If 2 a + b + 4 c + 2 d 2 a - b - 4 c + 2 d = 2 a - b + 4 c - 2 d 2 a + b - 4 c - 2 d , prove that a b = c d .

Given that, 2 a + b + 4 c + 2 d 2 a - b - 4 c + 2 d = 2 a - b + 4 c - 2 d 2 a + b - 4 c - 2 d ⇒ 2 a + b + 4 c + 2 d 2 a + b - 4 c - 2 d = 2 a - b + 4 c - 2 d 2 a - b - 4 c + 2 d Using the Componendo-Dividendo rule, we get, 2 a + b + 4 c + 2 d + 2 a + b - 4 c - 2 d 2 a + b + 4 c + 2 d - 2 a + b - 4 c - 2 d = 2 a - b + 4 c - 2 d + 2 a - b - 4 c + 2 d 2 a - b + 4 c - 2 d - 2 a - b - 4 c + 2 d ⇒ 2 a + b + 4 c + 2 d + 2 a + b - 4 c - 2 d 2 a + b + 4 c + 2 d - 2 a - b + 4 c + 2 d = 2 a - b + 4 c - 2 d + 2 a - b - 4 c + 2 d 2 a - b + 4 c - 2 d - 2 a + b + 4 c - 2 d ⇒ 4 a + 2 b 8 c + 4 d = 4 a - 2 b 8 c - 4 d ⇒ 4 a + 2 b 4 a - 2 b = 8 c + 4 d 8 c - 4 d [ By cross multiplication] Again, using the Componendo-Dividendo rule, we get, ⇒ 4 a + 2 b + 4 a - 2 b 4 a + 2 b - 4 a + 2 b = 8 c + 4 d + 8 c - 4 d 8 c + 4 d - 8 c + 4 d ⇒ 8 a 4 b = 16 c 8 d ⇒ 2 a b = 2 c d ⇒ a b = c d Hence, proved.

If 2 a + b + 4 c + 2 d 2 a - b - 4 c + 2 d = 2 a - b + 4 c - 2 d 2 a + b - 4 c - 2 d , prove that a b = c d .

Given that, 2 a + b + 4 c + 2 d 2 a - b - 4 c + 2 d = 2 a - b + 4 c - 2 d 2 a + b - 4 c - 2 d ⇒ 2 a + b + 4 c + 2 d 2 a + b - 4 c - 2 d = 2 a - b + 4 c - 2 d 2 a - b - 4 c + 2 d Using the Componendo-Dividendo rule, we get, 2 a + b + 4 c + 2 d + 2 a + b - 4 c - 2 d 2 a + b + 4 c + 2 d - 2 a + b - 4 c - 2 d = 2 a - b + 4 c - 2 d + 2 a - b - 4 c + 2 d 2 a - b + 4 c - 2 d - 2 a - b - 4 c + 2 d ⇒ 2 a + b + 4 c + 2 d + 2 a + b - 4 c - 2 d 2 a + b + 4 c + 2 d - 2 a - b + 4 c + 2 d = 2 a - b + 4 c - 2 d + 2 a - b - 4 c + 2 d 2 a - b + 4 c - 2 d - 2 a + b + 4 c - 2 d ⇒ 4 a + 2 b 8 c + 4 d = 4 a - 2 b 8 c - 4 d ⇒ 4 a + 2 b 4 a - 2 b = 8 c + 4 d 8 c - 4 d [ By cross multiplication] Again, using the Componendo-Dividendo rule, we get, ⇒ 4 a + 2 b + 4 a - 2 b 4 a + 2 b - 4 a + 2 b = 8 c + 4 d + 8 c - 4 d 8 c + 4 d - 8 c + 4 d ⇒ 8 a 4 b = 16 c 8 d ⇒ 2 a b = 2 c d ⇒ a b = c d 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 .

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 a ( y + z ) = b ( z + x ) = c ( x + y ) and out of a, b, c no two of them are equal then show that y - z a b - c = z - x b c - a = x - y c a - b

Given: a ( y + z ) = b ( z + x ) = c ( x + y ) Divide all terms with ‘ a b c ’ : a ( y + z ) a b c = b ( z + x ) a b c = c ( x + y ) a b c Cancel out the common factor: ⇒ ( y + z ) b c = ( z + x ) a c = ( x + y ) a b Rearrange the terms: ⇒ x + y a b = y + z b c = z + x a c Now, subtract third term from first term, subtract first term from second term and subtract second term from third term and obtain the equivalent: ⇒ x + y - ( z + x ) a b - a c = y + z - x + y b c - a b = z + x - y + z a c - b c Solve and cancel the opposite terms: ⇒ y - z a b - c = z - x b c - a = x - y c a - b Hence, proved.

