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

The number of boys was 320.

Let \(b\) be the number of boys, so there are \(480 - b\) girls. The total number passing is 75% of 480, which is 360:

\[ \begin{aligned} 0.70b + 0.85(480 - b) &= 360 \\ 408 - 0.15b &= 360 \\ b &= \frac{48}{0.15} = 320 \end{aligned} \]

Check: 70% of 320 boys is 224 and 85% of 160 girls is 136, a total of 360. A quick alligation gives the same: the overall rate of 75% lies 5 points above 70 and 10 below 85, so boys and girls are in the ratio 10 : 5, or 2 : 1. 370, 340 and 360 do not split 480 in that ratio.

Answer: A

81 litres of water must be added.

In 729 litres in the ratio 7:2, milk is \(\frac{7}{9} \times 729 = 567\) litres and water is 162 litres. Only water is added, so the milk stays at 567 litres. For the new ratio 7:3:

\[ \begin{aligned} \text{water needed} &= 567 \times \tfrac{3}{7} = 243 \\ \text{water added} &= 243 - 162 = 81 \end{aligned} \]

Check: \(567 : 243 = 7 : 3\). The other options leave the ratio short of or beyond 7:3.

Answer: C

The mixture costs Rs 18 per kg.

Take 2 kg of the Rs 15 sugar and 3 kg of the Rs 20 sugar, in the ratio 2 : 3:

\[ \begin{aligned} \text{cost} &= 2 \times 15 + 3 \times 20 = 90 \\ \text{price per kg} &= \frac{90}{2 + 3} = 18 \end{aligned} \]

A quick check: the mixture price must lie between 15 and 20, and closer to 20 because there is more of the dearer sugar. Rs 17 is the price if the ratio were reversed (3 : 2), and Rs 21 is impossible, being above both prices; Rs 19 is too close to the dearer sugar.