The Alligation Rule and Mixing for Profit

The Alligation Rule and the Cross

Mix two things at two different prices and the mixture's price lands somewhere between them. Finding that price from the quantities is the forward question, done in the Mixture Basics section. Alligation runs it backwards: you know the two prices and the price you want, and you need the ratio in which to mix.

The rule. Call the cheaper price C, the dearer price D and the mean price of the mixture M. Then

$$\frac{\text{quantity of cheaper}}{\text{quantity of dearer}} = \frac{D - M}{M - C}$$

The cross. Two kinds of pulses cost ₹72 and ₹96 per kg, and a shop wants a mixture worth ₹82 per kg. Write the two prices at the top, the mean in the middle, and subtract along each diagonal. Each difference ends up in the corner opposite the price it started from.

Cheaper : dearer = 14 : 10 = 7 : 5.

The number in the bottom-left corner is the cheaper variety's share, even though it came from the dearer price. That crossing over is the whole trick.

Why it works. Every kg of the cheaper pulse is ₹10 below the mean and every kg of the dearer one is ₹14 above it. In 7 kg of the cheaper pulse the shortfall is 7 × 10 = ₹70; in 5 kg of the dearer pulse the excess is 5 × 14 = ₹70. They cancel, so the 12 kg are worth exactly ₹82 a kg. Check: 7 × 72 + 5 × 96 = 504 + 480 = 984, and 984 ÷ 12 = 82.

When you can use it.

1. Two things are being mixed. Three or more need the pairing method shown later in this section.
2. The mean lies strictly between the two values. A mean of ₹100 from ₹72 and ₹96 is impossible, and that alone can strike out options.
3. The value is "per unit" of the thing you want the ratio of. Price per kg gives a ratio of kg; price per litre gives a ratio of litres.

Quick check. The mean sits nearer the variety you use more of. Here ₹82 is ₹10 from the cheaper price and ₹14 from the dearer one, so there must be more of the cheaper pulse, and 7 : 5 agrees. If the mean is exactly halfway, the ratio is 1 : 1. This check catches the swapped ratio (5 : 7 here), which is nearly always one of the options.

Using the Cross Backwards: Mean Price, a Missing Price and Quantities

The same cross answers four kinds of questions. In each one, fill in what you know and solve for the gap.

1. The mean price from a ratio. Two oils costing ₹110 and ₹150 per litre are mixed in the ratio 3 : 5. The mean moves up from the cheaper price by the dearer oil's share of the gap:

$$M = C + \frac{\text{parts of dearer}}{\text{total parts}} \times (D - C) = 110 + \frac{5}{8} \times 40 = 135$$

So the mixture costs ₹135 per litre. Check: (3 × 110 + 5 × 150) ÷ 8 = (330 + 750) ÷ 8 = 1080 ÷ 8 = 135.

2. A missing price. A grocer mixes rice at ₹64 per kg with another rice in the ratio 2 : 3, and the mixture is worth ₹76 per kg. The mixture costs more than ₹64, so the other rice must be the dearer one. On the cross, the ₹64 rice gets D − 76 and the other gets 76 − 64 = 12:

$$\frac{D - 76}{12} = \frac{2}{3} \Rightarrow D - 76 = 8 \Rightarrow D = 84$$

The second rice costs ₹84 per kg. Check: (2 × 64 + 3 × 84) ÷ 5 = (128 + 252) ÷ 5 = 380 ÷ 5 = 76.

3. One quantity from the other. How many kg of the ₹72 pulse must be mixed with 25 kg of the ₹96 pulse to make the ₹82 mixture above? The ratio is 7 : 5. The 25 kg of the dearer pulse is 5 parts, so 1 part is 5 kg and the cheaper pulse is 7 × 5 = 35 kg. Check: 35 × 72 + 25 × 96 = 2520 + 2400 = 4920, and 4920 ÷ 60 = 82.

4. A total to be split, or a difference. For 84 kg of the same mixture, split 84 in the ratio 7 : 5: one part is 7 kg, so 49 kg of the cheaper pulse and 35 kg of the dearer one. If the question says instead that the shop used 18 kg more of the cheaper pulse, that 18 kg is 7 − 5 = 2 parts, so 1 part is 9 kg: 63 kg and 45 kg.

Adding to a quantity you already have. A trader has 40 kg of tea at ₹260 per kg. How much tea at ₹340 per kg must he add so that the whole lot is worth ₹290 per kg?

₹260 tea : ₹340 tea = 50 : 30 = 5 : 3. His 40 kg is the 5 parts, so 1 part is 8 kg and he must add 3 × 8 = 24 kg. Check: 40 × 260 + 24 × 340 = 10,400 + 8,160 = 18,560, and 18,560 ÷ 64 = 290.

The lot he already has can itself be a mixture; all that matters is its price per kg.

[SSC Tip] When one quantity is given, divide it by its own share to get one part. Treating the 40 kg as the 3 parts gives $66\frac{2}{3}$ kg, a wrong option you will often see.

Alligation on Percentages: Mixing Two Strengths

The percentage of milk, acid or alcohol in a liquid is a "value per litre", just like a price. So the cross works on strengths too, with the volumes of the two liquids as the weights. Two fixed values help:

• plain water has 0% of the other liquid,
• the pure liquid is 100%.

