Ratio, Proportion and Partnership

If a dishonest dealer claims to sell goods at cost price but uses a false weight to gain 25%, it means he gives less than 1 kg while charging for 1 kg. Let the false weight be x grams. The gain percentage is given by ((1000 - x) / x) × 100 = 25%. Solving for x: (1000 - x) / x = 0.25 → 1000 - x = 0.25x → 1000 = 1.25x → x = 800 grams. Therefore, the false weight used is 800 grams to gain 25%. This matches option C.

The original alloy contains 50 gm with 80% gold, so gold weight is 40 gm. Let x be the gold added. The new gold percentage is (40 + x) / (50 + x) = 95%. Solving gives x = 200 gm. Therefore, 200 gm of gold should be added to reach 95% gold content.

Let the two numbers be 5x and 4x. Their difference is 5x - 4x = x = 10, so x = 10. The largest number is 5x = 50, which corresponds to option B.

Ramees sells 900 grams of goods but charges for 1 kilogram (1000 grams) at cost price. The gain arises because he gives less quantity but charges for the full kilogram. The gain percent is calculated as ((1000 - 900) / 900) × 100 = (100 / 900) × 100 = 11 1/9%. This matches option C, not B as originally stated.

The dealer uses 800 grams instead of 1000 grams but charges for 1000 grams, effectively selling 800 grams for the price of 1000 grams. The gain percent is ((1000 - 800) / 800) × 100 = 25%.

The ratio of Faheem's salary to Imran's salary to Naveed's salary is calculated by dividing each salary by the greatest common divisor. The salaries are 80,000, 70,000, and 50,000 rupees respectively. Dividing each by 10,000 gives 8:7:5, which matches option B.

To dilute 9 ml of lotion containing 50% alcohol to a lotion containing 30% alcohol, use the dilution formula C1V1 = C2V2, where C1 = 50%, V1 = 9 ml, C2 = 30%, and V2 is the final volume after dilution. Calculating: 50% × 9 ml = 30% × V2 → 4.5 = 0.3 × V2 → V2 = 4.5 / 0.3 = 15 ml. The volume of water to add is the final volume minus the initial volume: 15 ml - 9 ml = 6 ml. Therefore, 6 ml of water should be added, which corresponds to option D.

Ramees bought 4 dozen apples at Rs. 12 per dozen (cost = 4 × 12 = Rs. 48) and 2 dozen apples at Rs. 16 per dozen (cost = 2 × 16 = Rs. 32). Total cost = Rs. 80 for 6 dozen apples. To earn 20%, selling price = 80 × 1.2 = Rs. 96. Price per dozen = 96 ÷ 6 = Rs. 16.80.

Given the ratio 4:5:7, let the common multiplier be x. The difference between the costliest and cheapest cars is 7x - 4x = 3x = 60000, so x = 20000. The modest price corresponds to 5x = 5 * 20000 = Rs. 100000.

The profit share is divided in the ratio of investments. The ratio of A:B:C is 6300:4200:10500, which simplifies to 3:2:5. Total parts = 3+2+5=10. A's share = (3/10)*12100 = Rs.3630.

First, calculate the cost price per kg of the mixture: ((24 kg × Rs.150) + (36 kg × Rs.125)) / (24 + 36) = (3600 + 4500) / 60 = Rs.135 per kg. To make a 40% profit, selling price = 135 × 1.40 = Rs.189 per kg.

A's share of the capital is 1/3. The total profit is Rs.2460, so A's share of the profit = (1/3) × 2460 = Rs.820. However, the sum of the given fractions 1/3 + 1/4 + 1/5 = 47/60, so D's share is 13/60. Calculating A's share of profit correctly: A's fraction is 1/3 = 20/60, so A's profit share = (20/60) × 2460 = Rs.820. Therefore, option C (Rs.820) is correct, not B.

Since the man sells the mixture at the same rate as milk but gains 20%, the cost price of the mixture is 20% less than the selling price. This means the mixture is 100% milk + water, where water is free. The ratio of water to milk is 1:5 to achieve a 20% gain.

A invests Rs. 2000 for 1 year, B invests Rs. 3000 for 2 years, and C invests Rs. 4000 for 2 years. The profit shares are proportional to the product of investment and time: A = 2000×1 = 2000, B = 3000×2 = 6000, C = 4000×2 = 8000. Total = 2000 + 6000 + 8000 = 16000. A's share of the profit = (2000/16000) × 3200 = Rs. 400. Therefore, A's share in the profit is Rs. 400, which corresponds to option D.

