If x = 2 a + 1 + 2 a - 1 2 a + 1 - 2 a - 1 , prove that x 2 - 4 a x + 1 = 0

We can write the given expression as 2 a + 1 + 2 a - 1 2 a + 1 - 2 a - 1 = x 1 Apply Componendo and dividendo, we get, ⇒ 2 2 a + 1 2 2 a - 1 = x + 1 x - 1 ⇒ 2 a + 1 2 a - 1 = x + 1 x - 1 Squaring both sides, we get, 2 a + 1 2 a - 1 = x + 1 2 x - 1 2 Again applying componendo and dividendo. 2 a + 1 + 2 a - 1 2 a + 1 - 2 a + 1 = ( x + 1 ) 2 + ( x - 1 ) 2 ( x + 1 ) 2 - ( x - 1 ) 2 = 2 x 2 + 1 4 x ⇒ 2 a 1 = x 2 + 1 2 x ⇒ 4 ax = x 2 + 1 ⇒ x 2 - 4 ax + 1 = 0 Hence, proved.

If a + b - c a + b = b + c - a b + c = c + a - b c + a and a + b + c ≠ 0 then prove that a = b = c .

The given equation is a + b - c a + b = b + c - a b + c = c + a - b c + a . We can write the given equation as a + b a + b - c a + b = b + c b + c - a b + c = c + a c + a - b c + a ⇒ 1 - c a + b = 1 - a b + c = 1 - b c + a ⇒ - c a + b = - a b + c = - b c + a ⇒ a + b c = b + c a = c + a b ⇒ a + b + c c = b + c + a a = c + a + b b [Componendo] ⇒ 1 c = 1 a = 1 b ⇒ a = b = c [Invertendo] Hence, proved.

If cos 2 θ – sin 2 θ = 1 2 , then find the value of tan 2 θ .

The given equation is cos 2 θ – sin 2 θ = 1 2 By applying componendo and dividendo rule we get, cos 2 θ - sin 2 θ + 1 cos 2 θ - sin 2 θ - 1 = 1 + 2 1 - 2 ⇒ cos 2 θ - sin 2 θ + sin 2 θ + cos 2 θ cos 2 θ - sin 2 θ - sin 2 θ + cos 2 θ = 1 + 2 1 - 2 ⇒ 2 cos 2 θ - 2 sin 2 θ = 3 - 1 ⇒ sin 2 θ cos 2 θ = 1 3 (By invertendo) ⇒ tan 2 θ = 1 3 Hence, the value of tan 2 θ is 1 3 .

Using properties of proportion find x : y , given: x 2 + 2 x 2 x + 4 = y 2 + 3 y 3 y + 9

Using properties of proportion we have to find x : y , given: x 2 + 2 x 2 x + 4 = y 2 + 3 y 3 y + 9 Applying componendo and dividendo, ⇒ x 2 + 2 x + 2 x + 4 x 2 + 2 x - 2 x - 4 = y 2 + 3 y + 3 y + 9 y 2 + 3 y - 3 y - 9 ⇒ ( x + 2 ) 2 ( x - 2 ) ( x + 2 ) = ( y + 3 ) 2 ( y - 3 ) ( y + 3 ) ⇒ ( x + 2 ) 2 ( x - 2 ) 2 = ( y + 3 ) 2 ( y - 3 ) 2 [ because when we split ( x + 2 ) 2 and ( y + 3 ) 2 we will get, x 2 + 4 x + 4 and y 2 + 6 y + 9 ] ⇒ x + 2 x - 2 = y + 3 y - 3 Applying componendo and dividendo again, ⇒ x + 2 + x - 2 x + 2 - x + 2 = y + 3 + y - 3 y + 3 - y + 3 ⇒ 2 x / 4 = 2 y / 6 ⇒ x / 2 = y / 3 ⇒ x / y = 2 / 3 Hence, the ratio is x : y = 2 : 3 .

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The average age of a woman and her daughter is 46 years. The ratio of their ages is 15 : 8 respectively. What will be the respective ratio of their ages after 8 years ?

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Important Questions on Ratio and Proportion

MEDIUM

Quantitative Aptitude>Arithmetic>Ratio and Proportion>Ratio

Divide

27

into two parts so that

5

times the first and

11

times the second together equal to

195 .

The ratio of the first and second part is _____.

HARD

Quantitative Aptitude>Arithmetic>Ratio and Proportion>Ratio

\frac{\sqrt{x + 20} + \sqrt{x - 1}}{\sqrt{x + 20} - \sqrt{x - 1}} = \frac{7}{3}

, then what is the value of

\sqrt{(x + 20) (x - 1)}

MEDIUM

Quantitative Aptitude>Arithmetic>Ratio and Proportion>Ratio

Ranu’s monthly salary is four-fifth of Ali’s monthly salary. Ranu and Ali save one-fourth and two-fifth amount from their respective monthly salary. If the difference between the amount saved by Ranu and that saved by Ali is Rs.

