Veda Skills Quantitative Aptitude Unit 12: Mixtures and Alligation
- 2 topics
- 128 MCQs
- 8 short answers
- 2 long answers
- 27 flashcards
- 2 games
- ~100 min read
- English
Checked before publishing: 23 automated checks, including a plagiarism check against the textbook's own words. How we check our content
Rule of Alligation
Use the alligation cross to find the ratio in which two ingredients at different prices or strengths must be mixed, find the mean price from a ratio, treat water as a free ingredient, handle profit twists, and apply the same rule to averages of marks, interest rates and speeds.
What you will be able to do
- I can find the mixing ratio from two prices and a mean price.
- I can find the mean price when the mixing ratio is given.
- I can mix solutions of different strengths and treat water as costing ₹0.
- I can apply alligation to averages: marks, interest rates and speeds.
Key points
Alligation finds the ratio in which two things must be mixed to reach a given mean.
Cheaper : dearer = (d − m) : (m − c), where c < m < d.
The mean always lies between the two values; it sits nearer the one used in the larger quantity.
Given the ratio a : b, the mean is (a × c + b × d) ÷ (a + b).
Water is an ingredient with price ₹0 (or 0% milk).
With a profit, first turn the selling price into the cost price of the mixture, then use the cross.
The ratio is in the unit of the "per": price per kg gives kg, rate per rupee gives rupees, speed per hour gives hours.
Strengths in % mix the same way as prices: (q − r) : (r − p).
Full explanation
Gupta ji's tea blend
Gupta ji buys tea at ₹200 per kg and at ₹300 per kg. Customers like a blend that costs him ₹240 per kg. In what ratio should he mix the two teas?
Guessing 1 : 1 gives ₹250, too costly. He needs more of the cheap tea, but how much more? One small cross on paper answers it.
Mixing is a weighted average
If $a$ kg at price $c$ is mixed with $b$ kg at price $d$, the mean price is
$$m = \frac{a c + b d}{a + b}$$
The mean always lies between $c$ and $d$, and it is pulled towards the price used in the larger quantity.
The rule of alligation
For a cheaper price $c$, a dearer price $d$ and a mean $m$:
$$\text{cheaper} : \text{dearer} = (d - m) : (m - c)$$
Draw it as a cross. Each difference lands under the other price.
Worked example: from ratio back to price
Rice at ₹45 and ₹60 per kg is mixed in the ratio 2 : 3. Mean $= \frac{2 \times 45 + 3 \times 60}{5} = \frac{270}{5} = ₹54$.
Check with the cross: $(60 - 54) : (54 - 45) = 6 : 9 = 2 : 3$. The two methods agree.
Strengths mix the same way
Milk that is 30% pure and milk that is 60% pure are mixed to get 40% milk. Treat the percentages like prices:
$(60 - 40) : (40 - 30) = 20 : 10 = 2 : 1$.
Water is free
A milkman buys milk at ₹50 per litre and adds water so the mixture is worth ₹40 per litre. Water costs ₹0, so it goes on the cheaper side.
The profit twist
A grocer mixes sugar at ₹40 and ₹55 per kg and sells the mixture at ₹60, gaining 20%. The cost of the mixture is $60 \div 1.2 = ₹50$.
Now the cross: $(55 - 50) : (50 - 40) = 5 : 10 = 1 : 2$. Using ₹60 as the mean would be wrong: it is not between 40 and 55 at all.
Averages of groups
The same cross works for any two groups with their own averages. Boys average 60 marks, girls 75, the whole class 66:
boys : girls $= (75 - 66) : (66 - 60) = 9 : 6 = 3 : 2$. In a class of 40, there are 24 boys.
Speeds: a ratio of times
Ravi covers 61 km in 9 hours, partly walking at 4 km/h and partly cycling at 9 km/h. The average speed is $\frac{61}{9}$ km/h.
Time walking : time cycling $= (9 - \frac{61}{9}) : (\frac{61}{9} - 4) = 20 : 25 = 4 : 5$. So he walks for 4 hours, that is 16 km.
Back to Gupta ji
$(300 - 240) : (240 - 200) = 60 : 40 = 3 : 2$. For every 3 kg of the ₹200 tea he adds 2 kg of the ₹300 tea. Check: $(3 \times 200 + 2 \times 300) \div 5 = 1200 \div 5 = ₹240$.
Concepts and terms
When 3 kg at ₹200 is mixed with 2 kg at ₹300, the mean is (600 + 600) ÷ 5 = ₹240, not the plain average ₹250.
Write the two prices on top, the mean in the middle, and the differences crosswise below. The bottom numbers give the ratio.
Water costs nothing and has 0% milk, so it goes on the cheaper side with value 0.
Any two groups with their own averages mixing into one overall average follow the same cross: boys and girls, two loans, two speeds.
Did you know
Tea companies blend leaves from several gardens to keep the taste and the cost per kg the same all year; that blending is alligation.
Gold purity mixes the same way: 18-carat and 22-carat gold in equal parts give 20-carat gold.
Memory tricks
The cross — Top, middle, cross, read — Prices on top, mean in the middle, subtract crosswise, read the ratio at the bottom.
