A trader has two varieties of rice, type A & type B whose selling prices are Rs. 100 per kg and Rs. 180 per kg respectively. The possible quantities of type A rice to be mixed with 500 kg of type B rice, such that the selling price of the resulting mixture is ranging between Rs. 120 per kg to Rs. 150 per kg, are:
(A) 1400 kg
(B) 200 kg
(C) 1000 kg
(D) 800 kg
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), (C) and (D) only
Let x kg of type A be mixed with 500 kg of type B. The price of the mixture per kg is
\( \displaystyle \frac{100x+180\times 500}{x+500}=\frac{100x+90000}{x+500} \)
It must satisfy two linear inequalities (x + 500 > 0, so we may multiply through).
At least Rs. 120:
\( \displaystyle 100x+90000\ge 120x+60000 \)
\( \displaystyle 30000\ge 20x \)
\( \displaystyle x\le 1500 \)
At most Rs. 150:
\( \displaystyle 100x+90000\le 150x+75000 \)
\( \displaystyle 15000\le 50x \)
\( \displaystyle x\ge 300 \)
So the quantity of type A must satisfy \( 300\le x\le 1500 \) kg.
Checking the statements
(A) 1400 kg — Correct. 1400 lies between 300 and 1500.
(B) 200 kg — Incorrect. 200 is less than 300; the price would be \( \frac{20000+90000}{700}\approx 157 \), above Rs. 150.
(C) 1000 kg — Correct. 1000 lies between 300 and 1500.
(D) 800 kg — Correct. 800 lies between 300 and 1500.