MEDIUM

Earn 100

If $(2 a + b + 4 c + 2 d) (2 a - b - 4 c + 2 d) = (2 a - b + 4 c - 2 d) (2 a + b - 4 c - 2 d)$ , prove that $\frac{a}{b} = \frac{c}{d}$ .

Important Questions on Ratio and Proportion

EASY

Mathematics>Arithmetic>Ratio and Proportion>Introduction to Ratio and Proportion

The compound ratio of $a b : c^{2}$ , $b c : a$ and $c a : b^{2}$ is $1 : 1 .$

HARD

Mathematics>Arithmetic>Ratio and Proportion>Introduction to Ratio and Proportion

\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

Mathematics>Arithmetic>Ratio and Proportion>Introduction to Ratio and Proportion

If the product of three positive continued proportional numbers is

64

, what is their mean proportional?

HARD

Mathematics>Arithmetic>Ratio and Proportion>Introduction to Ratio and Proportion

p : q = 5 : 7

and

p + q = - 4

then what is the value of

(3 p + 2 q)

MEDIUM

Mathematics>Arithmetic>Ratio and Proportion>Introduction to Ratio and Proportion

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

Mathematics>Arithmetic>Ratio and Proportion>Introduction to Ratio and Proportion

\cos^{2} θ - \sin^{2} θ = \frac{1}{2}

, then find the value of

\tan^{2} θ

EASY

Mathematics>Arithmetic>Ratio and Proportion>Introduction to Ratio and Proportion

\frac{a}{2} = \frac{b}{3} = \frac{c}{4} = \frac{[2 a - 3 b + 4 c]}{p}

, then find

p

EASY

Mathematics>Arithmetic>Ratio and Proportion>Introduction to Ratio and Proportion

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

Mathematics>Arithmetic>Ratio and Proportion>Introduction to Ratio and Proportion

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

Mathematics>Arithmetic>Ratio and Proportion>Introduction to Ratio and Proportion

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

Mathematics>Arithmetic>Ratio and Proportion>Introduction to Ratio and Proportion

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

Mathematics>Arithmetic>Ratio and Proportion>Introduction to Ratio and Proportion

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

EASY

Mathematics>Arithmetic>Ratio and Proportion>Introduction to Ratio and Proportion

In a school, there are 1000 students. Out of whom 15% are boys. Each girl’s monthly fee is Rs 220 and each boy’s monthly fee is 15% less than that of girl. What is the total monthly fee of girls and boys together ?

EASY

Mathematics>Arithmetic>Ratio and Proportion>Introduction to Ratio and Proportion

A sum of money is divided among

W, X, Y

and

Z

in the ratio of

3 : 7 : 9 : 13 .

If the share of

W

and

Y

together is

R s . 11172,

then what is the difference between the amounts of

X

and

Z ?

EASY

Mathematics>Arithmetic>Ratio and Proportion>Introduction to Ratio and Proportion

12% of Anuj’s monthly salary is equal to 16% of Druv’s monthly salary. Sham’s monthly salary is half of Druv’s monthly salary. If Sham’s earns Rs 1.08 lac annually, what is Anuj’s monthly salary ?

EASY

Mathematics>Arithmetic>Ratio and Proportion>Introduction to Ratio and Proportion

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

Mathematics>Arithmetic>Ratio and Proportion>Introduction to Ratio and Proportion

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 ?

MEDIUM

Mathematics>Arithmetic>Ratio and Proportion>Introduction to Ratio and Proportion

Read the following information carefully and answer the questions given below it.

Five sports - Hockey, Cricket, Tennis, Badminton and Baseball are included in sports competition. The total number of players in this sports competition is 800. The ratio between the men and women players is

1 : 3.25 %

players in Cricket of total players. 110 players plays Badminton, 10% players in Tennis of total players. Hockey players are two times than Badminton players, remaining players play Baseball. 30% Cricket players are women. Half women cricketers are equal to Badminton players. 10% players of Hockey players are equal to women Tennis players. Hockey and Baseball have equal women players.

What is the ratio between the women Hockey players and men Badminton players?

EASY

Mathematics>Arithmetic>Ratio and Proportion>Introduction to Ratio and Proportion

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?

MEDIUM

Mathematics>Arithmetic>Ratio and Proportion>Introduction to Ratio and Proportion

If $a (y + z) = b (z + x) = c (x + y)$ and out of a, b, c no two of them are equal then show that

$\frac{y - z}{a (b - c)} = \frac{z - x}{b (c - a)} = \frac{x - y}{c (a - b)}$

If 2a+b+4c+2d2a-b-4c+2d=2a-b+4c-2d2a+b-4c-2d, prove that ab=cd.

Important Questions on Ratio and Proportion

Learn and Prepare for any exam you want !

If $(2 a + b + 4 c + 2 d) (2 a - b - 4 c + 2 d) = (2 a - b + 4 c - 2 d) (2 a + b - 4 c - 2 d)$ , prove that $\frac{a}{b} = \frac{c}{d}$ .