Two strengths, one target. Acid solutions of 22% and 37% are to be mixed to make a 27% solution.

22% solution : 37% solution = 10 : 5 = 2 : 1. Check: 2 litres of the first hold 0.44 litre of acid and 1 litre of the second holds 0.37 litre, so the 3 litres hold 0.81 litre of acid, which is 27%.

How much of a weak solution to add to a strong one. How many litres of a 20% solution must be added to 15 litres of a 50% solution to get a 26% solution?

20% : 50% = (50 − 26) : (26 − 20) = 24 : 6 = 4 : 1.

The 15 litres of the 50% solution are the 1 part, so you add 4 × 15 = 60 litres of the 20% solution. Check: 12 + 7.5 = 19.5 litres of the chemical in 75 litres, which is 26%.

Adding water. How much water must be added to 36 litres of a 75% alcohol solution to bring it down to 60%? Water is 0%:

water : solution = (75 − 60) : (60 − 0) = 15 : 60 = 1 : 4.

So 36 litres need 9 litres of water. Check: 27 litres of alcohol in 45 litres is 60%. The Mixture Basics section does this by keeping the alcohol fixed; both ways give the same answer, so use whichever is quicker for you.

Adding the pure liquid. How much pure milk must be added to 40 litres of a mixture that is 70% milk to make it 76% milk? Pure milk is 100%: mixture : milk = (100 − 76) : (76 − 70) = 24 : 6 = 4 : 1, so add 10 litres. Check: 28 + 10 = 38 litres of milk in 50 litres is 76%.

Fractions work the same way. A strength given as a fraction ($\frac{2}{3}$ milk, $\frac{3}{4}$ milk) goes on the cross exactly like a percentage. Bring the fractions to one denominator first, as in the part on ratios below.

Taking some liquid out and putting another strength in its place uses this same cross; that is covered in the Removal and Replacement section.

The ratio you get is between the volumes of the two liquids, not between the amounts of acid in them. Keep to one component all the way: if the question gives water percentages, put water percentages on the cross, or convert every one of them to the other liquid.

Alligation on Ratios of Two Mixtures

When each vessel is described by a ratio (milk : water = 7 : 3), the cross needs one number per vessel. Use the fraction of one component in the whole mixture:

• milk : water = 7 : 3 means milk is $\frac{7}{10}$ of the mixture,
• milk : water = 2 : 3 means milk is $\frac{2}{5}$ of the mixture.

Then allige those fractions exactly as you would percentages.

Same denominators. Vessel A has milk and water in the ratio 7 : 3 and vessel B in the ratio 2 : 3. In what ratio should they be mixed to get milk and water in the ratio 3 : 2?

Milk fractions: A = $\frac{7}{10}$, B = $\frac{2}{5} = \frac{4}{10}$, target = $\frac{3}{5} = \frac{6}{10}$. Working in tenths:

B : A = 1 : 2, so A : B = 2 : 1. Check: 2 litres of A hold 1.4 litres of milk and 1 litre of B holds 0.4 litre, so the 3 litres hold 1.8 litres of milk and 1.2 litres of water, which is 3 : 2.

The LCM trick. With different denominators, bring all three fractions to their LCM. A has milk : water = 5 : 3 (milk $\frac{5}{8}$), B has 3 : 4 (milk $\frac{3}{7}$) and the target is 1 : 1 (milk $\frac{1}{2}$). The LCM of 8, 7 and 2 is 56, so the fractions are $\frac{35}{56}$, $\frac{24}{56}$ and $\frac{28}{56}$. Allige the numerators: A : B = (28 − 24) : (35 − 28) = 4 : 7. Check: 4 litres of A hold 2.5 litres of milk and 7 litres of B hold 3 litres, so 11 litres hold 5.5 litres of milk, exactly half.

The trap: alliging the ratios themselves. Do not put 7 : 3 on the cross as $\frac{7}{3}$ (milk per litre of water). In the first question that gives $\frac{3}{2} - \frac{2}{3} = \frac{5}{6}$ and $\frac{7}{3} - \frac{3}{2} = \frac{5}{6}$, a ratio of 1 : 1. But 1 litre of each actually gives 1.1 litres of milk and 0.9 litre of water, which is 11 : 9, not 3 : 2. "Milk per litre of water" uses water as the weight, while you are mixing litres of mixture. Always use part of the whole.

Alloys work the same way. How many kg of an alloy with copper and zinc in the ratio 7 : 5 must be melted with 24 kg of an alloy with copper and zinc in the ratio 1 : 2 to get an alloy with equal amounts of copper and zinc? Copper fractions: $\frac{7}{12}$, $\frac{1}{3} = \frac{4}{12}$ and target $\frac{1}{2} = \frac{6}{12}$. First : second = (6 − 4) : (7 − 6) = 2 : 1, so 24 kg of the second needs 48 kg of the first. Check: 28 + 8 = 36 kg of copper in 72 kg.

