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[Concept: Concentration] In a mixture, the ratio of alcohol to water is 5:3. What is the concentration of alcohol in the mixture?
- 62.5%
- 60%
- 37.5%
- 50%
Explanation: Book definition: Concentration is the percentage of a particular quantity in the full mixture. Alcohol = 5 units, Water = 3 units. Total = 8 units. Concentration of alcohol = (5 / 8) × 100% = 62.5%.
[Concept: Concentration] A mixture contains milk and water in the ratio 7:13. What is the concentration of water in the mixture?
- 65%
- 35%
- 60%
- 70%
Explanation: Total mixture = 7 + 13 = 20 units. Quantity of water = 13 units. Concentration of water = (13 / 20) × 100% = 13 × 5% = 65%.
[Concept: Replacement – Same Quantity] A vessel contains 100L of pure milk. 10L of milk is withdrawn and replaced with water. This process is repeated 2 more times. How much pure milk is left?
- 72.9L
- 70L
- 81L
- 72L
Explanation: Book Formula: Left Quantity = x(1 – p/x)^n. Here, x = 100L, p = 10L, n = 3 (1 initial + 2 more times). Milk left = 100 × (1 – 10/100)^3 = 100 × (9/10)^3 = 100 × (729/1000) = 72.9L.
[Concept: Replacement – Same Quantity] From 50L of pure alcohol, 10L is withdrawn and replaced with water. The process is repeated one more time. What is the amount of alcohol left in the final mixture?
- 32L
- 36L
- 40L
- 25L
Explanation: Using the formula: x(1 – p/x)^n. Here x = 50L, p = 10L, n = 2. Left alcohol = 50 × (1 – 10/50)^2 = 50 × (4/5)^2 = 50 × (16/25) = 2 × 16 = 32L.
[Concept: Replacement – Different Quantity] A vessel has 200L of pure milk. First, 20L is replaced by water. Then 40L of the mixture is replaced by water. Finally, 50L is replaced by water. How much pure milk is left?
- 108L
- 120L
- 112L
- 100L
Explanation: Book Formula for different quantities: x[(x-p)/x][(x-q)/x][(x-r)/x]. Milk left = 200 × (180/200) × (160/200) × (150/200) = 200 × (9/10) × (4/5) × (3/4). Solving this: 200 × (108 / 200) = 108L.
[Concept: Replacement – Different Quantity] From 300L of pure juice, 30L is removed and replaced with water. Then 60L of the mixture is removed and replaced with water. Find the final quantity of juice.
- 216L
- 240L
- 200L
- 210L
Explanation: Formula: x[(x-p)/x][(x-q)/x]. Final juice = 300 × (270/300) × (240/300) = 300 × (9/10) × (4/5) = 300 × (36/50) = 6 × 36 = 216L.
[Concept: Basic Alligation – Cost Price] In what ratio must wheat at Rs 42/kg be mixed with wheat at Rs 54/kg so that the mixture is worth Rs 45/kg?
- 3:1
- 1:3
- 2:1
- 4:3
Explanation: Using the Alligation cross method: Cheap (C) = 42, Dear (D) = 54, Mixture (M) = 45. Ratio = (D – M) : (M – C) = (54 – 45) : (45 – 42) = 9 : 3 = 3:1.
[Concept: Basic Alligation – Cost Price] The cost price of Type A rice is Rs 60/kg and Type B is Rs 85/kg. To get a mixture worth Rs 75/kg, in what ratio should they be mixed?
- 2:3
- 3:2
- 1:2
- 3:4
Explanation: Using Alligation: Type A = 60, Type B = 85, Mixture = 75. Ratio (Quantity A : Quantity B) = (85 – 75) : (75 – 60) = 10 : 15 = 2:3.
[Concept: Alligation in Population] The population of a village is 8000. Males increase by 6% and females by 10%. If the new population becomes 8600, what was the initial number of males?
- 5000
- 3000
- 4000
- 6000
Explanation: Overall increase = 600. Overall % increase = (600/8000) × 100 = 7.5%. By Alligation: Males = 6%, Females = 10%, Avg = 7.5%. Ratio = (10 – 7.5) : (7.5 – 6) = 2.5 : 1.5 = 5:3. Initial Males = (5/8) × 8000 = 5000.
[Concept: Alligation in Population] In a town of 5000 people, males increase by 10% and females by 15%. The population becomes 5600. Find the initial number of females.
- 2000
- 3000
- 2500
- 1500
Explanation: Overall increase = 600. Overall % increase = (600/5000) × 100 = 12%. By Alligation: Males = 10%, Females = 15%, Avg = 12%. Ratio (Males : Females) = (15 – 12) : (12 – 10) = 3:2. Females = (2/5) × 5000 = 2000.
[Concept: Alligation in Income & Expenditure] A man spends 80% of his income. If his income increases by 25% and his expenditure increases by 15%, what is the percentage change in his savings?