The profit share among partners is proportional to the product of the amount invested and the time for which it is invested. Let A's investment be x. Then B invests 2x after 6 months, so for 6 months, and C invests 3x after 8 months, so for 4 months. Calculating the weighted investments: A = x * 12 = 12x, B = 2x * 6 = 12x, C = 3x * 4 = 12x. The ratio of their investments multiplied by time is 1:1:1, so the profit is shared equally. The total profit is Rs.18,000, so A's share is one-third of Rs.18,000, which is Rs.6,000. Therefore, the correct answer is option C (Rs.6000).

Since A and B invest Rs.6000 and Rs.8000 respectively for the same duration (one year), their profit sharing ratio is directly proportional to their investments, which is 6000:8000 or 3:4.

A receives 30% of the total profit as a managing partner, and the remaining 70% is divided among A, B, and C in the ratio of their capitals (5000:6000:4000). Let total profit be P. A's total profit = 0.3P + (5000/15000)*0.7P = 0.3P + (1/3)*0.7P = 0.3P + 0.2333P = 0.5333P. B's profit = (6000/15000)*0.7P = 0.28P. C's profit = (4000/15000)*0.7P = 0.1867P. Given A's profit is Rs.200 more than B's and C's combined profit: 0.5333P = 0.28P + 0.1867P + 200 => 0.5333P = 0.4667P + 200 => 0.0666P = 200 => P = 3000. Hence, total profit is Rs.3000.

First, calculate the working partner A's share: 5% of Rs.7400 = Rs.370. Remaining profit = Rs.7400 - Rs.370 = Rs.7030. Calculate each partner's capital-time product: A = 6500*6=39000, B=8400*5=42000, C=10000*3=30000. Total = 39000+42000+30000=111000. C's share = (30000/111000)*7030 ≈ Rs.1900. Therefore, the correct share of C in the total profit is approximately Rs.1900, which corresponds to option B.

In a partnership where partners invest different amounts for different periods, the profit is shared in proportion to the product of the capital invested and the time for which it is invested. For example, if Partner A invests Rs. 10,000 for 12 months and Partner B invests Rs. 15,000 for 6 months, their profit shares are proportional to 10,000×12 = 120,000 and 15,000×6 = 90,000 respectively. The total units are 210,000. If the total profit is Rs. 21,000, Partner A's share is (120,000/210,000)×21,000 = Rs. 12,000 and Partner B's share is (90,000/210,000)×21,000 = Rs. 9,000. This confirms that profit sharing is proportional to capital invested multiplied by the time invested.

A receives 10% of the total profit as a managing partner, which is Rs.960 (10% of Rs.9,600). The remaining profit is Rs.8,640 (Rs.9,600 - Rs.960), which is divided in the ratio of their capitals (A:B = 15,000:25,000 = 3:5). A's share of this remaining profit is (3/8) × 8,640 = Rs.3,240. Therefore, total money received by A = Rs.960 + Rs.3,240 = Rs.4,200. Hence, option D (Rs.4,200) is correct.

A's initial investment is Rs.300 and B's is Rs.400. After the first year, A's share of profit is (300/(300+400)) * 210 = Rs.90, which he reinvests, making his new investment 300 + 90 = Rs.390. B does not reinvest, so his investment remains Rs.400. The ratio of their investments for the second year is 390:400, which simplifies to 39:40. Therefore, the second year's profit should be divided in the ratio 39:40.

A's investment for 12 months is 70000 × 12 = 840000; B's investment for 6 months is 120000 × 6 = 720000. The ratio of their investments is 840000:720000 = 7:6. Total profit is Rs. 52000, so B's share = (6/13) × 52000 = Rs. 24000. However, this contradicts the options given. Rechecking: 7 + 6 = 13 parts; B's share = (6/13)*52000 = Rs. 24000, which matches option B. Therefore, the original correct answer B is correct. The initial calculation was correct; the explanation confirms B's share is Rs. 24000.

Given the capitals of A and B are in the ratio 7:9. A withdraws after 8 months, and the profit ratio is 8:9. Let B's capital be used for x months. The profit ratio is proportional to capital × time, so: A's contribution = 7 × 8 = 56 B's contribution = 9 × x Given profit ratio A:B = 8:9, so: 56 : 9x = 8 : 9 Cross-multiplied: 56 × 9 = 8 × 9x 504 = 72x x = 7 months Therefore, B's capital was used for 7 months, which corresponds to option D.

To find A's share of the rent, we calculate the total feed consumption for A and B based on the given equivalences. Using the ratios, we convert horses and cows into sheep equivalents, sum the total feed-months for both A and B, and then find A's proportion of the total. This results in A paying 2/3 of the rent.

The rent should be divided in proportion to the product of the number of horses and the months each person kept them. A's share is 12×8=96, B's share is 16×9=144, and C's share is 18×6=108. Total = 96+144+108=348. C's share of the rent = (108/348)×870 = Rs.270.