Which share is bigger? — Near means more — The price closer to the mean is used in the larger quantity.
Practice MCQs with answers (68)
- A 3 : 2
- B 2 : 5
- C 3 : 5
- D 2 : 3
Show answer
Why:
Cheaper : dearer = (300 − 240) : (240 − 200) = 60 : 40 = 3 : 2.
- A 1 : 3
- B 3 : 1
- C 1 : 4
- D 3 : 4
Show answer
Why:
Cheaper : dearer = (50 − 35) : (35 − 30) = 15 : 5 = 3 : 1.
- A 2 : 5
- B 3 : 2
- C 2 : 3
- D 3 : 5
Show answer
Why:
Cheaper : dearer = (60 − 52) : (52 − 40) = 8 : 12 = 2 : 3.
- A 2 : 3
- B 1 : 2
- C 1 : 3
- D 2 : 1
Show answer
Why:
Cheaper : dearer = (90 − 70) : (70 − 60) = 20 : 10 = 2 : 1.
- A 2 : 3
- B 3 : 2
- C 3 : 5
- D 2 : 5
Show answer
Why:
Cheaper : dearer = (40 − 34) : (34 − 25) = 6 : 9 = 2 : 3.
- A 1 : 3
- B 1 : 2
- C 2 : 1
- D 2 : 3
Show answer
Why:
Cheaper : dearer = (180 − 160) : (160 − 120) = 20 : 40 = 1 : 2.
- A 1 : 3
- B 2 : 3
- C 2 : 1
- D 1 : 2
Show answer
Why:
Cheaper : dearer = (24 − 20) : (20 − 18) = 4 : 2 = 2 : 1.
- A 3 : 1
- B 3 : 4
- C 1 : 4
- D 1 : 3
Show answer
Why:
Cheaper : dearer = (96 − 90) : (90 − 72) = 6 : 18 = 1 : 3.
- A 7 : 5
- B 7 : 12
- C 5 : 7
- D 5 : 12
Show answer
Why:
Cheaper : dearer = (47 − 40) : (40 − 35) = 7 : 5.
- A ₹55
- B ₹54
- C ₹52.5
- D ₹51
Show answer
Why:
Mean = (2 × 45 + 3 × 60) ÷ 5 = 270 ÷ 5 = ₹54.
- A ₹29
- B ₹24
- C ₹23
- D ₹26
Show answer
Why:
Mean = (3 × 20 + 1 × 32) ÷ 4 = 92 ÷ 4 = ₹23.
- A ₹84
- B ₹97
- C ₹90
- D ₹96
Show answer
Why:
Mean = (1 × 80 + 4 × 100) ÷ 5 = 480 ÷ 5 = ₹96.
- A ₹43
- B ₹44
- C ₹42
- D ₹41
Show answer
Why:
Mean = (5 × 36 + 7 × 48) ÷ 12 = 516 ÷ 12 = ₹43.
- A ₹186
- B ₹174
- C ₹175
- D ₹180
Show answer
Why:
Mean = (3 × 150 + 2 × 210) ÷ 5 = 870 ÷ 5 = ₹174.
- A 30 kg
- B 50 kg
- C 20 kg
- D 45 kg
Show answer
Why:
Cheaper : dearer = 9 : 6, so dearer = 30 × 6 ÷ 9 = 20 kg.
- A 24 kg
- B 60 kg
- C 16 kg
- D 36 kg
Show answer
Why:
Cheaper : dearer = 8 : 12, so dearer = 24 × 12 ÷ 8 = 36 kg.
- A 20 kg
- B 40 kg
- C 60 kg
- D 80 kg
Show answer
Why:
Cheaper : dearer = 20 : 10, so dearer = 40 × 10 ÷ 20 = 20 kg.
- A 60 kg
- B 45 kg
- C 5 kg
- D 15 kg
Show answer
Why:
Cheaper : dearer = 30 : 90, so dearer = 15 × 90 ÷ 30 = 45 kg.
- A 180 kg
- B 60 kg
- C 20 kg
- D 80 kg
Show answer
Why:
Cheaper : dearer = 15 : 5, so dearer = 60 × 5 ÷ 15 = 20 kg.
- A 1 : 3
- B 1 : 2
- C 2 : 3
- D 2 : 1
Show answer
Why:
Treat the percentages like prices: (60 − 40) : (40 − 30) = 20 : 10 = 2 : 1.
- A 2 : 1
- B 2 : 3
- C 2 : 5
- D 1 : 2
Show answer
Why:
Treat the percentages like prices: (50 − 30) : (30 − 20) = 20 : 10 = 2 : 1.
- A 3 : 2
- B 2 : 3
- C 2 : 5
- D 12 : 17
Show answer
Why:
Treat the percentages like prices: (85 − 75) : (75 − 60) = 10 : 15 = 2 : 3.
- A 3 : 5
- B 1 : 4
- C 3 : 2
- D 2 : 3
Show answer
Why:
Treat the percentages like prices: (40 − 22) : (22 − 10) = 18 : 12 = 3 : 2.