- 65%
- 50%
- 60%
- 75%
Explanation: Expenditure = 80%, Savings = 20%. Ratio Exp:Sav = 80:20 = 4:1. Alligation: Exp Inc = 15%, Sav Inc = x%, Overall Inc = 25%. Equation: (x – 25) / (25 – 15) = 4 / 1. => (x – 25) / 10 = 4 => x – 25 = 40 => x = 65%.
[Concept: Alligation in Income & Expenditure] A person spends 60% of his income. His income increases by 20% and expenditure increases by 10%. Find the percentage increase in his savings.
- 35%
- 30%
- 40%
- 25%
Explanation: Exp:Sav ratio = 60:40 = 3:2. Alligation: Exp Inc = 10%, Sav Inc = x%, Overall Inc = 20%. Equation: (x – 20) / (20 – 10) = 3 / 2. => (x – 20) / 10 = 3/2 => x – 20 = 15 => x = 35%.
[Concept: Alligation in Profit & Loss] A shopkeeper bought two items for Rs 1500. He sold the first at a 15% profit and the second at a 10% loss. If he makes no profit no loss overall, what is the CP of the first item?
- Rs 600
- Rs 900
- Rs 750
- Rs 800
Explanation: Profit/Loss Alligation: Item 1 = +15%, Item 2 = -10%, Overall = 0%. Ratio of CP = (0 – (-10)) : (15 – 0) = 10 : 15 = 2:3. CP of first item = (2/5) × 1500 = Rs 600.
[Concept: Alligation in Profit & Loss] A dealer sells one watch at 20% profit and another at 10% profit. The overall profit is 14%. If the cost price of the first watch is Rs 400, find the cost price of the second watch.
- Rs 600
- Rs 800
- Rs 500
- Rs 700
Explanation: Alligation: Watch 1 = 20%, Watch 2 = 10%, Avg = 14%. Ratio of CP = (14 – 10) : (20 – 14) = 4 : 6 = 2:3. If 2 units = Rs 400, then 1 unit = 200, so 3 units (CP of second watch) = Rs 600.
[Concept: Alligation in Simple Interest] A sum of Rs 12000 is lent into two parts, one at 8% and the other at 12% SI. If the overall annual interest rate is 11%, find the amount lent at 8%.
- Rs 3000
- Rs 9000
- Rs 4000
- Rs 6000
Explanation: Alligation on Interest Rates: Part 1 = 8%, Part 2 = 12%, Overall = 11%. Ratio of Principals = (12 – 11) : (11 – 8) = 1:3. Amount lent at 8% (Part 1) = (1/4) × 12000 = Rs 3000.
[Concept: Alligation in Simple Interest] Rs 20000 is lent out, a part at 6% p.a. and the rest at 10% p.a. SI. If the total interest after 1 year is Rs 1600, what sum was lent at 10%?
- Rs 10000
- Rs 12000
- Rs 8000
- Rs 15000
Explanation: Overall rate = (1600 / 20000) × 100 = 8%. Alligation: Part 1 = 6%, Part 2 = 10%, Overall = 8%. Ratio = (10 – 8) : (8 – 6) = 2:2 = 1:1. The amount is equally divided. Sum at 10% = Rs 10000.
[Concept: Alligation in Speed/Time] A man travels 100 km in 10 hours, partly on foot at 5 km/hr and partly on bicycle at 15 km/hr. Find the distance traveled on foot.
- 25 km
- 50 km
- 75 km
- 40 km
Explanation: Avg speed = 100 / 10 = 10 km/hr. Alligation on speeds gives the ratio of TIME: S1 = 5, S2 = 15, Avg = 10. Time Ratio = (15 – 10) : (10 – 5) = 5:5 = 1:1. Total time 10h is divided equally: 5h on foot, 5h on bicycle. Distance on foot = Speed × Time = 5 × 5 = 25 km.
[Concept: Alligation in Averages] A class has an overall average score of 40. The boys’ average is 35 and the girls’ average is 50. If there are 30 boys, find the number of girls.
- 15
- 20
- 10
- 25
Explanation: Alligation: Boys = 35, Girls = 50, Avg = 40. Ratio (Boys : Girls) = (50 – 40) : (40 – 35) = 10 : 5 = 2:1. If 2 units = 30 boys, then 1 unit = 15 girls.
[Concept: Replacement Ratio Formula] A container contains pure milk. 10% of the milk is taken out and replaced with water. This process is done 2 times in total. Find the ratio of milk to water in the final mixture.
- 81:19
- 90:10
- 80:20
- 72:28
Explanation: Formula: Final Milk / Initial Volume = (1 – 10/100)^2 = (9/10)^2 = 81/100. This means if total volume is 100, milk is 81. So, water = 100 – 81 = 19. Ratio = 81:19.
[Concept: Replacement Ratio Formula] From a vessel filled with alcohol, 20% is removed and replaced with water. The process is repeated 3 times. What is the ratio of alcohol to the total mixture finally?
- 64:125
- 16:25
- 81:125
- 48:100
Explanation: Formula: a / (a+b) = ((x-p)/x)^n. Here, 20% is removed, which means (1 – 1/5) = 4/5 remains each time. After 3 times, Alcohol / Total = (4/5)^3 = 64 / 125. The ratio is 64:125.