Chapter 10: Direct and Inverse Proportions

Ans: False, If x and y are in inverse proportion,

then xy = k (constant) e.g. Let x= 2 and y = 3

∴ xy = 2 x 3= 6.

Now, x + 1=2 + 1 = 3 and y+ 1 = 3 + 1 = 4

Then, (x + 1)(y+1) = 3 x 4 = 12 [not in inverse proportion]

Hence, (x+ 1)and (y + 1) cannot be in inverse proportion.

Ans: False, If p and q are in inverse proportion, then xy = k (constant)

e.g. Let p = 3 and q = 4

Then, pq = 3×4 = 12

Now, p+ 2 = 3+ 2 = 5 and q-2 = 4-2 =2

(p + 2) (q – 2) = 5 x 2 = 10 [not in inverse proportion]

Hence, (p+2) and (q -2) cannot be in inverse proportion.

Ans: False,If one angle of a triangle is kept fixed, then the measure of the remaining two angles cannot vary inversely with each other.

Let’s assume a triangle-

$\angle A$+ $\angle B$+ $\angle C$ = 180 (sum of all angles)

if $\angle $A=50 then $\angle $B+$\angle $C=180-50=130

Angle sum properties of a triangle don’t depend on any proportion.

Ans: True, When two quantities are related in such a manner that if one increases the other also increases, then they always vary directly.

Above statement is correct for direct proportion. It is a basic property of direct proportion.

Ans: True, When two quantities are related in such a manner that if one increases and the other decreases, then they always vary inversely. Above statement is correct for inverse proportion. It is a basic property of inverse proportion.

Ans: False

If x varies inversely as y i.e. $xy = k$ (constant)

If $x = 6$ and $y =8$ ,

$xy = 6\times 8=48$.

But if $x=8$ and $y= 10$,

$xy = 8\times 10=80$.

Here $48\ne 80,$

Hence, the value of y is not 10.

Ans: False.
The number of workers and the time to complete a job is a case of indirect proportion, e.g. If 60 workers can complete a work in 10 days. Then, 120 workers can complete the same work in 5 days.

Ans: True. For fixed time period (T) and rate of interest (R), the simple interest is directly proportional to the principal.

We know that,

$SI=\dfrac{P\times R\times T}{100}$

$\Rightarrow   \dfrac{SI}{P}=R\times \dfrac{T}{100}=Constant \left(  As R \And  T are constants  \right)$

Therefore, Simple interest is directly proportional to the principal.

Answer:  True
The area of cultivated land and the crop harvested is a case of direct proportion.
Since, the quantity of crop harvested is dependent upon the area of cultivated land.

In questions 60 to 62, which of the following vary directly and which vary inversely with each other, and which are neither of the two?

Answer: The time taken by a train to cover a fixed distance and the speed of the train are inversely proportional.

e.g. Let a train cover 200 km in 1 h with speed 200 km/h.

Then, the same train covers 200 km in 30 min with speed 400 km/h.

Ans: The distance traveled by CNG bus and the amount of CNG used are directly proportional.
e.g. Let a CNG bus travel 10 km in 1 kg of CNG.

Then, the same CNG bus traveled 20 km in $2 \times 1 = 2$ kg of CNG.

Ans: The number of people working and the time to complete a given work are inversely proportional to each other.

e.g. Let 20 workers complete a work in 2day.

Then, 10 workers can complete the same work in 4 days.

Ans: Income tax and the income are directly proportional to each other, e.g. Let Mr. X have$4.5$ lakh annual income.

Then, he pays $10%$ income tax on his income.

But if Mr. X has $5.5$ lakh annual income, then he has to pay $30%$ income tax on his salary/income.

Ans: Distance traveled by an auto rickshaw and time taken are directly proportional to each other.e.g. Let an auto rickshaw takes 2 h to travel 10 km. Then, it will take 4 h to travel 20 km.