Chapter 10: Direct and Inverse Proportions

Q1Written answer

Both u and v vary directly with each other. When u is 10, v is 15, which of the following is not a possible pair of corresponding values of u and v?

(a) 2 and 3

(b) 8 and 12

(c) 15 and 20

(d) 25 and 37.5

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Ans: u and v vary directly, hence,

$\dfrac{u}{v}= k$ $\left( constant \right)$

Hence, $\dfrac{u}{v} =\dfrac{10}{15} =\dfrac{2}{3}$

Now,

Option (a) $\to \dfrac{2}{3}$

Option (b) $\to \dfrac{8}{12} =\dfrac{2}{3}$

Option (c) $\to $ $\dfrac{15}{20} =\dfrac{3}{4}$

Option (d) $\to $ $\dfrac{25}{37.5} =\dfrac{2}{3}$

Therefore, option (c) is not the possible pair of corresponding values of u and v

Q2Written answer

Both x and y vary inversely with each other. When x is 10, y is 6, which of the following is not a possible pair of corresponding values of x and y? (a) 12 and 5

(b) 15 and 4

(c) 25 and 2.4

(d) 45 and 1.3

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Worked solution

Ans: Since x and y vary inversely, so

$xy = k$ (constant)

Putting the value of $x$ and $y$, we get,

$10\times 6 = 60$

Now,

Option (a) $\to $ $12 $and$ 5 12\times 5 = 60$

Option (b) $\to 15 $and$ 4 15\times 4 = 60$

Option (c) $\to $ $25 $and$ 2.4 25\times 2.4 = 60$

Option (d) $\to $ $45 $and$ 1.3 = 45\times 1.3 = 58.3$

Therefore, option (d) is not a possible pair of corresponding values of $x$ and $y$

Q3Written answer

Assuming land to be uniformly fertile, the area of land and the yield on it vary

(a) directly with each other.

(b) inversely with each other.

(c) neither directly nor inversely with each other.

(d) sometimes directly and sometimes inversely with each other.

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Worked solution

Ans: Since, the land is uniformly fertile, thus more the area of land will give more yield and vice versa. Therefore, area and yield on land vary (a) directly with each other.

Q4Written answer

The number of teeth and the age of a person vary

(a) directly with each other.

(b) inversely with each other.

(c) neither directly nor inversely with each other.

(d) sometimes directly and sometimes inversely with each other.

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Ans: Correct option is (d)

The number of teeth and the age of a person vary sometimes directly and sometimes inversely with each other; we cannot predict about the number of teeth with exactly the age of a person. It changes from person-to-person.

Q5Written answer

A truck needs 54 liters of diesel for covering a distance of $297$ km. The diesel required by the truck to cover a distance of $550$ km is

(a) 100 liters

(b) 50 liters

(c) 25.16 liters

(d) 25 liters

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Worked solution

Ans: Correct option is (a)

Distance covered by truck using 54 liters diesel = 297 km

Distance covered by truck using 1 liter diesel $=\dfrac{297}{54}$ = 5.5 km

Hence, for 550 km, diesel required $=\dfrac{550}{5.5} $= 100 liters (a)

Q6Written answer

By traveling at a speed of 48 kilometers per hour, a car can finish a certain journey in 10 hours. To cover the same distance in 8 hours, the speed of the car should be

(a) 60 km/h

(b) 80 km/h

(c) 30 km/h

(d) 40 km/h

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Ans: Correct option is (a)

Speed of car = 48 km/hr

Time taken = 10 hr

As we know,

Distance = speed $\times $ time $= 48 \times 10=480$ km

Speed required by car to cover 480 km in 8 hours $=\dfrac{480}{8}$ = 60 km/hr (a)

Q7Written answer

In which of the following cases, do the quantities vary directly with each other?

(a)

$x$	$0.5$	$2$	$8$	$32$
$y$	$2$	$8$	$32$	$128$

(b)

$p$	${{1}^{2}}$	${{2}^{2}}$	${{3}^{2}}$	${{4}^{2}}$
$q$	${{1}^{3}}$	${{2}^{3}}$	${{3}^{3}}$	${{4}^{3}}$

(c)

$r$	$2$	$5$	$25$	$50$
$s$	$25$	$10$	$2$	$0.5$

(d)

$u$	$2$	$4$	$6$	$9$	$12$
$v$	$18$	$9$	$6$	$4$	$3$

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Worked solution

Ans: Correct option is (a)

The value of x is directly proportional to values of y, such as,

$y = 4x$

If we put the values of $x = 0.5, 2, 8 $and $32,$

we get the values of $y$ as $2, 8, 32$ and $128$ respectively.

i.e If we multiply x with 4, we get the directly required result as the same as shown in corresponding y. In this case, as the value of x increases, the value of y also increases.

Q8Written answer

Which quantities in the previous question vary inversely with each other

(a) x and y

(b) p and q

(c) r and s

(d) u and v

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Worked solution

Ans: Correct option is (d)

When the value of u is increasing, the value of v is decreasing. Hence, u and v are inversely proportional.

Q9Written answer

Which of the following vary inversely with each other?

(a) speed and distance covered.

(b) distance covered and taxi fare.

(c) distance traveled and time taken.

(d) speed and time taken.

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Worked solution

Ans: Correct option is (d)

By the formula of speed we know,

$Speed =\dfrac{distance}{time}$

As we can see here, when the time decreases, the speed increases. Hence, they are inversely proportional.

Q10Written answer

Both x and y are in direct proportion, then 1 x and 1 y are

(a) in direct proportion.

(b) in inverse proportion.

(c) neither in direct nor in inverse proportion.

(d) sometimes in direct and sometimes in inverse proportion.

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Worked solution

Ans: Correct option is (a)

$x\propto y$

Two quantities x and y are said to be in Direct proportion, if an increase in x causes proportional increase in y and vice-versa

$\Rightarrow $ $\dfrac{1}{x}\propto \dfrac{1}{y}$

Thus, $\dfrac{1}{x},\dfrac{1}{y}$ are in direct proportion.

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