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 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 .

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.

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 .

MEDIUM

Earn 100

Study the following information to answer the questions given below :
i) The ratio of the populations of males, females and children 10 years old and above is 11 : 10 : 9 in state 'A'. Out of which 40% males or 8800 liters 20% children (10 years old and above) are illiterate while 30% females are literate.
ii) The no. of children below 10 years of age is 10% of the no. of females. 5% of the total population of the state is below poverty line and 80% of them are illiterate.

What is the total population of the state ?

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Important Questions on Ratio and Proportion

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

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

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

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

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

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

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

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

Out of the total population,

40 %

were females. Out of the total number of males,

75 %

were literates and the remaining

225

were illiterates. If

70 %

of the total number of females are literates, how many females were illiterate?

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

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

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

The difference between

\frac{4}{5}

of a number and

45 %

of the number is

56

. What is

65 %

of the number?

EASY

Quantitative Aptitude>Arithmetic>Ratio and Proportion>Ratio

Gaurav spent

Rs . 38460

on the renovation of his home,

Rs . 24468

on buying a home theatre and the remaining

28 %

of the total amount he had as cash with him. What was the total amount?

EASY

Quantitative Aptitude>Arithmetic>Ratio and Proportion>Ratio

Anil invested a certain amount in Debt and Equity Funds in the ratio of 4:5. At the end of one year, he earned a total dividend of 30% on his investment. After one year, he re-invested the amount including the dividend in the ratio of 6:7 in Debt and Equity Funds. If the amount re-invested in equity funds was Rs 94,500, What was the original amount invested in Equity Funds ?

EASY

Quantitative Aptitude>Arithmetic>Ratio and Proportion>Ratio

The population of a town two years ago was 45, 000. It increased by 12% in the first year and decreased by 15% in the second year. What was the population of the town at the end of two years?

EASY

Quantitative Aptitude>Arithmetic>Ratio and Proportion>Ratio

From a class, total $360$ students appeared in an exam. Three fourth of these are boys. $40 %$ of boys failed in the exam, two-third of the girls are failed in the exam and the remaining passed.

What is the difference between the number of boys who are passed and the number of girls who are failed?

Important Questions on Ratio and Proportion

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