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 C and B varies as C , then A ± B and A B will each vary as C .

If A ∝ C then A = K C . . . i where, K is the constant of proportionality. Also if B ∝ C then B = m C . . . ii where, m is the constant of proportionality. Now, from i and ii , we have A ± B = K C ± m C = C ( K ± m ) ⇒ A ± B = N C where N = K ± m is a constant. ∴ A ± B ∝ C Similarly, A B = K m C 2 ⇒ A B = K m C ⇒ A B ∝ C

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

MEDIUM

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$195$ men working $10$ hours a day can finish a job in $20$ days. How many men are employed to finish the job in $15$ days if they work $13$ hours a day?

50% studentsanswered this correctly

Important Questions on Life Mathematics

EASY

Mathematics>Arithmetic>Life Mathematics>Compound Variation

a \propto b

and

b \propto c

, then prove that

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

MEDIUM

Mathematics>Arithmetic>Life Mathematics>Compound Variation

\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>Arithmetic>Life Mathematics>Compound Variation

x \propto y, y \propto z

and

z \propto x

, find the product of three variation constants.

EASY

Mathematics>Arithmetic>Life Mathematics>Compound Variation

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>Arithmetic>Life Mathematics>Compound Variation

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

, then show that

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

MEDIUM

Mathematics>Arithmetic>Life Mathematics>Compound Variation

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>Arithmetic>Life Mathematics>Compound Variation

x \propto y^{2}

and

y = 2 a

when

x = a

, then show that

y^{2} = 4 a x

EASY

Mathematics>Arithmetic>Life Mathematics>Compound Variation

x

varies as

y

, such that

x = 8

when

y = 15

. Find

x,

y = 10 .

MEDIUM

Mathematics>Arithmetic>Life Mathematics>Compound Variation

If A varies as

C

and

B

varies as

C,

then

A \pm B

and

\sqrt{A B}

will each vary as

C .

MEDIUM

Mathematics>Arithmetic>Life Mathematics>Compound Variation

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>Arithmetic>Life Mathematics>Compound Variation

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>Arithmetic>Life Mathematics>Compound Variation

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>Arithmetic>Life Mathematics>Compound Variation

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>Arithmetic>Life Mathematics>Compound Variation

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>Arithmetic>Life Mathematics>Compound Variation

25

horses consume

18

quintal in

36

days, how long will

28

quintal last for

30

horses?

EASY

Mathematics>Arithmetic>Life Mathematics>Compound Variation

Write the quadrants in which the following points lie:

$(- 4, 7)$

MEDIUM

Mathematics>Arithmetic>Life Mathematics>Compound Variation

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>Arithmetic>Life Mathematics>Compound Variation

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>Arithmetic>Life Mathematics>Compound Variation

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>Arithmetic>Life Mathematics>Compound Variation

5

persons can do

5

jobs in

5

days, then

50

persons can do

50

jobs in _____ days.

195 men working 10 hours a day can finish a job in 20 days. How many men are employed to finish the job in 15 days if they work 13 hours a day?

Important Questions on Life Mathematics

Learn and Prepare for any exam you want !

$195$ men working $10$ hours a day can finish a job in $20$ days. How many men are employed to finish the job in $15$ days if they work $13$ hours a day?