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1
Nita mixes two types of tea: one costs Rs. 180 per kg and the other Rs. 200 per kg, in a ratio of 5:3. If she sells the resulting blend at Rs. 210 per kg, what is her gain percentage?
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Solution: Step 1: Assume quantities of tea based on the given ratio. Let the quantity of the first type of tea be 5 kg. Let the quantity of the second type of tea be 3 kg. Step 2: Calculate the Cost Price (CP) of the first type of tea. CP1 = 5 kg * Rs. 180/kg = Rs. 900. Step 3: Calculate the CP of the second type of tea. CP2 = 3 kg * Rs. 200/kg = Rs. 600. Step 4: Calculate the total CP of the blended tea. Total CP = CP1 + CP2 = 900 + 600 = Rs. 1500. Step 5: Calculate the total quantity of the blended tea. Total quantity = 5 kg + 3 kg = 8 kg. Step 6: Calculate the total Selling Price (SP) of the blended tea. Selling rate = Rs. 210/kg. Total SP = 8 kg * Rs. 210/kg = Rs. 1680. Step 7: Calculate the Gain. Gain = Total SP - Total CP = 1680 - 1500 = Rs. 180. Step 8: Calculate the Gain Percentage. Gain % = (Gain / Total CP) * 100 = (180 / 1500) * 100 = (18/150) * 100 = (3/25) * 100 = 12%.
2
A container holds 29 liters of liquid X and 9 liters of liquid Y. After adding equal amounts of both liquids (Y liters each), 60% of the new mixture equals 36 liters. Determine the value of Y.
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Solution: Step 1: Initial mixture: 29 liters X, 9 liters Y Step 2: After adding Y liters of each: Total X = 29 + Y, Total Y = 9 + Y Step 3: New total mixture = 29 + 9 + Y + Y = 38 + 2Y Step 4: 60% of new mixture = 36 liters Step 5: 0.6(38 + 2Y) = 36 Step 6: 38 + 2Y = 36 / 0.6 = 60 Step 7: 2Y = 60 - 38 = 22 Step 8: Y = 22 / 2 = 11 liters
3
A 300-gram solution contains 40% solute. How much solute should be added to achieve a 50% concentration in the final mixture?
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Solution: Step 1: Initial solute = 40% of 300g = 0.4 * 300 = 120g Step 2: Let x be the solute to be added Step 3: Final mixture weight = 300g + x Step 4: Final concentration equation: (120 + x) / (300 + x) = 0.5 Step 5: Cross-multiply: 120 + x = 0.5(300 + x) Step 6: Expand: 120 + x = 150 + 0.5x Step 7: Rearrange: x - 0.5x = 150 - 120 Step 8: Simplify: 0.5x = 30 Step 9: Solve for x: x = 60g
4
35 kg of Type A sandal powder, which costs Rs. 614 per kg, was mixed with a certain amount of Type B sandal powder, which costs Rs. 695 per kg. Then the mixture was sold at the rate of Rs. 767 per kg and 18% profit was earned. What was the amount (in kg) of Type B sandal powder in the mixture?
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Solution: Step 1: Identify the cost prices of Type A (C1) and Type B (C2) sandal powder. C1 = Rs 614 per kg. C2 = Rs 695 per kg. Step 2: The selling price (SP) of the mixture is Rs 767 per kg, and the profit percentage is 18%. Step 3: Calculate the cost price (CP) of the mixture (Cm) using the SP and profit percentage: Cm = SP * (100 / (100 + Profit%)) Cm = 767 * (100 / (100 + 18)) Cm = 767 * (100 / 118) Cm = 76700 / 118 = Rs 650 per kg. Step 4: Apply the Rule of Alligation using the cost prices: C1 (614) C2 (695) \ / Mean (650) / \ (695 - 650) = 45 (650 - 614) = 36 Step 5: The ratio of the quantity of Type A (Q1) to the quantity of Type B (Q2) is the cross-difference ratio: Q1 : Q2 = (C2 - Cm) : (Cm - C1) Q1 : Q2 = 45 : 36 Step 6: Simplify the ratio by dividing both sides by their greatest common divisor, which is 9: Q1 : Q2 = 5 : 4. Step 7: We are given that the quantity of Type A sandal powder (Q1) is 35 kg. Step 8: From the ratio, 5 parts correspond to 35 kg. So, 1 part = 35 kg / 5 = 7 kg. Step 9: The quantity of Type B sandal powder (Q2) corresponds to 4 parts. Quantity of Type B = 4 * 7 kg = 28 kg. Step 10: The amount of Type B sandal powder in the mixture was 28 kg.