Either component gives the same answer. In the first question the water fractions are $\frac{3}{10}$, $\frac{6}{10}$ and target $\frac{4}{10}$, and A : B = (6 − 4) : (4 − 3) = 2 : 1 again.

Mixing two mixtures in a given ratio and finding the new ratio (the forward question) is in the Mixture Basics section.

Alligation Outside Mixtures

The cross works for any average of two rates, as long as you know what the rate is measured per. The thing after "per" is the weight, and the ratio you get is a ratio of those weights.

Two rates of interest. ₹48,000 is lent in two parts, at 6% and 11% a year, and the total yearly interest is ₹3,840. The average rate is 3,840 ÷ 48,000 × 100 = 8%. Sum at 6% : sum at 11% = (11 − 8) : (8 − 6) = 3 : 2, so ₹28,800 is at 6% and ₹19,200 at 11%. Check: 1,728 + 2,112 = 3,840.

[Tier 2] Interest for several years. Divide by the number of years first. ₹11,520 earned in 3 years on the same ₹48,000 is ₹3,840 a year, the same 8%.

Profit on part, loss on the rest. A trader has 600 kg of sugar. He sells part at 12% profit and the rest at 4% loss, and gains 6% overall. A loss is a negative profit, so −4 goes on the cross:

profit part : loss part = (6 − (−4)) : (12 − 6) = 10 : 6 = 5 : 3.

He sold 375 kg at 12% profit. Check with a cost of ₹c per kg: 375 × 0.12c − 225 × 0.04c = 45c − 9c = 36c, which is 6% of 600c. Strictly the weights are cost prices; they become kg here only because every kg cost the same.

Two speeds. A man covers 54 km in 5 hours, partly on foot at 6 km/h and the rest on a cycle at 18 km/h. His average speed is 10.8 km/h, and the cross gives

time on foot : time cycling = (18 − 10.8) : (10.8 − 6) = 7.2 : 4.8 = 3 : 2.

So he walked for 3 hours, which is 18 km. Speed is km per hour, so the weights are hours; taking 3 : 2 as the ratio of distances (32.4 km on foot) is the classic slip.

Two groups. In an exam 70% of the boys and 55% of the girls passed, and 64% of all students passed. Boys : girls = (64 − 55) : (70 − 64) = 9 : 6 = 3 : 2.

Coins. A bag has 120 coins of ₹2 and ₹5 worth ₹348. The average coin is worth 348 ÷ 120 = ₹2.90, so ₹2 coins : ₹5 coins = (5 − 2.9) : (2.9 − 2) = 2.1 : 0.9 = 7 : 3, which is 84 coins and 36 coins. Check: 168 + 180 = 348.

Three or More Varieties

With three or more prices, pair each price below the mean with a price above it, use the cross on each pair, and add up the shares of any variety that is used twice.

Two cheap, one dear. Rice at ₹31, ₹37 and ₹55 per kg is to be mixed to get rice worth ₹43 per kg. Here ₹31 and ₹37 are below ₹43 and ₹55 is above it.

• Pair ₹31 with ₹55: the shares are 55 − 43 = 12 for ₹31 and 43 − 31 = 12 for ₹55.
• Pair ₹37 with ₹55: the shares are 55 − 43 = 12 for ₹37 and 43 − 37 = 6 for ₹55.

So ₹31 : ₹37 : ₹55 = 12 : 12 : 18 = 2 : 2 : 3. Check: 2 × 31 + 2 × 37 + 3 × 55 = 62 + 74 + 165 = 301, and 301 ÷ 7 = 43.

Many answers are correct. Any amount of the first pair mixed with any amount of the second pair keeps the mean at ₹43, so there are endless valid ratios. For instance 1 : 2 : 2 also works: 31 + 74 + 110 = 215, and 215 ÷ 5 = 43. That is why SSC asks "which of the following can be the ratio" and gives four options. Don't work out one ratio and hunt for it; test the options.

The balance test. Each kg below the mean is short by its distance from the mean, and each kg above it has an excess. A ratio works when total short = total excess.

• 2 : 2 : 3: short = 2 × 12 + 2 × 6 = 36, excess = 3 × 12 = 36. It works.
• 3 : 2 : 2: short = 3 × 12 + 2 × 6 = 48, excess = 2 × 12 = 24. It fails.

This is faster than finding the full average of each option.

One term of the ratio missing. Pulses at ₹66, ₹78 and ₹102 per kg are mixed in the ratio 2 : 3 : x, and the mixture is worth ₹84 per kg. Short: 2 × 18 + 3 × 6 = 54. Excess: x × 18. So 18x = 54 and x = 3. Check: (132 + 234 + 306) ÷ 8 = 672 ÷ 8 = 84.

Two mixed first, then a third. Teas at ₹210 and ₹270 per kg are mixed in the ratio 2 : 1, and this blend is then mixed with tea at ₹310 per kg to get a blend worth ₹250 per kg. Find the final ratio of the three teas.

• First blend: (2 × 210 + 270) ÷ 3 = ₹230 per kg.
• Blend : third tea = (310 − 250) : (250 − 230) = 60 : 20 = 3 : 1.
• The 3 parts of blend split as 2 : 1, so the teas are in the ratio 2 : 1 : 1. Check: (420 + 270 + 310) ÷ 4 = 1000 ÷ 4 = 250.

