Chapter 10: Direct and Inverse Proportions

Q71Written answer

It is given that l varies directly as m.

(i) Write an equation which relates l and m.

(ii) Find the constant of proportion (k), when l is 6 then m is 18.

(iii) Find l, when m is 33.

(iv) Find m when l is 8.

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Ans: Since l varies directly as m

Equation related to l and m is $\dfrac{l}{m}$ = k(constant)

Ans: If l = 6 and m = 18 then k= $\dfrac{6}{18}$ = $\dfrac{1}{3}$

Ans: If m = 33, then

$\dfrac{l}{m}$ =$\dfrac{1}{3}$

$\Rightarrow $ $\dfrac{l}{33}$ =$\dfrac{1}{3}$

$\Rightarrow $ l = 33 ($\dfrac{1}{3}$) $\Rightarrow $ l = 11

Ans: If l = 8, then

$\dfrac{l}{m}$ =$\dfrac{1}{3}$

$\Rightarrow $ $\dfrac{8}{m}$ =$\dfrac{1}{3}$

$\Rightarrow $ m = 8 (3) $\Rightarrow $ l = 24

Q72Written answer

If a deposit of Rs 2,000 earns an interest of Rs 500 in 3 years, how much interest would a deposit of Rs 36,000 earn in 3 years with the same rate of simple interest?

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Ans: If deposit of Rs 2000 earns in 3yr with an interest = Rs500

Then, a deposit of Rs1000 earns in 3yr with an interest =$\dfrac{500}{2}$ = 250

Similarly, deposit of Rs 3600 i.e.

Rs. 36 × 1000 earns in 3yr with an interest = 250 × 36 = Rs 9000

Q73Written answer

The mass of an aluminum rod varies directly with its length. If a 16 cm long rod has a mass of 192 g, find the length of the rod whose mass is 105 g.

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Ans: In direct proportion,$\dfrac{m}{l}$ = k $\Rightarrow $ $\dfrac{192}{16}$ = 12 =k

If mass of the rod=105g

$\dfrac{m}{l}$ = k $\Rightarrow $ $\dfrac{105}{l}$ = 12 $\Rightarrow $ $\dfrac{105}{12}$ = l $\Rightarrow $ l =8.75 cm

Q74Written answer

Find the values of x and y if a and b are in inverse proportion:

a. 12 × 8

b. 305 y

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Worked solution

Ans: Given,

a	12	x	8
b	30	5	y

In part a, the value of x

x=12×6=72

In part b the value of y

y = 5×9=45

Q75Written answer

If Naresh walks 250 steps to cover a distance of 200 meters, find the distance traveled in 350 steps.

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Ans: 250 steps covers a distance =200 meter

1 step covers = $\dfrac{200}{250}$

350 steps cover = $\dfrac{200}{250}$ $\times $ 350 = $\dfrac{4}{5}$ $\times $ 350 = 4 $\times $ 70 = 280

Q76Written answer

If Naresh walks 250 steps to cover a distance of 200 meters, find the distance traveled in 350 steps.

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Worked solution

Ans: 225 km travels in =25 ltr

1 km travel = $\dfrac{25}{225}$

540 km will require = $\dfrac{25}{225}$ $\times $ 540 = $\dfrac{20}{60}\dfrac{1}{9}$ $\times $ 540 = 60

Q77Written answer

From the following table, determine if x and y are in direct proportion or not.

