Mixture questions look as if they are about rice, tea, milk and acid, but they are about one idea: a mixture's value is a weighted average of its ingredients. Alligation is a fast way to find the weights, and repeated replacement is multiplying by the same fraction again and again. This lesson covers the alligation rule, mean prices, solutions and water, averages, profits and speeds, mixtures sold at a profit, replacement, and combining mixtures, with worked examples and traps.
A mixture's value, whether a price per kg or a strength in percent, is a weighted average of its ingredients, weighted by how much of each goes in. Every question in this chapter is that one idea seen from a different side.
If you know the amounts and want the mixture's value, multiply, add and divide. If you know the mixture's value and want the amounts, use alligation. And if liquid is drawn out and replaced again and again, multiply by the fraction kept each time.
It gives the ratio in which two ingredients must be mixed to reach a given mean value. With the cheaper value , the dearer value and the mean between them, the cheaper part to the dearer part is .
Alligation
Watch out
Swapping the two gaps is the commonest slip. Say it to yourself: the cheaper one gets the dearer side's gap. And the mean must lie between and ; if it does not, a profit or loss is hidden in the price, or something has gone wrong.
Around the mean, the gains and losses must balance. Mixing kg at price with kg at price to get price means , which rearranges to .
Key idea
The balance gives a quick check: the mean sits nearer the ingredient there is more of. A mean of ₹46 between ₹42 and ₹54 is much nearer ₹42, so there must be more of the ₹42 rice.
Multiply each price by its quantity, add, and divide by the total quantity. This is alligation in reverse, and it needs no cross.
Example 1 · Mean price
Pooja mixes 4 kg of rice at ₹35 a kg with 6 kg at ₹50 a kg. Find the price of the mixture.
Cost: .
Mean price: .
₹44 per kg
The plain average of 35 and 50 is 42.5, right only for equal quantities, and usually among the options.
Treat each solution's strength, its percentage of acid, sugar, milk or alcohol, exactly like a price. The two strengths are the two values, and the target strength is the mean.
Example 2 · Two syrups
In what ratio should a 15% sugar syrup and a 45% syrup be mixed to get a 25% syrup?
15% part to 45% part: .
Check: 2 litres at 15% and 1 litre at 45% hold litres of sugar in 3 litres, which is 25%.
| Ingredient | Value to use |
|---|---|
| Pure water, in a price question | ₹0 per litre |
| Pure water, in a strength question | 0% |
| Pure acid, milk or juice | 100% |
Example 3 · Diluting with water
How much water must be added to 30 litres of a 40% salt solution to make it 25%?
Solution to water: .
5 parts are 30 litres, so 1 part is 6 litres and the water is litres.
Check: 12 litres of salt in 48 litres is 25%.
18 litres
Yes. It works for any weighted average: average marks, salaries, interest rates, profit percentages, speeds. The only question is what the weights are, because the ratio you get is always a ratio of those weights.
| Situation | The two values | The ratio you get is of |
|---|---|---|
| Class average | Boys' and girls' averages | Number of boys to girls |
| Profit on parts of a stock | The two profit percentages | The cost price of each part |
| Interest on two loans | The two rates | The two sums lent |
| Average speed | The two speeds | The two times, not distances |
Example 4 · Two loans
Rafiq lends ₹20,000 in two parts at 6% and 11% simple interest and earns ₹1,600 in a year. How much did he lend at 6%?
Average rate: .
6% part to 11% part: .
Amount at 6%: .
₹12,000
Check: .
Losses and speeds
First turn the selling price into the cost price of the mixture, then use alligation on cost prices. Profit is on cost, so divide by , never multiply by .
Example 5 · Selling the mixture at a profit
A shopkeeper mixes pulses costing ₹80 and ₹110 a kg and sells the mixture at ₹108 a kg, making a 20% profit. Find the mixing ratio.
Cost of the mixture: rupees a kg.
Cheaper to dearer: .
Exam shortcut
A milkman who adds free water and sells at the cost price of milk gains water over milk, as a percentage. With milk to water he gains 20%, because he pays for 5 litres and sells 6.
