If a ∝ b and b ∝ c , then prove that a 3 + b 3 + c 3 ∝ 3 a b c .

Given, a ∝ b ∴ a = k b where k is non-zero constant. And b ∝ c ∴ b = l c where l is non-zero constant. ∴ a = k b = k l c = m c , where m = k l = constant Now, a 3 + b 3 + c 3 3 a b c = m c 3 + l c 3 + c 3 3 × m c × l c × c = c 3 m 3 + l 3 + 1 c 3 . 3 m l = m 3 + l 3 + 1 3 m l = n where n = m 3 + l 3 + 1 3 m l non-zero constant. Thus, a 3 + b 3 + c 3 = n . 3 a b c ∴ a 3 + b 3 + c 3 ∝ 3 a b c Hence, proved.

If b + c - a y + z - x = c + a - b z + x - y = a + b - c x + y - z , then prove it a x = b y = c z

Given, b + c - a y + z - x = c + a - b z + x - y = a + b - c x + y - z b + c - a y + z - x = c + a - b z + x - y = a + b - c x + y - z = k , where k = constant ∴ b + c - a = k y + z - x -----------(i) c + a - b = k z + x - y ---------------(ii) a + b - c = k x + y - z -------------------(iii) Adding (i) and (ii), we get b + c - a + c + a - b = k y + z - x + z + x - y ⇒ 2 c = 2 k z ⇒ c z = k Similarly, adding (ii) and(iii), we get c + a - b + a + b - c = k z + x - y + x + y - z ⇒ 2 a = 2 k x ⇒ a x = k Adding (i) and (iii), we get b + c - a + a + b - c = k y + z - x + x + y - z ⇒ 2 b = 2 k y ⇒ b y = k So, we can write a x = b y = c z Hence proved.

If x : a = y : b = z : c , then show that x 3 a 3 + y 3 b 3 + z 3 c 3 = 3 x y z a b c .

Let x a = y b = z c = k . Thus, x = a k , y = b k , z = c k ______________ 1 L . H . S : x 3 a 3 + y 3 b 3 + z 3 c 3 = a k 3 a 3 + b k 3 b 3 + c k 3 c 3 [From 1 ] = a 3 k 3 a 3 + b 3 k 3 b 3 + c 3 k 3 c 3 = k 3 + k 3 + k 3 = 3 k 3 R . H . S : 3 x y z a b c = 3 a k b k c k a b c [From 1 ] = 3 k 3 a b c a b c = 3 k 3 Therefore, L . H . S = R . H . S Hence, Proved

If 1 x - 1 y ∝ 1 x - y , then show that x 2 + y 2 ∝ x y .

Given, 1 x - 1 y ∝ 1 x - y Can be written as, 1 x - 1 y = k · 1 x - y , where k is the constant of variation ⇒ y - x x y = k x - y ⇒ y - x x - y = k x y ⇒ x y - x 2 - y 2 + x y = k x y ⇒ 2 x y - x 2 + y 2 = k x y ⇒ x 2 + y 2 = 2 x y - k x y ⇒ x 2 + y 2 = 2 - k x y ⇒ x 2 + y 2 ∝ x y , where 2 - k is a constant Hence proved

If x ∝ y and y ∝ z , then prove that: x 2 + y 2 + z 2 ∝ x y + y z + x z .

Given, x ∝ y and y ∝ z Then, x ∝ z ⇒ x = k z , where k is a constant And y ∝ z ⇒ y = m z , where m is a constant Now, x 2 + y 2 + z 2 x y + y z + z x = k z 2 + m z 2 + z 2 k z × m z + m z × z + k z × z = k 2 z 2 + m 2 z 2 + z 2 k m z 2 + m z 2 + k z 2 = z 2 k 2 + m 2 + 1 z 2 k m + m + k = k 2 + m 2 + 1 k m + m + k = Constant Hence, x 2 + y 2 + z 2 ∝ x y + y z + z x

x ∝ y 2 and y = 2 a when x = a , then show that y 2 = 4 a x

Given, x ∝ y 2 ⇒ x = k y 2 ........... i Now, if we put the given value in equation i , we get a = k × 2 a 2 ⇒ a = k × 4 a 2 ⇒ k = a 4 a 2 ⇒ k = 1 4 a Now, put the value of k in equation i , we get x = 1 4 a y 2 y 2 ⇒ 4 a x Hence proved

If x varies as y , such that x = 8 when y = 15 . Find x , y = 10 .

