Alligation or Mixture Questions and Answers

Answer: B

Water makes up 20% of the mixture.

Selling the mixture at the cost price of milk and gaining 25% means that for every 100 units of milk he pays for, he sells 125 units. The extra 25 units are water.

\[ \text{water} \% = \frac{25}{125} \times 100 = 20\% \]

The trap is 25%, which is water as a share of the milk, not of the mixture. The question asks for its share of the mixture.

Answer: C

The pure milk to be added is 9.2 litres. Milk in the three mixtures is 3 x 4/5 = 2.4, 4 x 3/4 = 3 and 5 x 5/7 = 25/7 litres, so the total milk is about 8.971 litres in a 12-litre mixture. After adding x litres of pure milk the milk fraction must be 6/7, so (8.971 + x)/(12 + x) = 6/7, which gives 62.8 + 7x = 72 + 6x and x = 9.2. The values 8.2, 8.7 and 9.7 do not satisfy this equation. A check on the final mixture: the milk is 8.971 + 9.2 = 18.171 litres in a total of 21.2 litres, and 18.171 divided by 21.2 is 0.857, which is 6/7 as required.

Answer: B

The mixture has 44 % spirit. Solution A has spirit and water in the ratio 2 : 3, so 6 litres of A contain 6 x 2/5 = 2.4 litres of spirit. Solution B has the ratio 7 : 8, so 9 litres of B contain 9 x 7/15 = 4.2 litres of spirit.

\[ \text{Spirit} = 2.4 + 4.2 = 6.6 \text{ litres} \]

\[ \text{Mixture} = 6 + 9 = 15 \text{ litres} \]

\[ \frac{6.6}{15} \times 100 = 44\% \]

The options 40 %, 52 % and 56 % are not the share of 6.6 litres in the 15 litres of the mixture. As a check, the water in the mixture is 3.6 + 4.8 = 8.4 litres, and 6.6 + 8.4 = 15 litres, the volume of the mixture.

Answer: A

The spirit in 780 mL of the mixture is 285 mL. In solution A the spirit is \(\frac{6}{13}\), so 3 L has \(\frac{18}{13}\) L of spirit. In solution B the ratio of water to spirit is 9 : 4, so spirit is \(\frac{4}{13}\), and 5 L has \(\frac{20}{13}\) L. \[ \begin{aligned} \text{Spirit} &= \frac{18}{13} + \frac{20}{13} \\ &= \frac{38}{13}\text{ L} \end{aligned} \] \[ \text{Total} = 3 + 5 = 8\text{ L} \] \[ \begin{aligned} \text{Fraction of spirit} &= \frac{38}{13\times 8} \\ &= \frac{19}{52} \end{aligned} \] \[ 780\times\frac{19}{52} = 15\times 19 = 285\text{ mL} \] Reading B as spirit to water (4 : 9 swapped) would give a different value, so the order in the stem matters; 304, 270 and 266 do not result.

Answer: A

The ratio is 5:4.

Variety P costs 800/5 = ₹ 160 per kg and variety Q costs 800/4 = ₹ 200 per kg. Selling at ₹ 192 with 8% profit means the cost price of the mixture is:

\[ \frac{192}{1.08} = 177.78\ \text{per kg} = \frac{1600}{9} \]

By alligation:

\[ \begin{aligned} \frac{P}{Q} &= \frac{200-177.78}{177.78-160} \\ &= \frac{22.22}{17.78} = \frac{5}{4} \end{aligned} \]

Check: 5 kg of P and 4 kg of Q cost 800 + 800 = ₹ 1600 for 9 kg, and selling 9 kg at 192 gives ₹ 1728, exactly 8% more. 1:1 would give a cost of ₹ 180 per kg and only 6.7% profit.

Answer: C

The sugar to be added is 60 gm.

The solution has \(0.4 \times 300 = 120\) gm of sugar and 180 gm of water. Adding sugar leaves the water unchanged, so at 50% the water must be half the new total:

\[ \begin{aligned} 180 &= 0.5\,(300 + x) \\ 300 + x &= 360 \Rightarrow x = 60 \end{aligned} \]

Check: 180 gm sugar in 360 gm of solution is exactly 50%. 50 gm would give 170/350, about 48.6%, and 40 gm gives 160/340, about 47%.

Answer: A

Water and wine must be mixed in the ratio 1 : 2.

Selling at Rs 50 per litre gives a 25% profit, so the cost price of the mixture is

\[ \frac{50}{1.25} = 40 \text{ rupees per litre} \]

Water costs nothing and wine costs Rs 60, so by alligation:

\[ \begin{aligned} \text{water} : \text{wine} &= (60 - 40) : (40 - 0) \\ &= 20 : 40 = 1 : 2 \end{aligned} \]

Check: 1 litre of water and 2 litres of wine cost Rs 120 for 3 litres, Rs 40 a litre. The other ratios give a cost of Rs 36 (2 : 3), about Rs 34.3 (3 : 4) and about Rs 33.3 (4 : 5), so the profit would not be 25%.

Answer: C

Take the quantities as 1, 1 and 2 kg, four kilograms in all.

\[ 126 + 135 + 2p = 4 \times 153 = 612 \]

\[ 2p = 612 - 261 = 351 \quad\Longrightarrow\quad p = \text{Rs. } 175.50 \]

The third variety carries half the weight of the mixture, which is why its price does not have to be far above 153 to lift the average.

Answer: A

Correct answer: 12 : 13

The original 50 litres contains \( 50 \times \frac{3}{5} = 30 \) litres of milk and 20 of water. Removing 10 litres of the mixture removes milk and water in their current 3 : 2 proportion - 6 litres of milk and 4 of water - leaving 24 of milk and 16 of water. Adding 10 litres of water brings the water to \( 16 + 10 = 26 \), so the new ratio is \[ 24 : 26 = 12 : 13. \] The essential point is that what is drawn off is a sample of the mixture, carrying both components in proportion. Removing 10 litres of pure milk would be a different problem and would leave 20 : 30, or 2 : 3. As a check, the total stays at 50 litres throughout, and \( 24 + 26 = 50 \).

Answer: A

No profit and no loss means the total cost equals the total revenue.

\[ 32(60) + 23x = 28(60 + x) \]

\[ 1920 + 23x = 1680 + 28x \]

\[ 240 = 5x \quad\Longrightarrow\quad x = 48 \text{ litres} \]

By alligation the same result follows from \( \tfrac{60}{x} = \tfrac{28-23}{32-28} = \tfrac{5}{4} \).