- A 4 : 5
- B 9 : 14
- C 1 : 4
- D 4 : 1
Show answer
Why:
Treat the percentages like prices: (70 − 50) : (50 − 45) = 20 : 5 = 4 : 1.
- A 10 litres
- B 90 litres
- C 30 litres
- D 15 litres
Show answer
Why:
Amount of 20% : amount of 60% = (60 − 30) : (30 − 20) = 30 : 10, so the 60% solution needed = 30 × 10 ÷ 30 = 10 litres.
- A 54 litres
- B 6 litres
- C 18 litres
- D 11 litres
Show answer
Why:
Amount of 40% : amount of 80% = (80 − 50) : (50 − 40) = 30 : 10, so the 80% solution needed = 18 × 10 ÷ 30 = 6 litres.
- A 4 : 1
- B 1 : 5
- C 1 : 4
- D 4 : 5
Show answer
Why:
Water is ₹0: water : milk = (50 − 40) : (40 − 0) = 10 : 40 = 1 : 4.
- A 1 : 4
- B 3 : 4
- C 3 : 1
- D 1 : 3
Show answer
Why:
Water is ₹0: water : milk = (60 − 45) : (45 − 0) = 15 : 45 = 1 : 3.
- A 1 : 8
- B 8 : 9
- C 8 : 1
- D 1 : 9
Show answer
Why:
Water is ₹0: water : milk = (36 − 32) : (32 − 0) = 4 : 32 = 1 : 8.
- A 5 : 1
- B 1 : 5
- C 1 : 6
- D 5 : 6
Show answer
Why:
Water is ₹0: water : milk = (48 − 40) : (40 − 0) = 8 : 40 = 1 : 5.
- A 12.5 litres
- B 12 litres
- C 10 litres
- D 40 litres
Show answer
Why:
Water : milk = 8 : 32, so water = 50 × 8 ÷ 40 = 10 litres.
- A 7.2 litres
- B 8 litres
- C 30 litres
- D 6 litres
Show answer
Why:
Water : milk = 10 : 50, so water = 36 × 10 ÷ 60 = 6 litres.
- A 1 : 1
- B 3 : 2
- C 1 : 2
- D 2 : 1
Show answer
Why:
Cost of the mixture = ₹27.5 × 100 ÷ 110 = ₹25. Then (30 − 25) : (25 − 20) = 1 : 1.
- A 1 : 2
- B 2 : 1
- C 3 : 1
- D 5 : 4
Show answer
Why:
Cost of the mixture = ₹78 × 100 ÷ 120 = ₹65. Then (75 − 65) : (65 − 60) = 2 : 1.
- A 11 : 8
- B 2 : 1
- C 1 : 2
- D 3 : 2
Show answer
Why:
Cost of the mixture = ₹60 × 100 ÷ 120 = ₹50. Then (55 − 50) : (50 − 40) = 1 : 2.
- A 15 : 14
- B 2 : 1
- C 3 : 2
- D 1 : 2
Show answer
Why:
Cost of the mixture = ₹165 × 100 ÷ 125 = ₹132. Then (135 − 132) : (132 − 126) = 1 : 2.
- A 1 : 4
- B 3 : 4
- C 4 : 1
- D 1 : 5
Show answer
Why:
Every 100 parts of milk are sold as 125 parts, so water : milk = 25 : 100 = 1 : 4.
- A 5 : 1
- B 1 : 5
- C 1 : 6
- D 4 : 5
Show answer
Why:
Every 100 parts of milk are sold as 120 parts, so water : milk = 20 : 100 = 1 : 5.
- A 10 : 1
- B 1 : 11
- C 1 : 10
- D 9 : 10
Show answer
Why:
Every 100 parts of milk are sold as 110 parts, so water : milk = 10 : 100 = 1 : 10.
- A 26
- B 16
- C 20
- D 24
Show answer
Why:
Ratio = (75 − 66) : (66 − 60) = 9 : 6 = 3 : 2, so boys = 40 × 3 ÷ 5 = 24.
- A 30
- B 20
- C 25
- D 32
Show answer
Why:
Ratio = (450 − 360) : (360 − 300) = 90 : 60 = 3 : 2, so unskilled workers = 50 × 3 ÷ 5 = 30.
- A 8
- B 10
- C 12
- D 6
Show answer
Why:
Ratio = (32 − 27) : (27 − 24) = 5 : 3, so bowlers = 16 × 5 ÷ 8 = 10.
- A 15
- B 14
- C 12
- D 18
Show answer
Why:
Ratio = (70 − 62) : (62 − 50) = 8 : 12 = 2 : 3, so boys = 30 × 2 ÷ 5 = 12.
- A 29
- B 25
- C 18
- D 27
Show answer
Why:
Ratio = (80 − 65) : (65 − 55) = 15 : 10 = 3 : 2, so unskilled workers = 45 × 3 ÷ 5 = 27.
- A ₹5,000
- B ₹6,000
- C ₹4,000
- D ₹7,000
Show answer
Why:
Average rate = 1000 × 100 ÷ 10000 = 10%. Ratio = (12 − 10) : (10 − 8) = 1 : 1, so ₹5,000 at 8%.