If x ∝ y then x = k y i.e. k = x y . . . i Here, k is the constant of proportionality. Now, it is given that x = 8 when y = 15 . Putting these values in i , we get k = 8 15 Now, when y = 10 then we have x = k × 10 = 8 15 × 10 = 5 1 3 .

If A varies as B C , then B varies inversely as C / A .

If A ∝ B C then we have A = k B C where, k is the constant of proportionality. ⇒ B = A k C ⇒ B = 1 k C A ⇒ B ∝ 1 C A

Given that x + y varies as z + 1 z and that x - y varies z - 1 z . Find the relation between x and z , provided that z = 2 when x = 3 and y = 1 .

Given: x + y ∝ z + 1 z x + y = k z + 1 z . . . i where, k is the constant of proportionality. And, x - y ∝ z - 1 z x - y = k ' z - 1 z . . . ii Now when x = 3 , y = 1 then z = 2 . Putting these values in i and ii , we get 4 = k 5 2 ⇒ k = 8 5 2 = k ' 3 2 ⇒ k ' = 4 3 Therefore, by i and ii , we have x + y = 8 5 z + 1 z . . . iii x - y = 4 3 z - 1 z . . . iv Eliminating y between these, we get 2 x = 44 z 15 + 4 15 z ⇒ 2 x = 22 15 z + 2 15 z

P varies directly as Q and inversely as R , also P = 2 3 when Q = 3 / 7 and R = 9 14 . Find Q , when P = 48 and R = 75 .

It is given that P varies directly as Q and inversely as R then we have P ∝ Q P ∝ 1 R Therefore, we have P = K Q R . . . i where K is any constant. Now, we have P = 2 3 when Q = 3 7 and R = 9 14 . Then from i , we have 2 3 = K 3 7 9 14 ⇒ K = 2 3 × 9 14 × 7 3 = 1 Putting this value in i , we get P = Q R ⇒ 48 = 1 · Q 75 ⇒ Q = 60

EASY

Earn 100

The relationship between $x$ and $y$ are proportional. When $x$ is $10$ , $y$ is $100$ . The equation for this relationship is

50% studentsanswered this correctly

Important Questions on Real-life Graphs and Direct Proportion

EASY

Mathematics>Geometry>Real-life Graphs and Direct Proportion>Direct Proportion

a \propto b

and

b \propto c

, then prove that

a^{3} + b^{3} + c^{3} \propto 3 a b c

MEDIUM

Mathematics>Geometry>Real-life Graphs and Direct Proportion>Direct Proportion

\frac{b + c - a}{y + z - x} = \frac{c + a - b}{z + x - y} = \frac{a + b - c}{x + y - z}

, then prove it

\frac{a}{x} = \frac{b}{y} = \frac{c}{z}

EASY

Mathematics>Geometry>Real-life Graphs and Direct Proportion>Direct Proportion

x \propto y, y \propto z

and

z \propto x

, find the product of three variation constants.

EASY

Mathematics>Geometry>Real-life Graphs and Direct Proportion>Direct Proportion

x : a = y : b = z : c

, then show that

\frac{x^{3}}{a^{3}} + \frac{y^{3}}{b^{3}} + \frac{z^{3}}{c^{3}} = \frac{3 x y z}{a b c}

MEDIUM

Mathematics>Geometry>Real-life Graphs and Direct Proportion>Direct Proportion

(\frac{1}{x} - \frac{1}{y}) \propto \frac{1}{x - y}

, then show that

x^{2} + y^{2} \propto x y

MEDIUM

Mathematics>Geometry>Real-life Graphs and Direct Proportion>Direct Proportion

If $x \propto y$ and $y \propto z$ , then prove that: $(x^{2} + y^{2} + z^{2}) \propto (x y + y z + x z)$ .

