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.

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 .

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 .

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The total number of students in a school is 2140. If the number of girls in the school is 1200, then what is the respective ratio of the total number of boys to the total number of girls in the school ?

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Important Questions on Ratio and Proportion

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

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 $x$ , where $x$ is a positive number:

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

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

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 $a = b = c$ .

MEDIUM

Quantitative Aptitude>Arithmetic>Ratio and Proportion>Ratio

Using properties of proportion solve for $x$ , given

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

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

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 _____.

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

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:

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

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

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?

MEDIUM

Quantitative Aptitude>Arithmetic>Ratio and Proportion>Ratio

A certain sum is divided among A, B and C in such way that A gets Rs.

40

more than half of the sum. B gets Rs.

120

less than three-eighths of the sum and C gets Rs.

200

. What is the total sum?

EASY

Quantitative Aptitude>Arithmetic>Ratio and Proportion>Ratio

20 boys and 32 girls form a group for social work. During their membership drive same number of boys and girls joined the group. How many members does the group have now, if the ratio of boys to girls is 3 : 4 respectively?

EASY

Quantitative Aptitude>Arithmetic>Ratio and Proportion>Ratio

Six men can complete a piece of work in 35 hours. In how many hours will 15 men complete the same piece of work?

EASY

Quantitative Aptitude>Arithmetic>Ratio and Proportion>Ratio

52% students from a college participated in a survey. What is the respective ratio between the number of students who did not participate in the survey to the number of students who participated?

HARD

Quantitative Aptitude>Arithmetic>Ratio and Proportion>Ratio

In a college, the ratio of boys to girls is 31 : 23 respectively. When 75 more girls join the college, this ratio becomes 124 : 107. How many more girls should join the college to make the number of boys and girls equal ?

EASY

Quantitative Aptitude>Arithmetic>Ratio and Proportion>Ratio

A single person takes 10 minutes to stitch a bag. If from 10.00 am. to 12.30 pm, 1245 bags are to be stitched, how many persons should be employed on this job ?

The total number of students in a school is 2140. If the number of girls in the school is 1200, then what is the respective ratio of the total number of boys to the total number of girls in the school ?

Important Questions on Ratio and Proportion

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