One cheap, two dear. Pair the single cheap price with each dear price and add its two shares. Nothing else changes.

Mixing for Profit: Find the Cost of the Mixture First

CGL papers often add a selling price to an alligation question: "the mixture is sold at ₹x per kg at a gain of y%". The two varieties are given at their cost prices, so the cross needs the cost price of the mixture, not its selling price. Find that first, then allige.

$$\text{CP of mixture} = \text{SP} \times \frac{100}{100 + \text{gain}\%} \quad \text{or} \quad \text{SP} \times \frac{100}{100 - \text{loss}\%}$$

With a gain. Two varieties of rice cost ₹54 and ₹69 per kg. The mixture is sold at ₹72 per kg at a gain of 20%. In what ratio were they mixed?

• CP of the mixture = 72 × $\frac{100}{120}$ = ₹60 per kg.
• Cross on cost prices: (69 − 60) : (60 − 54) = 9 : 6 = 3 : 2.

Check by mixing forward: (3 × 54 + 2 × 69) ÷ 5 = (162 + 138) ÷ 5 = ₹60, and 60 plus 20% is 72.

With a loss. Two oils cost ₹155 and ₹200 per litre. The mixture is sold at ₹168 per litre at a loss of 4%.

• CP of the mixture = 168 × $\frac{100}{96}$ = ₹175 per litre.
• Cross: (200 − 175) : (175 − 155) = 25 : 20 = 5 : 4.

Check: (5 × 155 + 4 × 200) ÷ 9 = (775 + 800) ÷ 9 = 1575 ÷ 9 = 175, and 4% less than 175 is 168.

Two slips the options are built on.

1. Alliging on the selling price. Here ₹168 lies between ₹155 and ₹200, so it can be put on the cross, and it gives (200 − 168) : (168 − 155) = 32 : 13, which is wrong.
2. Taking the gain off the selling price. In the rice question, 72 − 20% of 72 = 57.6 is not the cost. The 20% is on the cost, so divide by 1.2. Using 57.6 gives (69 − 57.6) : (57.6 − 54) = 11.4 : 3.6 = 19 : 6.

Fraction gains. Common gains are quicker as fractions:

• 25% = $\frac{1}{4}$, so CP = SP × $\frac{4}{5}$
• 20% = $\frac{1}{5}$, so CP = SP × $\frac{5}{6}$
• $16\frac{2}{3}\%$ = $\frac{1}{6}$, so CP = SP × $\frac{6}{7}$
• $12\frac{1}{2}\%$ = $\frac{1}{8}$, so CP = SP × $\frac{8}{9}$
• a loss of 10% means CP = SP × $\frac{10}{9}$

The three steps.

1. Selling price and gain or loss give the cost price of the mixture.
2. Allige the two cost prices around it.
3. Use the ratio for whatever is asked: a ratio, a quantity or a price.

[SSC Tip] The cost of the mixture must lie between the two cost prices. If your CP falls outside, you have used the gain or loss the wrong way round.

Mixing for Profit: Quantity, Price and Gain

The same three steps answer every version. Only the unknown changes.

A quantity. How many kg of tea at ₹225 per kg must be mixed with 36 kg of tea at ₹180 per kg so that selling the mixture at ₹231 per kg gives a 10% profit?

• CP of the mixture = 231 × $\frac{100}{110}$ = ₹210.
• ₹180 tea : ₹225 tea = (225 − 210) : (210 − 180) = 15 : 30 = 1 : 2.
• The 36 kg of ₹180 tea is 1 part, so the ₹225 tea is 72 kg.

Check: 36 × 180 + 72 × 225 = 6,480 + 16,200 = 22,680 for 108 kg, which is ₹210 per kg, and 10% more is ₹231.

The gain, when the ratio is given. Wheat at ₹23 and ₹32 per kg is mixed in the ratio 5 : 4 and sold at ₹32.40 per kg. Find the gain percent.

• CP = (5 × 23 + 4 × 32) ÷ 9 = (115 + 128) ÷ 9 = 243 ÷ 9 = ₹27, or 23 + $\frac{4}{9}$ × 9 = 27.
• Gain = ₹5.40 on ₹27, which is 20%. The gain is always on the cost; ₹5.40 on ₹32.40 ($16\frac{2}{3}\%$) is the trap.

The selling price for a target gain. To gain 25% on that wheat, sell at 27 × 1.25 = ₹33.75 per kg.

A missing price. A grocer mixes rice at ₹44 per kg with another rice in the ratio 3 : 1, sells the mixture at ₹57 per kg and gains 20%. Find the price of the second rice.

• CP of the mixture = 57 × $\frac{5}{6}$ = ₹47.50.
• 4 kg of mixture cost 4 × 47.5 = ₹190, and the 3 kg of the first rice cost ₹132, so 1 kg of the second costs ₹58.

Check with the cross: (58 − 47.5) : (47.5 − 44) = 10.5 : 3.5 = 3 : 1.