- A ₹8,200
- B ₹7,200
- C ₹4,800
- D ₹6,000
Show answer
Why:
Average rate = 840 × 100 ÷ 12000 = 7%. Ratio = (10 − 7) : (7 − 5) = 3 : 2, so ₹7,200 at 5%.
- A ₹12,000
- B ₹10,000
- C ₹8,000
- D ₹9,000
Show answer
Why:
Average rate = 1560 × 100 ÷ 20000 = 7.8%. Ratio = (9 − 7.8) : (7.8 − 6) = 2 : 3, so ₹8,000 at 6%.
- A 4 km
- B 45 km
- C 36 km
- D 16 km
Show answer
Why:
Average speed 61/9 km/h. Time on foot : by cycle = (9 − 61/9) : (61/9 − 4), so time on foot = 4 h and distance = 4 × 4 = 16 km.
- A 10 km
- B 30 km
- C 2 km
- D 60 km
Show answer
Why:
Average speed 35/3 km/h. Time on foot : by cycle = (15 − 35/3) : (35/3 − 5), so time on foot = 2 h and distance = 5 × 2 = 10 km.
- A 24 km
- B 8 km
- C 2 km
- D 36 km
Show answer
Why:
Average speed 8.8 km/h. Time on foot : by cycle = (12 − 8.8) : (8.8 − 4), so time on foot = 2 h and distance = 4 × 2 = 8 km.
- A 1 : 2
- B 7 : 3
- C 2 : 1
- D 9 : 4
Show answer
Why:
Milk = 8 × 3/4 + 4 × 1/2 = 8 L; water = 4 L; ratio 2 : 1.
- A 2 : 3
- B 13 : 8
- C 8 : 3
- D 3 : 2
Show answer
Why:
Milk = 10 × 2/5 + 10 × 4/5 = 12 L; water = 8 L; ratio 3 : 2.
- A 1 : 3
- B 1 : 1
- C 3 : 1
- D 4 : 3
Show answer
Why:
Milk fractions: 5/7, 3/7, target 1/2. First : second = (difference of 3/7 and 1/2) : (difference of 1/2 and 5/7) = 1/14 : 3/14 = 1 : 3.
- A 3 : 8
- B 3 : 5
- C 8 : 5
- D 5 : 3
Show answer
Why:
Milk fractions: 2/3, 3/5, target 5/8. First : second = (difference of 3/5 and 5/8) : (difference of 5/8 and 2/3) = 1/40 : 1/24 = 3 : 5.
- A ₹100
- B ₹38
- C ₹52
- D ₹64
Show answer
Why:
The mean must lie between ₹40 and ₹60, and ₹52 is the only option in that range.
- A The ₹50 item
- B Both equally
- C It cannot be told
- D The ₹30 item
Show answer
Why:
₹33 is much nearer ₹30, so the ₹30 item is the bigger share (17 : 3).
- A ₹0
- B the mean price
- C the price of milk
- D half the price of milk
Show answer
Why:
Water costs nothing, so its price is 0.
- A the pass marks
- B the number of students
- C the total marks
- D the highest marks
Show answer
Why:
An average is marks per student, so the weights are students.
- A the fuel used
- B the distances at each speed
- C the times spent at each speed
- D the two speeds
Show answer
Why:
Speed is distance per hour, so the weights are hours.
- A ₹72
- B ₹48
- C ₹60
- D ₹50
Show answer
Why:
Cost = 60 ÷ 1.2 = ₹50. The cross always uses cost prices.
- A Both A and R are true, and R is the correct explanation of A
- B Both A and R are true, but R is not the correct explanation of A
- C A is true but R is false
- D A is false but R is true
Show answer
Why:
By R, the price = (2 × 40 + 1 × 70) ÷ 3 = 150 ÷ 3 = ₹50, so A is true and R is exactly why.
- A Both A and R are true, and R is the correct explanation of A
- B Both A and R are true, but R is not the correct explanation of A
- C A is true but R is false
- D A is false but R is true
Show answer
Why:
With water at ₹0: water : milk = (50 − 40) : (40 − 0) = 1 : 4, so A is false. R is true.
- A 1 only
- B 2 only
- C Both 1 and 2
- D Neither 1 nor 2
Show answer
Why:
60 : 40 = 3 : 2, true. For ₹280: 20 : 80 = 1 : 4, so more of the ₹300 tea; 2 is false.
- A 1 only
- B 2 only
- C Both 1 and 2
- D Neither 1 nor 2
Show answer
Why:
Speeds are averages per hour, so 1 is true. A mean always lies between the two values, so 2 is false.
- A 2 : 1
- B 3 : 2
- C 1 : 2
- D 3 : 1
Show answer
Why:
(50 − 38) : (38 − 32) = 12 : 6 = 2 : 1.
- A ₹35
- B ₹33
- C ₹37
- D ₹30
Show answer
Why:
(3 × 25 + 2 × 45) ÷ 5 = 165 ÷ 5 = ₹33. Check: (45 − 33) : (33 − 25) = 12 : 8 = 3 : 2.
- A 40%
- B 50%
- C 45%
- D 35%
Show answer
Why:
Milk = 3 + 15 = 18 L in 40 L, which is 45%.
