Direct and Inverse Variation

Q1: Fill in the blanks:

i. When x and y are in direct proportion then _____ = ______ = ______ = _______ and so on.

In direct proportion, the ratio \( \frac{x}{y} \) remains constant.
Answer: \(\frac{x_1}{y_1} = \frac{x_2}{y_2} = \frac{x_3}{y_3} = \frac{x_4}{y_4} \)

ii. If x men can do a work in y days, then y men can do the same work in ________ days.

This is inverse proportion (more men → fewer days).
So, \[ x \times y = y \times D \\ \Rightarrow D = \frac{x \times y}{y} = x \]Answer: x days

iii. If m articles cost ₹n in all, then the total cost of n articles is ₹______.

Cost of 1 article = ₹n ÷ m = \(\frac{n}{m}\)
So, cost of n articles = \(\frac{n}{m} \times n = \frac{n^2}{m}\)
Answer: ₹\(\frac{n^2}{m}\)

iv. If 8 oranges cost ₹52, then the number of oranges that can be bought for ₹169 is _______.

Cost of 1 orange = ₹52 ÷ 8 = ₹6.5
Oranges for ₹169 = ₹169 ÷ 6.5 = 26
Answer: 26 oranges

v. 6 pipes can fill a tank in 120 minutes. Then, 5 pipes will fill it in _______ minutes.

Fewer pipes → more time (inverse proportion)
Let time = \( x \) \[ 6 \times 120 = 5 \times x \\ \Rightarrow x = \frac{720}{5} = 144 \]Answer: 144 minutes

Q2: Write True (T) or False (F):

i. When x and y are in inverse proportion, then x₁y₁ = x₂y₂.

In inverse proportion, the product of the variables remains constant. \[ x \propto \frac{1}{y} \\ \Rightarrow x_1y_1 = x_2y_2 \]Answer: True (T)

ii. 3 persons can build a wall in 4 days. Then 4 persons can build it in \(3\frac{1}{4}\) days.

Use inverse proportion: \[ 3 \times 4 = 12 \\ \Rightarrow 4 \times x = 12 \\ \Rightarrow x = \frac{12}{4} = 3 \text{ days} \] But statement says \(3\frac{1}{4} = 3.25\) days ⇒ Incorrect
Answer: False (F)

iii. The speed and distance covered are in direct proportion.

For fixed time, more speed → more distance ⇒ direct proportion
Answer: True (T)

iv. The number of men and the time taken by them to do a job are in inverse proportion.

More men ⇒ less time needed (if work is same), so inverse proportion
Answer: True (T)