Sold at the price of the dearer variety. Sugar at ₹39 and ₹54 per kg is mixed, and the mixture is sold at ₹54 per kg for a 20% gain. CP = 54 × $\frac{5}{6}$ = ₹45, so ₹39 sugar : ₹54 sugar = (54 − 45) : (45 − 39) = 9 : 6 = 3 : 2. Check: (3 × 39 + 2 × 54) ÷ 5 = (117 + 108) ÷ 5 = 45.

[Tier 2] Changing the mixing ratio. If the same sugar were mixed 2 : 3 instead and sold at the same ₹54, the CP would be (78 + 162) ÷ 5 = ₹48, and the gain only ₹6 on ₹48, which is 12.5%. More of the dearer variety means a higher cost and a smaller gain at the same selling price.

Free Water: Milk and Juice Sold at Cost Price

Water costs nothing, so on the cross it sits at ₹0. A milkman who adds water and sells the mixture at the cost price of pure milk makes his profit only because the water is sold as milk.

Selling at cost price. Milk costs ₹63 per litre. A milkman adds water and sells the mixture at ₹63 per litre, gaining 12.5%.

• CP of the mixture = 63 × $\frac{8}{9}$ = ₹56 per litre.
• Water (₹0) : milk (₹63) = (63 − 56) : (56 − 0) = 7 : 56 = 1 : 8.

When he sells at the cost price there is a quicker way. The gain is the water as a percentage of the milk, so water : milk = gain% : 100 = 12.5 : 100 = 1 : 8.

$$\text{Gain}\% = \frac{\text{water}}{\text{milk}} \times 100 \quad \text{(selling at the cost price of milk)}$$

Going forwards, 1 litre of water with every 8 litres of milk, sold at cost price, gives a 12.5% gain.

Water as a part of the mixture. In the 1 : 8 mixture, water is $\frac{1}{9}$ of the whole, which is $11\frac{1}{9}\%$, not 12.5%. For "what percentage of the mixture is water", use $\frac{g}{100 + g} \times 100$, where g is the gain percent.

How much water to add. How much water must be added to 56 litres of milk costing ₹52 per litre so that selling the mixture at ₹52 per litre gives a gain of 25%? Water must be 25% of the milk: 14 litres. Check: 70 litres sell for 70 × 52 = ₹3,640 against a cost of 56 × 52 = ₹2,912, a gain of ₹728, which is 25%. Taking 25% of the mixture instead gives the wrong option $18\frac{2}{3}$ litres.

Selling above cost price. Milk costs ₹45 per litre. A milkman adds water and sells the mixture at ₹48 per litre, gaining 28%. CP of the mixture = 48 ÷ 1.28 = ₹37.50. Water : milk = (45 − 37.5) : (37.5 − 0) = 7.5 : 37.5 = 1 : 5. Check: 5 litres of milk cost ₹225; the 6 litres of mixture sell for ₹288, a gain of ₹63, which is 28%.

Selling below cost price and still gaining. A dairy sells diluted milk at 10% below the cost price of milk and still gains 20%. Take milk at ₹100 a litre. The mixture sells at ₹90, so its cost is 90 ÷ 1.2 = ₹75. Water : milk = (100 − 75) : (75 − 0) = 25 : 75 = 1 : 3.

A one-line check: (mixture ÷ milk) × (SP ÷ CP of milk) = 1 + gain. Here $\frac{4}{3} \times \frac{9}{10} = \frac{6}{5}$, a 20% gain.

Juice, oil, anything. The same applies to juice or any liquid diluted with water. If what is added is not free (a cheaper oil, say), it is an ordinary two-price question like the ones above.

[SSC Tip] At cost price, gain = water ÷ milk, not water ÷ mixture. Both answers are always among the options, so check which one the question asks for.

Solved Examples

Question 1

In what ratio must a grocer mix two varieties of pulses costing ₹68 per kg and ₹89 per kg so that the mixture costs ₹77 per kg? (a) 3 : 4 (b) 4 : 3 (c) 5 : 4 (d) 7 : 5

Answer. The mean, ₹77, is 9 above the cheaper price and 12 below the dearer one. On the cross the cheaper pulse gets the dearer one's distance: cheaper : dearer = (89 − 77) : (77 − 68) = 12 : 9 = 4 : 3. The answer is (b) 4 : 3. Check: 4 × 68 + 3 × 89 = 272 + 267 = 539, and 539 ÷ 7 = 77. ₹77 is nearer ₹68, so more of the cheaper pulse makes sense.

Question 2

How many kg of sugar costing ₹43 per kg must be mixed with 32 kg of sugar costing ₹52 per kg so that the mixture is worth ₹47 per kg? (a) 25.6 kg (b) 36 kg (c) 40 kg (d) 44 kg

Answer. Cheaper : dearer = (52 − 47) : (47 − 43) = 5 : 4. The 32 kg I already have is the dearer sugar, which is 4 parts, so 1 part is 8 kg and the cheaper sugar is 5 × 8 = 40 kg. The answer is (c) 40 kg. Check: 40 × 43 + 32 × 52 = 1720 + 1664 = 3384 for 72 kg, and 3384 ÷ 72 = 47. Treating the 32 kg as the 5 parts gives 25.6 kg.