If a vessel holds litres of a pure liquid, and each time litres are drawn out and replaced with water, the liquid left after operations is . Each operation keeps the same fraction of whatever liquid is there.
Replacement
Example 6 · Three replacements
A 50-litre can is full of milk. 5 litres are drawn out and replaced with water, three times in all. How much milk is left?
Fraction kept each time: .
Milk left: litres.
36.45 litres of milk, and 13.55 litres of water
Subtracting 5 litres three times gives 35 litres, which is wrong: after the first draw, what comes out is partly water.
Exam shortcut
Track the pure component step by step: take out its share, then add the new solution's share. There is no single power formula.
Example 7 · Topping up with a stronger acid
A 50-litre tank of 30% acid has 10 litres replaced with a 90% acid solution. Find the new strength.
Acid at first: 15 litres. Removed with the 10 litres: 3 litres. Added: 9 litres.
Acid now: litres in 50 litres.
42%
As a check by weighted average, the tank holds old and new solution in , and .
Hold the ingredient you are not adding fixed and rebuild the total around it. Adding water or letting it evaporate keeps the solute fixed; adding milk to milk and water keeps the water fixed.
Turn each mixture into actual quantities of each component, add, and simplify. Never add ratio terms unless the two mixtures have the same total.
Exam shortcut
To find the mixing ratio for a target, write one component as a fraction of each whole and use alligation on the fractions. Milk to water is milk and is . For , which is : first to second is .
Alligation
Mean value
Weights
Replacement
The same weighted-average idea runs through averages and ratio and proportion, and cost and selling prices through profit, loss and discount.
60 questions, grouped by type, easy to hard. Pick an answer and check it. The worked solution opens when you ask for it.
Kavya's grocery mixes rice costing ₹42 per kg with rice costing ₹54 per kg so that the mixture is worth ₹46 per kg. In what ratio (cheaper : dearer) must the two be mixed?
AnswerD ·
Cheaper share = dearer price − mean = .
Dearer share = mean − cheaper price = .
Cheaper : dearer = .
Worth noting
The difference on the dearer side belongs to the CHEAPER variety. Swapping them gives , the tempting slip.
Tenzin blends two teas, one at ₹180 per kg and the other at ₹240 per kg, in the ratio . What is the cost of the blend per kg?
AnswerB · ₹200
Cost of 3 kg = rupees.
Cost per kg = rupees.
Worth noting
₹210 is the plain average, which ignores the ratio; ₹220 attaches the weight 2 to the wrong price.
A sweet shop mixes 3 kg of sugar at ₹38 per kg with 5 kg of sugar at ₹46 per kg. What is the price of the mixture per kg?
AnswerC · ₹43
Total cost = rupees.
Price per kg = rupees.
Worth noting
₹42 is the simple average of 38 and 46; it would be right only for equal quantities. ₹41 comes from swapping the quantities.
A lab assistant has a 20% acid solution and a 50% acid solution. In what ratio should they be mixed to get a 30% acid solution?
AnswerA ·
20% part : 50% part = .
= .
Worth noting
The mean 30% sits closer to 20%, so more of the 20% solution is needed. That check rules out .
In a class test, the boys averaged 62 marks and the girls averaged 74 marks. The whole class averaged 70 marks. What is the ratio of boys to girls?
AnswerD ·
Boys : girls = .
= .
Worth noting
The class average is nearer the girls' average, so there must be more girls. is the reversed ratio.
How many kg of dal costing ₹30 per kg must be mixed with 12 kg of dal costing ₹45 per kg so that the mixture is worth ₹36 per kg?
AnswerC · 18 kg
Cheaper : dearer = .
2 parts = 12 kg, so 1 part = 6 kg.
Cheaper dal = kg.
Worth noting
8 kg comes from reversing the ratio to .
A jeweller melts together two alloys containing 70% gold and 85% gold in the ratio by weight. What percentage of the new alloy is gold?
AnswerC · 80%
Gold in 3 units = (percent-units).
Gold percentage = .
Worth noting
77.5% is the simple average; 75% attaches the larger weight to the 70% alloy.
Meenakshi mixes 10 litres of a 25% mango-pulp drink with 15 litres of a 40% mango-pulp drink. What is the mango-pulp percentage of the mixture?