7000

, what is Ranu’s monthly salary?

MEDIUM

Quantitative Aptitude>Arithmetic>Ratio and Proportion>Ratio

A box contains one rupee, fifty paisa and twenty five paisa coins. The total number of coins are $378$ . The ratio of values of the above coins is $13 : 11 : 7$ . The number of twenty five paise coins was:

MEDIUM

Quantitative Aptitude>Arithmetic>Ratio and Proportion>Ratio

Using properties of proportion, solve for $x$ , where $x$ is a positive number:

$\frac{2 x + \sqrt{4 x^{2} - 1}}{2 x - \sqrt{4 x^{2} - 1}} = 4$

MEDIUM

Quantitative Aptitude>Arithmetic>Ratio and Proportion>Ratio

A sum of Rs.

221

is divided among X, Y and Z such that X gets Rs.

52

more than Y. Y gets Rs.

26

more than Z. The ratio of the shares of X, Y and Z respectively is

MEDIUM

Quantitative Aptitude>Arithmetic>Ratio and Proportion>Ratio

The respective ratio between two positive numbers (X and Y) is

3 : 5

. When

2

is added to both the numbers, the ratio between them becomes

5 : 8

. What is the difference between both the numbers?

EASY

Quantitative Aptitude>Arithmetic>Ratio and Proportion>Ratio

A certain amount of money has to be divided between two persons P and Q in the ratio

5 : 7

. But it was divided in the ratio

2 : 5

and thereby Q gets an additional

Rs . 33

. Find the total amount.

MEDIUM

Quantitative Aptitude>Arithmetic>Ratio and Proportion>Ratio

x = \frac{\sqrt{2 a + 1} + \sqrt{2 a - 1}}{\sqrt{2 a + 1} - \sqrt{2 a - 1}}

, prove that

x^{2} - 4 a x + 1 = 0

MEDIUM

Quantitative Aptitude>Arithmetic>Ratio and Proportion>Ratio

If $\frac{[a + b - c]}{[a + b]} = \frac{[b + c - a]}{[b + c]} = \frac{[c + a - b]}{[c + a]}$ and $a + b + c \neq 0$ then prove that $a = b = c$ .

HARD

Quantitative Aptitude>Arithmetic>Ratio and Proportion>Ratio

\cos^{2} θ - \sin^{2} θ = \frac{1}{2}

, then find the value of

\tan^{2} θ

EASY

Quantitative Aptitude>Arithmetic>Ratio and Proportion>Ratio

Using properties of proportion find $x : y$ , given:

$\frac{x^{2} + 2 x}{2 x + 4} = \frac{y^{2} + 3 y}{3 y + 9}$

MEDIUM

Quantitative Aptitude>Arithmetic>Ratio and Proportion>Ratio

Using properties of proportion solve for $x$ , given

$\frac{\sqrt{5 x} + \sqrt{2 x - 6}}{\sqrt{5 x} - \sqrt{2 x - 6}} = 4$

MEDIUM

Quantitative Aptitude>Arithmetic>Ratio and Proportion>Ratio

In water, The ratio of the mass of hydrogen to the mass of oxygen is always _____.

EASY

Quantitative Aptitude>Arithmetic>Ratio and Proportion>Ratio

The average age of

11

players of a cricket team is increased by

2

months, when two of them aged

18

years and

20

years are replaced by two new players. The average age of the new players is:

EASY

Quantitative Aptitude>Arithmetic>Ratio and Proportion>Ratio

The age of a man is three times the sum of the ages of his two sons. After

5

years hence, his age will be double the sum of their ages. The present age of the man is:

MEDIUM

Quantitative Aptitude>Arithmetic>Ratio and Proportion>Ratio

The respective ratio between the present age of Ram, Rohan, and Raj is

3 : 4 : 5

. If the average of their present ages is

28

y e a r s

, then what would be the sum of the ages of Ram and Rohan together after

5

y e a r s

EASY

Quantitative Aptitude>Arithmetic>Ratio and Proportion>Ratio

The present age of vishal and shekhar are in the ratio of

14 : 17

respectively. Six years from now, their ages will be in the ratio of

17 : 20

. What is shekhar present age?

MEDIUM

Quantitative Aptitude>Arithmetic>Ratio and Proportion>Ratio

In a family, a couple has a son and a daughter. The age of the father is three times that of his daughter and the age of the son is half of his mother. The wife is

9 years

younger to her husband and the brother is

7 years

older than his sister. What is the age of the mother?

MEDIUM

Quantitative Aptitude>Arithmetic>Ratio and Proportion>Ratio

A father says to his son, "

7

years ago my age was seven times of your age at that time and after

3

years my age will be thrice of your age." The present age of the son is:

The average age of a woman and her daughter is 46 years. The ratio of their ages is 15 : 8 respectively. What will be the respective ratio of their ages after 8 years ?

Important Questions on Ratio and Proportion

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