Question 3

Two varieties of tea are mixed in the ratio 5 : 3. The first costs ₹240 per kg and the mixture is worth ₹270 per kg. What is the price per kg of the second variety? (a) ₹288 (b) ₹300 (c) ₹350 (d) ₹320

Answer. The mixture costs more than ₹240, so the second tea is the dearer one, and it makes up 3 of the 8 parts. The mean moves up from ₹240 by $\frac{3}{8}$ of the gap: 270 = 240 + $\frac{3}{8}$ × (D − 240), so D − 240 = 80 and D = 320. The answer is (d) ₹320. Check: 5 × 240 + 3 × 320 = 1200 + 960 = 2160, and 2160 ÷ 8 = 270.

Question 4

A shopkeeper wants 63 kg of a mixture worth ₹45 per kg from two kinds of rice costing ₹37 per kg and ₹61 per kg. How many kg of the cheaper rice should he use? (a) 42 kg (b) 21 kg (c) 31.5 kg (d) 36 kg

Answer. Cheaper : dearer = (61 − 45) : (45 − 37) = 16 : 8 = 2 : 1. The 63 kg is 3 parts, so 1 part is 21 kg and the cheaper rice is 2 × 21 = 42 kg. The answer is (a) 42 kg. Check: 42 × 37 + 21 × 61 = 1554 + 1281 = 2835, and 2835 ÷ 63 = 45.

Question 5

How many litres of an 8% acid solution must be added to 18 litres of a 38% acid solution to get a 20% acid solution? (a) 12 litres (b) 27 litres (c) 36 litres (d) 30 litres

Answer. On the cross: 8% solution : 38% solution = (38 − 20) : (20 − 8) = 18 : 12 = 3 : 2. The 18 litres of the 38% solution are 2 parts, so 1 part is 9 litres and I need 3 × 9 = 27 litres of the 8% solution. The answer is (b) 27 litres. Check: 2.16 + 6.84 = 9 litres of acid in 45 litres, which is 20%.

Question 6

Vessel A has milk that is 55% pure (the rest is water) and vessel B has milk that is 85% pure. How many litres should be taken from vessel A to make 45 litres of a mixture that is 75% milk? (a) 30 litres (b) 22.5 litres (c) 15 litres (d) 18 litres

Answer. A : B = (85 − 75) : (75 − 55) = 10 : 20 = 1 : 2. So A gives 1 part out of 3: $\frac{1}{3}$ × 45 = 15 litres. The answer is (c) 15 litres. Check: 15 × 0.55 + 30 × 0.85 = 8.25 + 25.5 = 33.75 litres of milk, and 33.75 ÷ 45 = 75%.

Question 7

Two vessels contain milk and water in the ratios 4 : 1 and 3 : 7. In what ratio should the contents of the two vessels be mixed to get a mixture with milk and water in the ratio 1 : 1? (a) 3 : 2 (b) 4 : 21 (c) 1 : 1 (d) 2 : 3

Answer. I turn each ratio into milk out of the whole. First vessel: $\frac{4}{5} = \frac{8}{10}$. Second: $\frac{3}{10}$. Target: $\frac{1}{2} = \frac{5}{10}$. In tenths the cross gives first : second = (5 − 3) : (8 − 5) = 2 : 3. The answer is (d) 2 : 3. Check: 2 litres of the first hold 1.6 litres of milk and 3 litres of the second hold 0.9 litre, so 5 litres hold 2.5 litres of milk, exactly half. You get 4 : 21 by alliging 4, $\frac{3}{7}$ and 1 (milk per litre of water), which is the wrong measure.

Question 8

Alloy P contains copper and tin in the ratio 7 : 2 and alloy Q contains them in the ratio 5 : 7. How many kg of alloy P must be melted with 48 kg of alloy Q to get an alloy with copper and tin in the ratio 2 : 1? (a) 96 kg (b) 108 kg (c) $21\frac{1}{3}$ kg (d) 120 kg

Answer. Copper out of the whole: P = $\frac{7}{9}$, Q = $\frac{5}{12}$, target = $\frac{2}{3}$. The LCM of 9, 12 and 3 is 36, so these are $\frac{28}{36}$, $\frac{15}{36}$ and $\frac{24}{36}$. Cross on 28, 15 and 24: P : Q = (24 − 15) : (28 − 24) = 9 : 4. The 48 kg of Q is 4 parts, so 1 part is 12 kg and P = 9 × 12 = 108 kg. The answer is (b) 108 kg. Check: copper = 84 + 20 = 104 kg out of 156 kg, which is $\frac{2}{3}$.

Question 9

A sum of ₹56,000 is lent in two parts, one at 7% and the other at 12% simple interest per annum. The total interest for one year is ₹5,040. How much was lent at 12%? (a) ₹33,600 (b) ₹28,000 (c) ₹24,000 (d) ₹22,400

Answer. Average rate = 5,040 ÷ 56,000 × 100 = 9%. Sum at 7% : sum at 12% = (12 − 9) : (9 − 7) = 3 : 2. So the 12% part is $\frac{2}{5}$ of ₹56,000, which is ₹22,400. The answer is (d) ₹22,400. Check: 7% of 33,600 is 2,352 and 12% of 22,400 is 2,688; together they make 5,040.