- A 2.4 litres
- B 28 litres
- C 6 litres
- D 36/7 litres
Show answer
Why:
Ratio 25% : 75% = (75 − 40) : (40 − 25) = 35 : 15 = 7 : 3, so 12 × 3 ÷ 7 = 36/7 litres (about 5.14 L).
Important short questions (4)
(54 − 45) : (45 − 42) = 9 : 3 = 3 : 1.
Water : milk = 5 : 40 = 1 : 8, so water = 40 ÷ 8 = 5 litres.
Cost = 99 ÷ 1.1 = ₹90; (100 − 90) : (90 − 80) = 1 : 1.
Speed is distance per hour, an average over hours, so the weights in the average are hours.
Long answer questions (1)
(a) (25 − 15) : (15 − 10) = 10 : 5 = 2 : 1.
(b) 40 kg at 10% and 20 kg at 25%.
(c) Cost ₹2,400. Profit = 1,600 × 0.10 + 800 × 0.25 = 160 + 200 = ₹360, which is 15% of ₹2,400. Correct.
Flashcards (14)
Play this topic
- Cross check
Replacement in Mixtures
Track what is left when part of a mixture is taken out and replaced again and again, work backwards to the vessel size or the amount drawn, and solve adding, boiling-off and ratio-change questions by holding on to the part that does not change.
What you will be able to do
- I can find the pure liquid left after n replacements using V(1 − x/V)ⁿ.
- I can work backwards to find the vessel size or the amount drawn out.
- I can find what fraction of a mixture to replace to reach a new strength.
- I can solve adding-water, adding-milk and evaporation questions by keeping one part fixed.
Key points
Taking out x litres from V litres removes the fraction x/V of every ingredient.
After one replacement, the pure liquid left is V(1 − x/V).
After n replacements, it is V(1 − x/V)ⁿ, not V − nx.
The fraction of pure liquid left is (1 − x/V)ⁿ; multiply by 100 for the percentage.
If a fraction f of a mixture is replaced by water, every other ingredient is multiplied by (1 − f).
Adding water keeps the milk fixed: V × p = (V + w) × q.
Adding milk keeps the water fixed; boiling off water keeps the salt fixed.
For a ratio change after adding one ingredient, write both ratios with the unchanged part equal.
Full explanation
Lakshmi's milk can
Lakshmi runs a dairy. Her helper takes 10 litres out of a 100-litre can of pure milk and tops it up with water. Later he does the same thing again, and then a third time.
The helper thinks 30 litres of milk are gone. Lakshmi knows it is less. Who is right, and how much pure milk is left?
One replacement
Taking out 10 L from 100 L removes $\frac{10}{100} = \frac{1}{10}$ of everything in the can. So $\frac{9}{10}$ of the milk stays: 90 L. Water is added to fill the can again.
The second time is different
Now the can holds 90 L milk and 10 L water. The next 10 L taken out is a mixture: it removes 9 L of milk and 1 L of water. So the milk falls to 81 L, not 80 L. Each time, the milk is multiplied by $\frac{9}{10}$.
The formula
A vessel of $V$ litres of pure liquid; $x$ litres are replaced each time, $n$ times:
$$\text{pure left} = V\left(1 - \frac{x}{V}\right)^n$$
The water (or other liquid) is $V$ minus this.
Worked example: ratio after replacement
A 40-litre can of milk; 4 litres are replaced with water twice. Milk left $= 40 \times \left(\frac{9}{10}\right)^2 = 32.4$ L, water $= 7.6$ L.
Milk : water $= 32.4 : 7.6 = 81 : 19$.
Going backwards
After two replacements of 9 litres, milk : water is 16 : 9. Find the vessel size.
$\left(1 - \frac{9}{V}\right)^2 = \frac{16}{25}$, so $1 - \frac{9}{V} = \frac{4}{5}$, $\frac{9}{V} = \frac{1}{5}$ and $V = 45$ litres.
Replacing part of a mixture
A vessel is full of a mixture that is $\frac{5}{8}$ syrup. What fraction must be drawn off and replaced with water to make it half syrup?
$\frac{5}{8}(1 - f) = \frac{1}{2}$, so $1 - f = \frac{4}{5}$ and $f = \frac{1}{5}$.
Adding, not replacing: hold the fixed part
30 L of milk is 80% pure. How much water makes it 60% pure? The milk, 24 L, does not change:
$24 = 0.6 \times (30 + w)$, so $30 + w = 40$ and $w = 10$ L.
If instead you boil off water, the salt or sugar stays fixed in the same way.
Ratio changes
Milk and water are 4 : 1. Adding 6 L of water makes it 4 : 3. The milk is fixed, so write $4k$ for it: water goes from $k$ to $3k$, so $2k = 6$ and $k = 3$.
Milk = 12 L and the original mixture was 15 L.
Back to Lakshmi
$100 \times \left(\frac{9}{10}\right)^3 = 72.9$ litres of pure milk are left. Only 27.1 L of milk went, not 30 L, because the later rounds also took out some water. Lakshmi was right.
Concepts and terms
Each round keeps the same fraction (1 − x/V) of the pure liquid, so n rounds multiply it n times.
