proportion

\(c={b^2\over a}\)

Calculation:

According to the question

⇒ 48 = \({({4xy})^2\over 3x^2}={16x^2y^2\over3x^2}={16y^2\over3}\)

⇒ 3 × 3 = y2

⇒ y =√(3 × 3) = 3

∴ The required result will be 3.

Given:

p is the third proportional to 3, 9

Calculation:

Let the fourth proportion be x

p is the third proportional to 3, 9

⇒ 3/9 = 9/p

⇒ 3p = 81

⇒ p = 27

Now,

The fourth proportion is

⇒ 6/27 = 4/x

⇒ 6x = (27 × 4)

⇒ 6x = 108

⇒ x = 18

∴ The value of fourth proportion is 18