Question 10

A trader bought two articles for ₹27,000 in all. He sold one at a profit of 15% and the other at a loss of 9%, and on the whole he gained 5%. What was the cost price of the article sold at a profit? (a) ₹15,750 (b) ₹11,250 (c) ₹13,500 (d) ₹16,200

Answer. A loss is a negative profit, so the two rates are +15% and −9%, and the overall rate is +5%. Cost of the first : cost of the second = (5 − (−9)) : (15 − 5) = 14 : 10 = 7 : 5. So the first article cost $\frac{7}{12}$ × 27,000 = ₹15,750. The answer is (a) ₹15,750. Check: 15% of 15,750 is a profit of 2,362.50 and 9% of 11,250 is a loss of 1,012.50; the net gain of 1,350 is 5% of 27,000.

Question 11

Ravi covers 69 km in 6 hours. He walks part of the way at 4.5 km/h and cycles the rest at 15 km/h. How far does he walk? (a) 23 km (b) 9 km (c) 60 km (d) 12 km

Answer. Average speed = 69 ÷ 6 = 11.5 km/h. Speed is per hour, so the cross gives a ratio of times: walking time : cycling time = (15 − 11.5) : (11.5 − 4.5) = 3.5 : 7 = 1 : 2. He walks for $\frac{1}{3}$ × 6 = 2 hours, which is 2 × 4.5 = 9 km. The answer is (b) 9 km. Check: he cycles 4 × 15 = 60 km, and 9 + 60 = 69. Taking 1 : 2 as the ratio of distances gives 23 km, the trap.

Question 12

Three varieties of tea costing ₹135, ₹165 and ₹225 per kg are mixed so that the mixture costs ₹185 per kg. Which of the following can be the ratio in which they are mixed? (a) 1 : 2 : 3 (b) 3 : 1 : 2 (c) 2 : 1 : 3 (d) 2 : 3 : 1

Answer. I use the balance test around ₹185. Each kg of ₹135 tea is 50 short, each kg of ₹165 tea is 20 short, and each kg of ₹225 tea is 40 extra. (a) 1 : 2 : 3: short = 50 + 40 = 90, extra = 120, no. (b) 3 : 1 : 2: short = 150 + 20 = 170, extra = 80, no. (c) 2 : 1 : 3: short = 100 + 20 = 120, extra = 120, yes. (d) 2 : 3 : 1: short = 100 + 60 = 160, extra = 40, no. The answer is (c) 2 : 1 : 3. Check: 2 × 135 + 165 + 3 × 225 = 270 + 165 + 675 = 1110, and 1110 ÷ 6 = 185.

Question 13

Rice costing ₹41 per kg and rice costing ₹53 per kg are mixed in the ratio 1 : 2. This blend is then mixed with rice costing ₹69 per kg to get a mixture worth ₹54 per kg. In what ratio are the three kinds of rice finally mixed? (a) 1 : 2 : 1 (b) 2 : 1 : 1 (c) 1 : 2 : 3 (d) 1 : 1 : 1

Answer. First blend: (41 + 2 × 53) ÷ 3 = 147 ÷ 3 = ₹49 per kg. Now the cross with ₹49 and ₹69 around ₹54: blend : third rice = (69 − 54) : (54 − 49) = 15 : 5 = 3 : 1. The 3 parts of blend are 1 part of ₹41 rice and 2 parts of ₹53 rice, so the final ratio is 1 : 2 : 1. The answer is (a) 1 : 2 : 1. Check: 41 + 106 + 69 = 216, and 216 ÷ 4 = 54.

Question 14

Two varieties of rice costing ₹46 per kg and ₹64 per kg are mixed, and the mixture is sold at ₹60.75 per kg at a profit of 12.5%. In what ratio were the two varieties mixed? (a) 4 : 5 (b) 13 : 59 (c) 5 : 4 (d) 4 : 3

Answer. First the cost of the mixture. 12.5% is $\frac{1}{8}$, so the selling price is $\frac{9}{8}$ of the cost, and CP = 60.75 × $\frac{8}{9}$ = ₹54 per kg. Now the cross on cost prices: (64 − 54) : (54 − 46) = 10 : 8 = 5 : 4. The answer is (c) 5 : 4. Check: 5 × 46 + 4 × 64 = 230 + 256 = 486, 486 ÷ 9 = 54, and 54 plus 12.5% is 60.75. Putting ₹60.75 on the cross gives (64 − 60.75) : (60.75 − 46) = 3.25 : 14.75 = 13 : 59, the trap.

Question 15

A dealer mixes two kinds of ghee costing ₹520 per kg and ₹640 per kg and sells the mixture at ₹532 per kg, making a loss of 5%. In what ratio did he mix them? (a) 2 : 1 (b) 1 : 2 (c) 9 : 1 (d) 3 : 1

Answer. With a 5% loss, the selling price is 95% of the cost, so CP = 532 × $\frac{100}{95}$ = ₹560 per kg. Cross: (640 − 560) : (560 − 520) = 80 : 40 = 2 : 1. The answer is (a) 2 : 1. Check: (2 × 520 + 640) ÷ 3 = 1680 ÷ 3 = 560, and 5% of 560 is 28, so he sells at 532.