AnswerC · 34%
Pulp = litres.
Percentage = .
Worth noting
32.5% is the plain average of 25% and 40%; 31% swaps the two volumes.
A milkman buys milk at ₹60 per litre. In what ratio (water : milk) must he add water, which costs nothing, so that the mixture costs him ₹48 per litre?
AnswerB ·
Treat water as a liquid priced ₹0 per litre.
Water : milk = .
= .
Worth noting
is the reversed ratio — it would make the mixture mostly water.
Gurpreet mixes 2 kg of cashews at ₹900 per kg with 3 kg of a second variety. The mixture is worth ₹1,080 per kg. What is the price of the second variety per kg?
AnswerC · ₹1,200
Cost of 5 kg of mixture = rupees.
Cost of the 2 kg = rupees.
Second variety: rupees per kg.
Worth noting
₹1,260 comes from assuming the mean is the plain average, so the second price is .
A sports hostel has 30 students with an average weight of 48 kg and 20 students with an average weight of 58 kg. What is the average weight of all 50 students?
AnswerB · 52 kg
Total weight = kg.
Average = kg.
Worth noting
53 kg is the simple average of 48 and 58, which ignores that the groups are of different sizes.
A café blends coffee beans costing ₹800 per kg with beans costing ₹500 per kg to make a blend worth ₹620 per kg. What is the ratio of the ₹800 beans to the ₹500 beans?
AnswerB ·
₹800 beans : ₹500 beans = .
= .
Worth noting
reverses the ratio. The mean ₹620 is nearer ₹500, so there are more of the ₹500 beans.
Arjun sells 50 kg of rice, part of it at a 10% profit and the rest at a 25% profit. Overall he makes a 16% profit. How many kg did he sell at a 10% profit?
AnswerC · 30 kg
Treat the profit percentages like prices: 10% and 25% around a mean of 16%.
10% part : 25% part = .
Quantity at kg.
Worth noting
20 kg is the quantity sold at 25% — it is what you get if the ratio is reversed.
Fatima mixes two teas costing ₹120 per kg and ₹150 per kg. She sells the mixture at ₹165 per kg and makes a 25% profit. In what ratio (cheaper : dearer) did she mix them?
AnswerD ·
Cost price of the mixture = rupees per kg.
Cheaper : dearer = .
= .
Worth noting
Taking the cost as 75% of ₹165 (₹123.75) is the common error — profit is on cost, so divide by 1.25. That error leads to .
A dairy has 20 litres of a mixture that is 60% milk. How many litres of a mixture that is 90% milk must be added to get a mixture that is 80% milk?
AnswerD · 40 litres
.
So the 90% mixture must be twice the 60% mixture.
Needed = litres.
Worth noting
10 litres comes from reversing the ratio. Check: milk = out of 60 litres = 80%.
A company has 120 employees with an average salary of ₹30,000 a month. Managers average ₹50,000 and all other employees average ₹25,000. How many managers are there?
AnswerA · 24
Managers : others = .
Total parts = .
Managers = .
Worth noting
30 comes from taking one-fourth of 120 — a ratio of means one part in FIVE. 96 is the number of other employees.
Lalremsiami lends ₹12,000 in two parts, one at 8% simple interest a year and the other at 12% a year. Her total interest for one year is ₹1,140. How much did she lend at 8%?
AnswerD · ₹7,500
Average rate = .
8% part : 12% part = .
Amount at rupees.
Worth noting
₹4,500 is the amount at 12% (the ratio reversed). Check: = ₹1,140.
Alloy P has copper and zinc in the ratio ; alloy Q has them in the ratio . In what ratio must P and Q be melted together to get an alloy with copper and zinc in the ratio ?
AnswerD ·
Copper fraction: P = , Q = , target = .
.
= .
Worth noting
Alligation works on the fraction of ONE component in each mixture, not on the ratios themselves. Convert each ratio to a fraction of the whole first.
Joseph mixes almonds costing ₹640 per kg and ₹800 per kg in the ratio and sells the mixture at ₹840 per kg. What is his profit percentage?
AnswerC · 20%
Cost of 8 kg = rupees, so ₹700 per kg.