Replacing a fraction f by water multiplies the old concentration by (1 − f): new = old × (1 − f).
When only one ingredient is added or removed, the other stays the same. Write the equation around it.
Did you know
Fish keepers change 20% of the aquarium water every week. After 3 changes only 0.8 × 0.8 × 0.8 = 51.2% of the original water is left: the same (1 − x/V)ⁿ rule.
Memory tricks
Replacement — Multiply, never subtract — Each round multiplies the milk by (1 − x/V); do not subtract x each time.
Adding or boiling — Hold the one that stays — Find the ingredient that does not change and equate its amounts before and after.
Practice MCQs with answers (60)
- A 81 litres
- B 90 litres
- C 72.9 litres
- D 80 litres
Show answer
Why:
100 × (1 − 10/100)² = 100 × (0.9)² = 81 litres.
- A 25.6 litres
- B 32 litres
- C 30 litres
- D 40 litres
Show answer
Why:
50 × (1 − 10/50)² = 50 × (0.8)² = 32 litres.
- A 16 litres
- B 36 litres
- C 27 litres
- D 20.25 litres
Show answer
Why:
64 × (1 − 16/64)³ = 64 × (0.75)³ = 27 litres.
- A 50 litres
- B 80 litres
- C 51.2 litres
- D 64 litres
Show answer
Why:
125 × (1 − 25/125)³ = 125 × (0.8)³ = 64 litres.
- A 32.4 litres
- B 29.16 litres
- C 32 litres
- D 36 litres
Show answer
Why:
40 × (1 − 4/40)² = 40 × (0.9)² = 32.4 litres.
- A 64.8 litres
- B 58.32 litres
- C 52.488 litres
- D 56 litres
Show answer
Why:
80 × (1 − 8/80)³ = 80 × (0.9)³ = 58.32 litres.
- A 48 litres
- B 54 litres
- C 48.6 litres
- D 43.74 litres
Show answer
Why:
60 × (1 − 6/60)² = 60 × (0.9)² = 48.6 litres.
- A 64 litres
- B 37 litres
- C 40 litres
- D 36 litres
Show answer
Why:
Pure juice left = 100 × (0.8)² = 64 L, so water = 100 − 64 = 36 litres.
- A 14.4 litres
- B 16 litres
- C 25.6 litres
- D 15.4 litres
Show answer
Why:
Pure honey left = 40 × (0.8)² = 25.6 L, so water = 40 − 25.6 = 14.4 litres.
- A 58 litres
- B 57 litres
- C 81 litres
- D 24 litres
Show answer
Why:
Milk left = 81 × (2/3)³ = 24 L, so water = 81 − 24 = 57 litres.
- A 18 litres
- B 72.9 litres
- C 17.1 litres
- D 18.1 litres
Show answer
Why:
Pure oil left = 90 × (0.9)² = 72.9 L, so water = 90 − 72.9 = 17.1 litres.
- A 9 : 1
- B 4 : 1
- C 19 : 81
- D 81 : 19
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Why:
Pure honey = 32.4 L and water = 7.6 L, so the ratio is 81 : 19.
- A 9 : 7
- B 7 : 9
- C 3 : 1
- D 1 : 1
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Why:
Milk = 11.25 L and water = 8.75 L, so the ratio is 9 : 7.
- A 7 : 9
- B 9 : 7
- C 3 : 1
- D 1 : 1
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Why:
Pure oil = 36 L and water = 28 L, so the ratio is 9 : 7.
- A 61 : 64
- B 16 : 9
- C 64 : 61
- D 2 : 3
Show answer
Why:
Milk = 51.2 L and water = 48.8 L, so the ratio is 64 : 61.
- A 54 litres
- B 90 litres
- C 36 litres
- D 45 litres
Show answer
Why:
Milk is now 16 parts out of 25, so (1 − 9/V)² = 16/25. Then 1 − 9/V = 4/5, 9/V = 1/5 and V = 45 litres.
- A 64 litres
- B 32 litres
- C 56 litres
- D 72 litres
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Why:
Milk is now 49 parts out of 64, so (1 − 8/V)² = 49/64. Then 1 − 8/V = 7/8, 8/V = 1/8 and V = 64 litres.
- A 15 litres
- B 20 litres
- C 25 litres
- D 30 litres
Show answer
Why:
Milk is now 27 parts out of 64, so (1 − 5/V)³ = 27/64. Then 1 − 5/V = 3/4, 5/V = 1/4 and V = 20 litres.
- A 36 litres
- B 44 litres
- C 40 litres
- D 16 litres
Show answer
Why:
Milk is now 81 parts out of 100, so (1 − 4/V)² = 81/100. Then 1 − 4/V = 9/10, 4/V = 1/10 and V = 40 litres.
- A 25 litres
- B 18 litres
- C 17.5 litres
- D 20 litres
Show answer
Why:
(1 − x/80)² = 45/80 = 9/16, so 1 − x/80 = 3/4 and x = 80 × 1/4 = 20 litres.