(i)

x	3	6	15	20	30
y	12	24	45	60	120

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Worked solution

Ans: $\dfrac{x}{y}$ = $\dfrac{3}{12}$, $\dfrac{6}{24}$,$\dfrac{15}{45}$ ,$\dfrac{20}{60}$ , $\dfrac{30}{120}$ $\Rightarrow $ $\dfrac{1}{4}$,$\dfrac{1}{4}$,$\dfrac{1}{3}$,$\dfrac{1}{3}$,$\dfrac{1}{4}$

Therefore, (i) is not in direct proportion

(ii)

x	4	7	10	16
y	24	42	60	96

Ans: $\dfrac{x}{y}$ = $\dfrac{4}{24}$, $\dfrac{7}{42}$,$\dfrac{15}{45}$ ,$\dfrac{10}{60}$ , $\dfrac{16}{96}$ $\Rightarrow $ $\dfrac{1}{6}$,$\dfrac{1}{6}$,$\dfrac{1}{6}$,$\dfrac{1}{6}$,$\dfrac{1}{6}$

Therefore (ii) is in direct proportion

(iii)

x	1	4	9	20
y	1.5	6	13.5	30

Ans: $\dfrac{x}{y}$ = $\dfrac{1}{1.5}$, $\dfrac{4}{6}$,$\dfrac{9}{13.5}$ ,$\dfrac{20}{30}$ $\Rightarrow $ $\dfrac{2}{3}$,$\dfrac{2}{3}$,$\dfrac{2}{3}$,$\dfrac{2}{3}$

Therefore (iii) is in direct proportion

Q78Written answer

If a and b vary inversely to each other, then find the values of p, q, r; x, y, z and l, m, n.

(i)

a	6	8	q	25
b	18	p	39	r

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Worked solution

Ans: a $\times $ b = 6 $\times $ 18 = 8 $\times $ p = q $\times $ 39 = 25 $\times $ r = k

$\Rightarrow $ 108 = 8 $\times $ p = q $\times $ 39 = 25 $\times $ r = k

Hence , p= $\dfrac{108}{8}$ , q = $\dfrac{108}{39}$ , r= $\dfrac{108}{25}$

$\Rightarrow $ p= $\dfrac{27}{2}$ , q = $\dfrac{36}{13}$ , r= $\dfrac{108}{25}$

(ii)

a	2	y	6	10
b	x	12.5	15	z

Ans: a $\times $ b = 2$ \times $ x = y $\times $ 12.5 = 6 $\times $ 15 = 10 $\times $ z = k

$\Rightarrow $ 90 = 2 $\times $ x = y $\times $ 12.5 = 10 $\times $ z

Hence , x= $\dfrac{90}{2}$ , y = $\dfrac{90}{12.5}$ , z= $\dfrac{90}{10}$

$\Rightarrow $ x=45 , y =7.2 , z=9

(iii)

a	l	9	n	6
b	5	m	25	10

Ans: a $\times $ b = l $\times $5 = 9 $\times $ m = n $\times $ 25 = 6 $\times $ 10 = k

$\Rightarrow $ 60 = 5l = 9m = 25n

Hence , x= $\dfrac{60}{5}$ , y = $\dfrac{60}{9}$ , z= $\dfrac{60}{25}$

$\Rightarrow $ x=12 , y =$\dfrac{20}{3}$ , z=$\dfrac{12}{5}$=2.4

Q79Written answer

If 25 meters of cloth costs Rs 337.50, then

(i) What will be the cost of 40 meters of the same type of cloth?

(ii) What will be the length of the cloth bought for Rs 810?

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Worked solution

Ans: 25 meter cloth costs = Rs. 337.50

1 meter cloth cost = $\dfrac{337.5}{25} $= Rs.13.5

40 meter clothe cost = 13.5×40 = Rs. 540

Ans: 25 meter cloth costs = Rs. 337.50

1 meter cloth cost = $\dfrac{337.5}{25} $= Rs.13.5

In Rs 810 clothe can be purchased = $\dfrac{1}{13.5}$ $\times $ 810=60meter

Q80Written answer

A swimming pool can be filled in 4 hours by 8 pumps of the same type. How many such pumps are required if the pool is to be filled in 2$\dfrac{2}{3}$ hours?

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Ans: To fill the tank in 1 hour total pumps required = 4×8=32

Total time = 2$\dfrac{2}{3}$ hours = $\dfrac{8}{3}$ hours

Total mumps required = 32 $\div \dfrac{8}{3}$ = $\dfrac{32 \times 3}{8}$ = 12 pumps

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