Question 16

How many kg of sugar costing ₹51 per kg must be mixed with 30 kg of sugar costing ₹63 per kg so that, by selling the mixture at ₹66 per kg, there is a gain of 20%? (a) 15 kg (b) 45 kg (c) 75 kg (d) 60 kg

Answer. CP of the mixture = 66 × $\frac{5}{6}$ = ₹55. Cross: ₹51 sugar : ₹63 sugar = (63 − 55) : (55 − 51) = 8 : 4 = 2 : 1. The 30 kg of ₹63 sugar is 1 part, so I need 2 × 30 = 60 kg of the ₹51 sugar. The answer is (d) 60 kg. Check: 60 × 51 + 30 × 63 = 3060 + 1890 = 4950 for 90 kg, which is ₹55 per kg, and 20% more is ₹66.

Question 17

Rice costing ₹52 per kg and rice costing ₹70 per kg are mixed in the ratio 4 : 5, and the mixture is sold at ₹74.40 per kg. What is the gain percent? (a) $16\frac{2}{3}\%$ (b) 24% (c) 20% (d) 22%

Answer. Cost of 9 kg = 4 × 52 + 5 × 70 = 208 + 350 = ₹558, so CP = 558 ÷ 9 = ₹62 per kg. Gain = 74.40 − 62 = ₹12.40 on ₹62, which is 20%. The answer is (c) 20%. The gain is worked out on the cost; ₹12.40 on ₹74.40 would give $16\frac{2}{3}\%$.

Question 18

A trader mixes dal costing ₹84 per kg with another dal in the ratio 3 : 2. He sells the mixture at ₹110.40 per kg and makes a profit of 15%. What is the cost of the second dal per kg? (a) ₹104 (b) ₹114 (c) ₹150 (d) ₹120

Answer. CP of the mixture = 110.40 ÷ 1.15 = ₹96 per kg. 5 kg of the mixture cost 5 × 96 = ₹480. The 3 kg of the first dal cost 3 × 84 = ₹252, so the 2 kg of the second cost ₹228, which is ₹114 per kg. The answer is (b) ₹114. Check on the cross: (114 − 96) : (96 − 84) = 18 : 12 = 3 : 2.

Question 19

A shopkeeper mixes two grades of tea costing ₹264 per kg and ₹354 per kg and sells the mixture at ₹354 per kg, the price of the costlier tea. He gains 18%. In what ratio did he mix the two grades? (a) 3 : 2 (b) 2 : 3 (c) 9 : 5 (d) 5 : 3

Answer. CP of the mixture = 354 ÷ 1.18 = ₹300 per kg. Cross: (354 − 300) : (300 − 264) = 54 : 36 = 3 : 2. The answer is (a) 3 : 2. Check: 3 × 264 + 2 × 354 = 792 + 708 = 1500 for 5 kg, which is ₹300 a kg, and 18% more is ₹354.

Question 20

A milkman mixes water with milk and sells the mixture at the cost price of pure milk. He gains $14\frac{2}{7}\%$. What percentage of the mixture is water? (a) $14\frac{2}{7}\%$ (b) 12.5% (c) $16\frac{2}{3}\%$ (d) 87.5%

Answer. Selling at the cost price, the gain is the water as a part of the milk. $14\frac{2}{7}\% = \frac{1}{7}$, so water : milk = 1 : 7. Out of 8 litres, 1 is water, which is $\frac{1}{8}$ = 12.5% of the mixture. The answer is (b) 12.5%. $14\frac{2}{7}\%$ compares water with the milk, not with the mixture.

Question 21

Milk costs ₹56 per litre. A milkman mixes water with it and sells the mixture at ₹63 per litre, making a profit of 31.25%. What is the ratio of water to milk in the mixture? (a) 5 : 16 (b) 1 : 7 (c) 1 : 5 (d) 1 : 6

Answer. CP of the mixture = 63 ÷ 1.3125 = ₹48 per litre. Water costs ₹0. Cross: water : milk = (56 − 48) : (48 − 0) = 8 : 48 = 1 : 6. The answer is (d) 1 : 6. Check: 6 litres of milk cost ₹336; with 1 litre of water he sells 7 litres for 7 × 63 = ₹441, a gain of ₹105, and 105 ÷ 336 = 31.25%. 5 : 16 would be right only if he sold at ₹56; here he also sells above the cost price.

Question 22

A vendor has 60 litres of juice that cost him ₹50 per litre. He adds water and sells the mixture at ₹48 per litre, and still makes a profit of 20%. How many litres of water did he add? (a) 12 litres (b) 20 litres (c) 15 litres (d) 16 litres

Answer. CP of the mixture = 48 ÷ 1.2 = ₹40 per litre. Water : juice = (50 − 40) : (40 − 0) = 10 : 40 = 1 : 4. With 60 litres of juice he adds $\frac{1}{4}$ × 60 = 15 litres of water. The answer is (c) 15 litres. Check: 75 litres sell for 75 × 48 = ₹3,600 against a cost of ₹3,000, a gain of ₹600, which is 20%.