EASY

Mathematics>Geometry>Real-life Graphs and Direct Proportion>Direct Proportion

x \propto y^{2}

and

y = 2 a

when

x = a

, then show that

y^{2} = 4 a x

EASY

Mathematics>Geometry>Real-life Graphs and Direct Proportion>Direct Proportion

x

varies as

y

, such that

x = 8

when

y = 15

. Find

x,

y = 10 .

EASY

Mathematics>Geometry>Real-life Graphs and Direct Proportion>Direct Proportion

A

varies as

B C,

then

B

varies inversely as

C / A .

MEDIUM

Mathematics>Geometry>Real-life Graphs and Direct Proportion>Direct Proportion

The weight of the circular disc varies as the square of the radius when the thickness remains the same, it also varies as the thickness when the radius remains the same. Two discs have their thicknesses in the ratio of

9 : 8,

find the ratio of their radii if the weight of the first is twice that of the second.

MEDIUM

Mathematics>Geometry>Real-life Graphs and Direct Proportion>Direct Proportion

A

varies directly as the square root of

B

and inversely as the cube of

C

. If

A = 3,

when

B = 256

and

C = 2,

find

B

when

A = 24

and

C = \frac{1}{2} .

MEDIUM

Mathematics>Geometry>Real-life Graphs and Direct Proportion>Direct Proportion

Given that

x + y

varies as

z + \frac{1}{z}

and that

x - y

varies

z - \frac{1}{z}

. Find the relation between

x

and

z,

provided that

z = 2

when

x = 3

and

y = 1 .

MEDIUM

Mathematics>Geometry>Real-life Graphs and Direct Proportion>Direct Proportion

A

and

B

together can do a piece of work in

12

days;

B

and

C

together can do it in

15

days. If A is twice as good a workman as

C

, in how many days

A

alone will do the same work?

MEDIUM

Mathematics>Geometry>Real-life Graphs and Direct Proportion>Direct Proportion

A

varies as

B

and

C

jointly, while

B

varies as

D^{2}

and

C

varies inversely as

A .

Show that

A

varies as

D .

MEDIUM

Mathematics>Geometry>Real-life Graphs and Direct Proportion>Direct Proportion

25

horses consume

18

quintal in

36

days, how long will

28

quintal last for

30

horses?

EASY

Mathematics>Geometry>Real-life Graphs and Direct Proportion>Direct Proportion

Write the quadrants in which the following points lie:

$(- 4, 7)$

MEDIUM

Mathematics>Geometry>Real-life Graphs and Direct Proportion>Direct Proportion

A group of labours promises to do a piece of work in

10

days but five of them become absent. If the remaining labours complete the work in

12

days, find their original number in the group

MEDIUM

Mathematics>Geometry>Real-life Graphs and Direct Proportion>Direct Proportion

P

varies directly as

Q

and inversely as

R,

also

P = \frac{2}{3}

when

Q = 3 / 7

and

R = \frac{9}{14}

. Find

Q

, when

P = \sqrt{48}

and

R = \sqrt{75} .

EASY

Mathematics>Geometry>Real-life Graphs and Direct Proportion>Direct Proportion

If the square of

x,

varies as cube of

y

and

x = 3

when

y = 4

. Find the value of

y

, when

x = \frac{1}{\sqrt{3}} .

HARD

Mathematics>Geometry>Real-life Graphs and Direct Proportion>Direct Proportion

5

persons can do

5

jobs in

5

days, then

50

persons can do

50

jobs in _____ days.

The relationship between x and y are proportional. When x is 10, y is 100. The equation for this relationship is

Important Questions on Real-life Graphs and Direct Proportion

Learn and Prepare for any exam you want !

The relationship between $x$ and $y$ are proportional. When $x$ is $10$ , $y$ is $100$ . The equation for this relationship is