- A 10 litres
- B 9 litres
- C 15 litres
- D 8 litres
Show answer
Why:
(1 − x/50)² = 32/50 = 16/25, so 1 − x/50 = 4/5 and x = 50 × 1/5 = 10 litres.
- A 61/3 litres
- B 25 litres
- C 30 litres
- D 23 litres
Show answer
Why:
(1 − x/125)³ = 64/125, so 1 − x/125 = 4/5 and x = 125 × 1/5 = 25 litres.
- A 43.75%
- B 50%
- C 56.25%
- D 75%
Show answer
Why:
Each time 0.75 of the juice stays: (0.75)² × 100 = 56.25%.
- A 40%
- B 64%
- C 48.8%
- D 51.2%
Show answer
Why:
Each time 0.8 of the juice stays: (0.8)³ × 100 = 51.2%.
- A 81%
- B 90%
- C 19%
- D 80%
Show answer
Why:
Each time 0.9 of the juice stays: (0.9)² × 100 = 81%.
- A 17.5%
- B 12.5%
- C 25%
- D 87.5%
Show answer
Why:
Each time 0.5 of the juice stays: (0.5)³ × 100 = 12.5%.
- A 1/2
- B 1/8
- C 1/5
- D 3/8
Show answer
Why:
Syrup goes from 5/8 to 1/2: 5/8 × (1 − f) = 1/2, so 1 − f = 0.8 and f = 1/5.
- A 1/2
- B 1/6
- C 1/4
- D 1/3
Show answer
Why:
Syrup goes from 3/4 to 1/2: 3/4 × (1 − f) = 1/2, so 1 − f = 2/3 and f = 1/3.
- A 2/7
- B 1/5
- C 3/10
- D 1/2
Show answer
Why:
Syrup goes from 7/10 to 1/2: 7/10 × (1 − f) = 1/2, so 1 − f = 5/7 and f = 2/7.
- A 3/5
- B 1/4
- C 1/8
- D 1/5
Show answer
Why:
Syrup goes from 4/5 to 3/5: 4/5 × (1 − f) = 3/5, so 1 − f = 0.75 and f = 1/4.
- A 5 : 7
- B 8 : 7
- C 1 : 1
- D 2 : 1
Show answer
Why:
Milk = 40 × (1 − 15/60) = 30 L; water = 60 − 30 = 30 L; ratio 1 : 1.
- A 2 : 3
- B 13 : 12
- C 1 : 1
- D 12 : 13
Show answer
Why:
Milk = 30 × (1 − 10/50) = 24 L; water = 50 − 24 = 26 L; ratio 12 : 13.
- A 1 : 1
- B 17 : 23
- C 5 : 4
- D 25 : 23
Show answer
Why:
Milk = 50 × (1 − 16/80) = 40 L; water = 80 − 40 = 40 L; ratio 1 : 1.
- A 6 litres
- B 10 litres
- C 22.5 litres
- D 40 litres
Show answer
Why:
Milk stays 24 L. New total = 24 ÷ 0.6 = 40 L, so add 10 litres.
- A 48 litres
- B 6 litres
- C 8 litres
- D 100/3 litres
Show answer
Why:
Milk stays 36 L. New total = 36 ÷ 0.75 = 48 L, so add 8 litres.
- A 75 litres
- B 6 litres
- C 48 litres
- D 15 litres
Show answer
Why:
Milk stays 30 L. New total = 30 ÷ 0.4 = 75 L, so add 15 litres.
- A 5 litres
- B 125/6 litres
- C 30 litres
- D 3 litres
Show answer
Why:
Milk stays 18 L. New total = 18 ÷ 0.6 = 30 L, so add 5 litres.
- A 22 litres
- B 20 litres
- C 12 litres
- D 30 litres
Show answer
Why:
Water stays 48 L and must be 60% of the new total: 48 ÷ 0.6 = 80 L, so add 20 litres.
- A 6 litres
- B 15 litres
- C 10 litres
- D 12 litres
Show answer
Why:
Water stays 30 L and must be 60% of the new total: 30 ÷ 0.6 = 50 L, so add 10 litres.
- A 18 litres
- B 8 litres
- C 4.5 litres
- D 6 litres
Show answer
Why:
Water stays 27 L and must be 75% of the new total: 27 ÷ 0.75 = 36 L, so add 6 litres.
- A 15 litres
- B 20 litres
- C 25 litres
- D 1.2 litres
Show answer
Why:
Salt stays 2 L. New total = 2 ÷ 0.08 = 25 L, so 40 − 25 = 15 litres evaporate.
- A 40 litres
- B 20 litres
- C 1.2 litres
- D 25 litres
Show answer
Why:
Salt stays 2.4 L. New total = 2.4 ÷ 0.06 = 40 L, so 60 − 40 = 20 litres evaporate.
- A 1 litre
- B 25 litres
- C 20 litres
- D 30 litres
Show answer
Why:
Salt stays 1.5 L. New total = 1.5 ÷ 0.05 = 30 L, so 50 − 30 = 20 litres evaporate.
- A 15 litres
- B 8 litres
- C 18 litres
- D 12 litres
Show answer
Why:
Milk is fixed at 4k. Water goes from 1k to 3k, so 2k = 6, k = 3 and milk = 12 litres.
