Question 1

A container has 60 litres of a mixture of alcohol and water in the ratio 7:5. How much pure alcohol must be added to make the ratio 3:1?

Accepted Answer

Step 1: Original ratio 7:5 means alcohol = 60 × (7/12) = 35 litres, water = 60 × (5/12) = 25 litres. Step 2: Let x litres of pure alcohol be added. New alcohol = 35 + x, water remains 25. Step 3: New ratio is 3:1, so (35 + x)/25 = 3/1. Step 4: 35 + x = 75 → x = 40 litres.

Question 2

In a partnership, A and B share profits in the ratio 8:5. If A's profit is ₹4,800, what is the total profit?

Accepted Answer

Step 1: Profit ratio A:B = 8:5, so total parts = 8 + 5 = 13. Step 2: A's profit = ₹4,800 corresponds to 8 parts. Step 3: 1 part = 4,800 ÷ 8 = ₹600. Step 4: Total profit = 13 × 600 = ₹7,800.

Question 3

Three partners M, N, and O invest capitals in the ratio 6:4:5 and for periods in the ratio 2:3:4 respectively. If the total profit is ₹3,450, what is N's share?

Accepted Answer

Step 1: M's effective capital = 6 × 2 = 12. Step 2: N's effective capital = 4 × 3 = 12. Step 3: O's effective capital = 5 × 4 = 20. Step 4: Ratio M:N:O = 12:12:20 = 3:3:5. Step 5: Total parts = 11. Step 6: N's share = (3/11) × 3,450 = ₹900.

Question 4

Two containers have sugar solutions of 20% and 50% concentration respectively. In what ratio should they be mixed to get a 30% solution?

Accepted Answer

Step 1: Use alligation method. Concentration 1 = 20%, Concentration 2 = 50%, Target = 30%. Step 2: Difference from target: |20 - 30| = 10 (for 20% solution), |50 - 30| = 20 (for 50% solution). Step 3: Ratio = 20:10 = 2:1 (inverse of differences). Step 4: Mix 20% and 50% solutions in ratio 2:1.

Question 5

A shopkeeper mixes two types of rice costing ₹40 per kg and ₹60 per kg in the ratio 3:2. What is the cost price of the mixture per kg?

Accepted Answer

Step 1: Ratio of rice = 3:2 (total parts = 5). Step 2: Cost of first type = ₹40/kg, cost of second type = ₹60/kg. Step 3: Total cost for 5 kg = (3 × 40) + (2 × 60) = 120 + 120 = 240. Step 4: Cost per kg = 240 ÷ 5 = ₹48/kg.

Question 6

A and B enter into a partnership with capital investments in the ratio 5:3. If the total capital is ₹80,000, how much did A invest?

Accepted Answer

Step 1: Capital ratio A:B = 5:3, so total parts = 5 + 3 = 8. Step 2: A's share = (5/8) × 80,000 = ₹50,000.

Question 7

X, Y, and Z start a business with investments of ₹12,000, ₹15,000, and ₹18,000 respectively. If the profit at the end of the year is ₹9,000, what is Y's share of the profit?

Accepted Answer

Step 1: Investment ratio X:Y:Z = 12,000:15,000:18,000 = 4:5:6 (dividing by 3,000). Step 2: Total parts = 4 + 5 + 6 = 15. Step 3: Y's share = (5/15) × 9,000 = ₹3,000.

Question 8

P and Q form a partnership where P invests ₹24,000 for 8 months and Q invests ₹32,000 for 6 months. The profit is ₹5,600. What is P's profit share?

Accepted Answer

Step 1: P's capital × time = 24,000 × 8 = 192,000. Step 2: Q's capital × time = 32,000 × 6 = 192,000. Step 3: Ratio P:Q = 192,000:192,000 = 1:1. Step 4: P's share = (1/2) × 5,600 = ₹2,800.

Question 9

A and B invest in a business in the ratio 7:5. After one year, A withdraws half of his capital. If the total profit at the end of two years is ₹2,400, what is A's profit share?

Accepted Answer

Step 1: Let A's capital = 7x, B's capital = 5x. Step 2: A's effective capital = 7x × 1 + (7x/2) × 1 = 7x + 3.5x = 10.5x. Step 3: B's effective capital = 5x × 2 = 10x. Step 4: Ratio A:B = 10.5x:10x = 21:20. Step 5: A's share = (21/41) × 2,400 = ₹1,200.

Question 10

A goldsmith mixes gold of 18 carats and 24 carats in the ratio 1:2. What is the carat value of the resulting mixture?

Accepted Answer

Step 1: Ratio of 18-carat gold to 24-carat gold = 1:2 (total parts = 3). Step 2: Weighted average carat value = (1 × 18 + 2 × 24)/3 = (18 + 48)/3 = 66/3 = 22 carats.