Profit per kg = = ₹140.
Profit % = .
Worth noting
16⅔% divides the profit by the selling price; profit % is always on cost price.
A shop has 30 litres of mustard oil at ₹150 per litre. How many litres of groundnut oil at ₹190 per litre must be mixed with it so that the blend costs ₹166 per litre?
AnswerB · 20 litres
Mustard : groundnut = .
3 parts = 30 litres, so 1 part = 10 litres.
Groundnut oil = litres.
Worth noting
45 litres comes from reversing the ratio to .
A coaching batch of 45 students averaged 64 marks in a mock test. The boys averaged 60 and the girls averaged 69. How many girls are in the batch?
AnswerA · 20
Boys : girls = .
Total parts = 9, so 1 part = students.
Girls = .
Worth noting
25 is the number of boys. Check: .
How many litres of water must be added to 24 litres of a 35% acid solution to make it a 21% acid solution?
AnswerB · 16 litres
Water is a 0% acid solution.
Solution : water = .
3 parts = 24 litres, so water = litres.
Worth noting
40 litres is the final volume, not the water added. Check: acid = litres, and .
Ramesh buys milk at ₹54 per litre, mixes water with it in the ratio milk : water = , and sells the mixture at 10% above ₹54 per litre. What is his profit percentage?
AnswerD · 37.5%
Cost of 5 litres of mixture = cost of 4 litres of milk = rupees, so ₹43.20 per litre.
Selling price = rupees per litre.
Profit % = .
Worth noting
The gains multiply, they do not add: 1.25 (from the free water) × 1.1 (from the higher price) = 1.375. Adding gives the tempting 35%, and 30% adds 10% to the 20% share of water in the mixture.
A goldsmith has 22-carat gold and 18-carat gold. She wants to make 60 g of 21-carat gold. How many grams of 18-carat gold should she use?
AnswerA · 15 g
22-carat : 18-carat = .
18-carat gold = g.
Worth noting
45 g is the 22-carat part. The target 21 is much closer to 22, so most of the mix must be 22-carat.
Deepa mixes 15 litres of a 40% sugar syrup with 10 litres of a second syrup. The mixture is 34% sugar. What is the sugar percentage of the second syrup?
AnswerB · 25%
Sugar in the mixture = litres.
Sugar from the first syrup = litres.
Second syrup: , i.e. 25%.
Worth noting
28% assumes 34% is the simple average of the two strengths; the volumes are unequal. 10% divides the 2.5 litres of sugar by 25 litres instead of 10.
A kirana store mixes three grades of rice costing ₹30, ₹36 and ₹45 per kg in the ratio . What is the cost of the mixture per kg?
AnswerA · ₹35.50
Cost of 6 kg = rupees.
Cost per kg = rupees.
Worth noting
₹37 is the plain average of the three prices. ₹38 attaches the weights 2 and 1 to the wrong prices.
Shalini bought a phone and a tablet for ₹8,000 in all. She sold the phone at a 10% loss and the tablet at a 15% profit, and made a 5% profit overall. What did the phone cost her?
AnswerB · ₹3,200
Write the loss as −10% and apply alligation to −10% and +15% around +5%.
Phone : tablet = : .
Phone = rupees.
Worth noting
Check: and +15% of 4,800 = +720, net +. ₹4,800 is the tablet's cost.
A milkman buys pure milk at ₹56 per litre. He adds water and sells the mixture at ₹60 per litre, making a profit of 25%. In what ratio (water : milk) is the mixture?
AnswerC ·
Cost price of the mixture = rupees per litre.
Water costs ₹0. Water : milk = .
= .
Worth noting
wrongly says 25% of the mixture's volume is water. comes from taking cost as 75% of the selling price (₹45) instead of dividing by 1.25. compares the ₹8 gap with ₹56 instead of ₹48.
Harish has 36 litres of a 30% acid solution. How many litres of pure acid must he add to raise the strength to 40%?
AnswerC · 6 litres
Pure acid is a 100% solution.
Solution : pure acid = .
Pure acid = litres.
Worth noting
Adding pure acid also increases the total volume. 3.6 litres simply takes 10% of 36; 4 litres makes the added acid 10% of the final volume, forgetting that the old acid already counts toward the 40%. Check: .