- A 21 litres
- B 24 litres
- C 15 litres
- D 30 litres
Show answer
Why:
Write 1 : 1 as 5 : 5 so that milk keeps its 5 parts. Milk keeps 5k and water goes from 2k to 5k: 3k = 9 gives k = 3, so the original mixture was (5 + 2) × 3 = 21 litres.
- A 15 litres
- B 20 litres
- C 30 litres
- D 25 litres
Show answer
Why:
Write 1 : 1 as 3 : 3 so that milk keeps its 3 parts. Milk keeps 3k and water goes from 1k to 3k: 2k = 10 gives k = 5, so the original mixture was (3 + 1) × 5 = 20 litres.
- A 45 litres
- B 17.5 litres
- C 35 litres
- D 10 litres
Show answer
Why:
Milk is fixed at 7k. Water goes from 2k to 4k, so 2k = 10, k = 5 and milk = 35 litres.
- A the vessel becomes smaller
- B water also contains milk
- C there is no mistake
- D later draws also remove some water
Show answer
Why:
After the first round the can holds a mixture, so 10 L drawn contains less than 10 L of milk. The answer is 72.9 L.
- A The quantity of milk
- B The quantity of water
- C The ratio
- D The total volume
Show answer
Why:
Only water is added, so the milk does not change.
- A The amount of salt falls
- B The salt percentage rises
- C The salt percentage falls
- D Nothing changes
Show answer
Why:
Salt stays and the total shrinks, so its share grows.
- A 1 − (x/V)ⁿ
- B 1 − nx/V
- C (1 − x/V)ⁿ
- D (x/V)ⁿ
Show answer
Why:
Each round keeps the fraction 1 − x/V of what was there.
- A 3
- B 5
- C 10
- D 4
Show answer
Why:
(1/2)³ = 12.5% and (1/2)⁴ = 6.25%, so after 4 operations.
- A Both A and R are true, and R is the correct explanation of A
- B Both A and R are true, but R is not the correct explanation of A
- C A is true but R is false
- D A is false but R is true
Show answer
Why:
100 × 9/10 × 9/10 = 81, and R is exactly the reason.
- A Both A and R are true, and R is the correct explanation of A
- B Both A and R are true, but R is not the correct explanation of A
- C A is true but R is false
- D A is false but R is true
Show answer
Why:
24 L milk in 40 L is 60%, so A is true. R is false: the milk stays fixed, not the water.
- A 1 only
- B 2 only
- C Both 1 and 2
- D Neither 1 nor 2
Show answer
Why:
40 × 0.81 = 32.4 and 32.4 : 7.6 = 81 : 19. Both true.
- A 1 only
- B 2 only
- C Both 1 and 2
- D Neither 1 nor 2
Show answer
Why:
Only acid is added, so water stays: 1 true. Boiling reduces the volume: 2 false.
- A 22.5 litres
- B 30 litres
- C 17.5 litres
- D 20 litres
Show answer
Why:
40 × (3/4)² = 40 × 9/16 = 22.5 litres. Subtracting 20 L would be the trap.
- A 15 litres
- B 10 litres
- C 4.5 litres
- D 40 litres
Show answer
Why:
Milk 18 L stays. 18 ÷ 0.45 = 40 L in all, so 10 L of water.
- A 2/3
- B 1/4
- C 1/3
- D 1/2
Show answer
Why:
3/4 × (1 − f) = 1/2, so 1 − f = 2/3 and f = 1/3.
- A 2 litres
- B 25 litres
- C 30 litres
- D 20 litres
Show answer
Why:
Salt 3 L stays. 3 ÷ 0.10 = 30 L remain, so 20 L boil away.
Important short questions (4)
50 × (9/10)³ = 50 × 0.729 = 36.45 litres.
(1 − 8/V)² = 49/64, so 1 − 8/V = 7/8 and V = 64 litres.
Salt 2.4 L; 2.4 ÷ 0.05 = 48 L must remain, so 12 L evaporate.
From the second round on, the 10 L drawn is a mixture, so it takes out less than 10 L of milk. The milk shrinks by 9/10 each time: 100 → 90 → 81 → 72.9.
Long answer questions (1)
(a) (1 − x/80)² = 45/80 = 9/16, so 1 − x/80 = 3/4 and x = 20 L.
(b) Milk 45 L, water 35 L, ratio 9 : 7.
(c) 45 × 3/4 = 33.75 litres.
Flashcards (13)
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- Replacement check
Veda Skills Quantitative Aptitude Unit 12 — frequently asked questions
(54 − 45) : (45 − 42) = 9 : 3 = 3 : 1.
Water : milk = 5 : 40 = 1 : 8, so water = 40 ÷ 8 = 5 litres.
Cost = 99 ÷ 1.1 = ₹90; (100 − 90) : (90 − 80) = 1 : 1.
50 × (9/10)³ = 50 × 0.729 = 36.45 litres.
(1 − 8/V)² = 49/64, so 1 − 8/V = 7/8 and V = 64 litres.
Salt 2.4 L; 2.4 ÷ 0.05 = 48 L must remain, so 12 L evaporate.
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