Question 11

A merchant mixes three types of tea costing ₹100, ₹120, and ₹150 per kg in the ratio 2:3:5 respectively. At what price per kg should he sell the mixture to make a 20% profit?

Accepted Answer

Step 1: Let quantities be 2x, 3x, and 5x kg respectively. Total = 10x kg. Step 2: Cost price = (2x × 100) + (3x × 120) + (5x × 150) = 200x + 360x + 750x = 1310x. Step 3: Cost per kg = 1310x / 10x = ₹131. Step 4: For 20% profit, selling price = 131 × 1.20 = ₹157.20 per kg.

Question 12

A container has 120 litres of a mixture of petrol and diesel in the ratio 7:5. How much pure petrol must be added so that the ratio becomes 3:2?

Accepted Answer

Step 1: Original mixture has 7 + 5 = 12 parts. Petrol = (7/12) × 120 = 70 litres, Diesel = (5/12) × 120 = 50 litres. Step 2: Let x litres of pure petrol be added. New petrol = 70 + x, diesel remains 50. Step 3: New ratio is 3:2, so (70 + x)/50 = 3/2. Step 4: 2(70 + x) = 3(50) → 140 + 2x = 150 → 2x = 10 → x = 5 litres.

Question 13

Two containers have alcohol solutions of 20% and 50% respectively. In what ratio should they be mixed to get a 35% alcohol solution?

Accepted Answer

Step 1: Use alligation method. Desired concentration = 35%. Step 2: Difference between 50% and 35% = 15. Difference between 35% and 20% = 15. Step 3: Ratio of 20% solution to 50% solution = 15:15 = 1:1.

Question 14

A shopkeeper mixes two types of rice costing ₹40 per kg and ₹60 per kg. If he sells the mixture at ₹55 per kg and makes no profit or loss, in what ratio were the two types mixed?

Accepted Answer

Step 1: Cost price of mixture = Selling price = ₹55 per kg (no profit/loss). Step 2: Use alligation. Difference between ₹60 and ₹55 = ₹5. Difference between ₹55 and ₹40 = ₹15. Step 3: Ratio of cheaper rice to costlier rice = 5:15 = 1:3.

Question 15

If A:B = 3:5 and B:C = 2:3, what is A:B:C?

Accepted Answer

Step 1: A:B = 3:5 (multiply by 2) → A:B = 6:10. Step 2: B:C = 2:3 (multiply by 5) → B:C = 10:15. Step 3: Combine using common B value: A:B:C = 6:10:15.

Question 16

A, B, and C enter into a partnership. A invests ₹24,000 for 8 months, B invests ₹32,000 for 10 months, and C invests ₹40,000 for 6 months. If the total profit at the end of the year is ₹13,200, what is B's share of the profit?

Accepted Answer

Step 1: Calculate capital × time for each partner. A: 24,000 × 8 = 192,000. B: 32,000 × 10 = 320,000. C: 40,000 × 6 = 240,000. Step 2: Find the ratio of contributions: 192,000 : 320,000 : 240,000. Divide by 16,000 to simplify: 12 : 20 : 15. Step 3: Total parts = 12 + 20 + 15 = 47. Step 4: B's share = (20/47) × 13,200 = ₹5,600.

Question 17

The ratio of boys to girls in a class is 4:5. If there are 12 more girls than boys, how many total students are in the class?

Accepted Answer

Step 1: Boys:Girls = 4:5, so Boys = 4x, Girls = 5x. Step 2: 5x - 4x = 12 → x = 12. Step 3: Total = 4x + 5x = 9x = 9(12) = 108.

Question 18

A recipe requires flour and sugar in the ratio 7:3. If 280 grams of flour is used, how much sugar (in grams) is needed?

Accepted Answer

Step 1: Flour:Sugar = 7:3. Step 2: 7 parts = 280 g → 1 part = 40 g. Step 3: Sugar = 3 × 40 = 120 g.

Question 19

Two numbers are in the ratio 5:8. If their difference is 36, what is the sum of the two numbers?

Accepted Answer

Step 1: Numbers are 5x and 8x. Step 2: 8x - 5x = 36 → 3x = 36 → x = 12. Step 3: Sum = 5x + 8x = 13x = 13(12) = 156.

Question 20

A chemist has two solutions: one with 30% acid and another with 60% acid. He mixes them in a certain ratio and obtains 50 litres of 48% acid solution. How many litres of the 30% solution were used?

Accepted Answer

Step 1: Let x litres of 30% solution and y litres of 60% solution be mixed. Step 2: x + y = 50 (total volume). Step 3: 0.30x + 0.60y = 0.48 × 50 = 24 (acid content). Step 4: From equation 1, y = 50 - x. Substitute: 0.30x + 0.60(50 - x) = 24. Step 5: 0.30x + 30 - 0.60x = 24 → -0.30x = -6 → x = 20 litres.

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