Venkat sells 80 kg of onions. He sells some of them at a 5% loss (they were going soft) and the rest at a 15% profit. Overall he gains 9%. How many kg did he sell at a loss?
AnswerA · 24 kg
Write the loss as −5%, and apply alligation to −5% and +15% around +9%.
Loss part : profit part = : .
Quantity sold at a loss = kg.
Worth noting
Using +5% instead of −5% gives and the wrong answer 48 kg. 56 kg is the quantity sold at a profit.
In a college quiz, the boys averaged 68 points and the girls averaged 78 points; the overall average was 72. There were 12 more boys than girls. How many students took the quiz?
AnswerD · 60
Boys : girls = .
The difference of part is 12 students.
Total = students (36 boys and 24 girls).
Worth noting
36 and 24 are the boys and the girls separately. Check: .
When two teas are mixed in the ratio , the blend costs ₹176 per kg. When they are mixed in the ratio , the blend costs ₹184 per kg. What is the price per kg of the cheaper tea?
AnswerA · ₹160
Let the prices be (cheaper) and . Then and .
Adding: , so .
Subtracting: .
So and ; the cheaper tea costs ₹160 per kg.
Worth noting
₹180 is the average of the two blend prices — it equals the average of the two teas, not the cheaper one. ₹176 is just the first blend's price.
A bus covers 330 km in 6 hours. It travels part of the way at 40 km/h and the rest at 60 km/h. How many km does it travel at 40 km/h?
AnswerA · 60 km
Average speed = km/h.
Time at 40 : time at .
Time at 40 km/h = hours.
Distance = km.
Worth noting
With speeds, alligation gives the ratio of TIMES, because average speed is total distance ÷ total time. Applying to the distance gives the wrong 82.5 km; 90 km uses 60 km/h for the 1.5 hours.
A tank holds a 40% sugar solution. Some of it is drained off and replaced with the same amount of a 16% sugar solution. The tank now holds a 25% solution. What fraction of the solution was replaced?
AnswerD ·
After replacement the tank is a mix of old 40% solution and new 16% solution, averaging 25%.
Old : new = .
Replaced fraction = .
Worth noting
is the fraction that stayed, and is the ratio old : new read as a fraction. Check: .
A can holds 50 litres of pure milk. Kiran takes out 10 litres and pours in 10 litres of water. What is the ratio of milk to water in the can now?
AnswerC ·
Milk left = litres.
Water = 10 litres (the amount poured in).
Milk : water = .
Worth noting
compares the original 50 litres with the 10 litres of water, forgetting that 10 litres of milk went out.
A 60-litre mixture of milk and water has milk and water in the ratio . How much water must be added so that the ratio becomes ?
AnswerA · 12 litres
Milk = litres; water = 18 litres.
Milk does not change, so in the new ratio 7 parts = 42 litres and 1 part = 6 litres.
New water = litres, so add litres.
Worth noting
30 litres is the new total of water, not the amount to be added. 18 litres is the water already there.
Neha has 30 litres of a 20% salt solution. How much water must she add to bring the salt concentration down to 12%?
AnswerD · 20 litres
Salt = litres, and adding water does not change it.
For 6 litres to be 12%, the total must be litres.
Water to add = litres.
Worth noting
Keep the salt fixed and find the new total. 8 litres just subtracts the percentages , which has no meaning here.
A jug holds 45 litres of a milk–water mixture in the ratio . Asha adds 5 litres of water. What percentage of the new mixture is water?
AnswerB · 30%
Water at first = litres.
After adding: water = 15 litres, total = 50 litres.
Water % = .
Worth noting
33⅓% divides 15 by the OLD total of 45 litres; the added water also enlarges the total.
A tank holds 64 litres of pure milk. 16 litres are taken out and replaced with water. This is done once more. How much milk is left in the tank?
AnswerC · 36 litres
Each operation leaves of the milk.
After two operations: litres.
Worth noting
32 litres wrongly subtracts 16 litres of milk twice; the second time the 16 litres taken out is only three-quarters milk.
A 40-litre mixture is 25% water and the rest milk. How much milk must be added so that water makes up only 20% of the mixture?
