Exercise: 10-B
Mental Maths
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)