AnswerA · 10 litres
Water = litres, and it does not change.
For 10 litres to be 20%, the total must be litres.
Milk to add = litres.
Worth noting
50 litres is the new total, and 40 litres is the milk in the final mixture — neither is the amount added.
A 56-litre mixture contains milk and water in the ratio . How much milk must be added to make the ratio ?
AnswerA · 8 litres
Milk = litres; water = 16 litres.
Water stays 16, so milk must become litres.
Milk to add = litres.
Worth noting
48 litres is the milk needed in the end, not the milk to be added.
A tray holds 40 kg of brine that is 5% salt. How many kg of water must evaporate for the brine to become 8% salt?
AnswerA · 15 kg
Salt = kg, which stays the same.
For 2 kg to be 8%, the total must be kg.
Water evaporated = kg.
Worth noting
25 kg is what remains, not what evaporates. 64 kg multiplies 40 by — the right idea for adding water, backwards here.
A beaker has 20 litres of a 60% acid solution. 5 litres are removed and replaced with water. What is the acid concentration now?
AnswerB · 45%
Acid at first = litres.
Removing 5 of 20 litres removes of the acid, leaving litres.
The volume is back to 20 litres, so concentration = .
Worth noting
48% adds the water without removing anything . 55% just subtracts 5 from 60.
A vessel contains 40 litres of pure milk. Each time, 4 litres are drawn out and replaced with water. How much milk remains after two such operations?
AnswerC · 32.4 litres
Each operation keeps of the milk.
After two operations: litres.
Worth noting
32 litres subtracts 4 litres of milk twice; 29.16 litres is the result after THREE operations.
A drum holds 80 litres of pure coconut oil. 8 litres are removed and replaced with a cheaper oil, and this is done three times in all. How much coconut oil is left in the drum?
AnswerB · 58.32 litres
Each operation keeps of the coconut oil.
After three operations: litres.
Worth noting
56 litres subtracts 8 litres three times, ignoring that later removals carry some cheaper oil. 64.8 litres stops after two operations.
A container is filled with 60 litres of pure milk. 12 litres are drawn out and replaced with water, and then this is repeated once. What is the ratio of milk to water now?
AnswerD ·
Each operation keeps of the milk.
Milk = litres; water = litres.
Milk : water = .
Worth noting
Shortcut: milk fraction = ² = , so milk : water = . comes from subtracting 12 litres of milk twice; is the ratio after one operation.
Jar P has milk and water in the ratio , and jar Q has them in the ratio . In what ratio should the contents of P and Q be mixed to get milk and water in the ratio ?
AnswerC ·
Milk fractions: P = , Q = , target = .
.
= .
Worth noting
Write every mixture as the fraction of milk in the WHOLE before applying alligation. is the reversed ratio; the target is much nearer P's .
A vessel is full of pure milk. 9 litres are drawn out and replaced with water, and the operation is repeated once. The milk and water are now in the ratio . What is the capacity of the vessel?
AnswerD · 45 litres
Milk fraction now = .
So , giving .
, so litres.
Worth noting
Take the square root of , not of — the formula uses milk as a fraction of the WHOLE. Using leads to 21 litres.
A 90-litre can holds milk and water in the ratio . Priya draws out 18 litres of the mixture and pours in 18 litres of water. What is the new ratio of milk to water?
AnswerB ·
At first: milk = 70 litres, water = 20 litres.
The 18 litres drawn out are in the ratio : milk 14 litres, water 4 litres.
Left: milk 56, water 16. After adding 18 litres of water: milk 56, water 34.
Milk : water = .
Worth noting
treats the 18 litres drawn out as pure milk; forgets that 4 litres of water also left the can.
A chemist pours 12 litres of a solution with acid and water in the ratio into 8 litres of a solution with acid and water in the ratio . What percentage of the combined solution is acid?
AnswerB · 56%
Acid from the first = litres.
Acid from the second = litres.
Acid % = .
Worth noting
60% is the average of 40% and 80%, correct only for equal volumes. 64% swaps the volumes; 44% is the water percentage.
Mohan has 12 litres of hand sanitiser that is 75% alcohol. How much water should he add to bring it down to 60% alcohol?
AnswerB · 3 litres
Alcohol = litres, which stays fixed.
For 9 litres to be 60%, the total must be litres.
Water to add = litres.
Worth noting
1.8 litres takes 15% of 12 litres, which is not how dilution works. 15 litres is the final volume.
A tank holds 60 litres of a 40% acid solution. 15 litres are removed and replaced with 15 litres of an 80% acid solution. What is the strength of the solution now?
AnswerC · 50%
Acid at first = litres.
Removed: litres of acid; added: litres.
Acid now = litres in 60 litres, i.e. 50%.
Worth noting
48% adds the 80% solution without removing the 15 litres first. 60% is the plain average of 40% and 80%, and 30% treats the added liquid as water.
A can holds 40 litres of pure juice. Some juice is drawn out and replaced with water, and the operation is repeated with the same amount. The can now has 25.6 litres of juice. How many litres were drawn out each time?
AnswerA · 8 litres
, so .
, so .
litres.
Worth noting
14.4 litres is the total juice lost , as if there were only one operation. 12.8 litres halves 25.6 instead of taking a square root.
A 50-litre mixture has milk and water in the ratio . By mistake, 10 litres of water are added. How much milk must now be added to bring the ratio back to ?
AnswerA · 15 litres
At first: milk 30 litres, water 20 litres. After the mistake: water 30 litres.
For , milk must be litres.
Milk to add = litres.
Worth noting
45 litres is the total milk needed, not the amount to add. Adding 10 litres of milk does not undo 10 litres of water, because the ratio is not .
Sunita pours 20 litres of a milk–water mixture in the ratio and 30 litres of a milk–water mixture in the ratio into one large pot. What is the ratio of milk to water in the pot?
AnswerA ·
First mixture: milk = , water = 5 litres.
Second mixture: milk = , water = 12 litres.
Total: milk 33, water 17, so milk : water = .
Worth noting
adds the ratio terms , which ignores the volumes. swaps the two volumes.
A 60-litre mixture has milk and water in the ratio . How many litres of the mixture must be drawn off and replaced with water so that milk and water are in the ratio ?
AnswerD · 22.5 litres
Milk now = 48 litres. For in 60 litres, milk must fall to 30 litres.
Drawing litres removes the fraction of the milk: .
, so litres.
Worth noting
18 litres assumes the liquid drawn off is pure milk; it is four-fifths milk, so more must be drawn. Check: 22.5 litres carries 18 litres of milk away.
A drum holds 125 litres of pure milk. 25 litres are drawn out and replaced with water; this is done three times in all. What is the final ratio of milk to water?
AnswerD ·
Each operation keeps of the milk.
Milk after three operations = litres.
Water = litres, so milk : water = .
Worth noting
comes from subtracting 25 litres of milk three times (leaving 50 litres). is the ratio after only two operations.
A 32-litre can is full of pure juice. Each time, 8 litres are drawn out and replaced with water. What is the fewest number of such operations after which juice makes up less than half of the can?
AnswerB · 3
Each operation keeps of the juice.
After 1 operation: 24 litres; after 2: 18 litres — still more than half of 32 (16 litres).
After 3 operations: litres, which is less than 16.
Worth noting
Two operations would be the answer only if 8 litres of JUICE left each time . After the first operation the 8 litres drawn out are partly water.
Mixture A has milk and water in the ratio and mixture B has them in the ratio . How many litres of A must be mixed with 18 litres of B to get a mixture with milk and water in the ratio ?
AnswerD · 36 litres
Milk fractions: A = , B = , target = .
.
A = litres.
Worth noting
9 litres comes from reversing the ratio. Check: milk = , water = , and .
A 40-litre can holds milk and water in the ratio . Anil draws out 8 litres and replaces them with water, then does the same once more. What percentage of the can is now water?
AnswerC · 52%
Milk at first = litres.
Each operation keeps of the milk.
Milk left = litres, so water = litres.
Water % = .
Worth noting
The formula applies to the milk even though the can did not start as pure milk. 48% is the milk percentage; 40% is the water after only one operation; 55% subtracts 6 litres of milk each time.
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