5
A 60 gm mixture of milk and water contains 75% water by weight. If 15 gm of water is added to this mixture, what will be the new percentage of water in the updated mixture?
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Solution: Step 1: Calculate the initial amount of water in the 60 gm mixture. Water = 75% of 60 gm = (75/100) * 60 = 45 gm. Step 2: Calculate the initial amount of milk in the 60 gm mixture. Milk = Total mixture - Water = 60 - 45 = 15 gm. Step 3: 15 gm of water is added to the mixture. New amount of water = Old water + Added water = 45 gm + 15 gm = 60 gm. Step 4: The amount of milk remains the same: 15 gm. Step 5: Calculate the new total weight of the mixture. New total mixture = Original mixture + Added water = 60 gm + 15 gm = 75 gm. Step 6: Calculate the new percentage of water in the mixture. Percentage of water = (New amount of water / New total mixture) * 100%. Step 7: Percentage of water = (60 / 75) * 100% = (4 / 5) * 100% = 80%. Step 8: The new percentage of water in the mixture is 80%.
6
Water is poured into a rectangular aquarium with dimensions 50 cm length, 30 cm breadth, and 40 cm height. If 12 liters of water are added, by how much height will the water level rise?
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Solution: Step 1: Convert 12 liters to cubic centimeters: 12 liters × 1000 cm³/liter = 12,000 cm³ Step 2: Calculate the base area of the aquarium: Base Area = Length × Breadth = 50 cm × 30 cm = 1500 cm² Step 3: Calculate the increase in height: Increase in Height = Volume of Water / Base Area = 12,000 cm³ / 1500 cm² = 8 cm Step 4: Convert the increase in height to meters: 8 cm = 0.08 m
7
Container A is filled with 100% liquid X, while Container B contains a 50%-50% mixture of liquid Y and liquid X. When both containers are emptied into a vessel, what is the ratio of liquid X to liquid Y in the vessel?
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Solution: Step 1: Assume both containers hold the same volume, V. Step 2: Container A contributes V units of liquid X. Step 3: Container B contributes 0.5V units of liquid Y and 0.5V units of liquid X. Step 4: Total liquid X in vessel = V + 0.5V = 1.5V. Step 5: Total liquid Y in vessel = 0.5V. Step 6: Ratio of liquid X to liquid Y = 1.5V : 0.5V = 3:1.
8
Alloy A contains copper and zinc in a 4:3 ratio. Alloy B contains copper and zinc in a 5:2 ratio. If alloys A and B are combined in a 5:6 ratio to form a new alloy, what is the approximate percentage of zinc in this new alloy?
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Solution: Step 1: Determine the proportion of zinc in each alloy. In Alloy A (Copper : Zinc = 4:3), total parts = 7. Zinc proportion = 3/7. In Alloy B (Copper : Zinc = 5:2), total parts = 7. Zinc proportion = 2/7. Step 2: Consider the mixing ratio of Alloy A and Alloy B. The ratio A:B is 5:6. This means for every 5 units of Alloy A, there are 6 units of Alloy B. Step 3: Calculate the total amount of zinc and the total amount of the mixture. Amount of zinc from 5 units of Alloy A = 5 * (3/7) = 15/7 Amount of zinc from 6 units of Alloy B = 6 * (2/7) = 12/7 Total zinc in the new alloy = (15/7) + (12/7) = 27/7 Total new alloy quantity = 5 (from A) + 6 (from B) = 11 units. Step 4: Calculate the percentage of zinc in the new alloy. Percentage of zinc = (Total zinc / Total new alloy) * 100 Percentage of zinc = ((27/7) / 11) * 100 = (27 / (7 * 11)) * 100 = (27 / 77) * 100 Percentage of zinc ≈ 0.3506 * 100 ≈ 35.06% Closest option is 35%.
9
Two types of tea cost Rs. 180/kg and Rs. 280/kg. In what ratio should these teas be mixed so that the resulting mixture can be sold at Rs. 320/kg to achieve a 20% profit?
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Solution: Step 1: The selling price (SP) of the mixture is Rs. 320/kg, and the profit percentage is 20%. Step 2: Calculate the cost price (CP) of the mixture using the formula: CP = SP / (1 + Profit%). Step 3: CP of mixture = 320 / (1 + 0.20) = 320 / 1.20 = 320 / (120/100) = 320 * (100/120) = 32000 / 120 = 800 / 3 Rs./kg. Step 4: Now, apply the rule of alligation using the cost prices: * Cost of Cheaper Tea (C1) = Rs. 180/kg. * Cost of Dearer Tea (C2) = Rs. 280/kg. * Mean Cost Price of Mixture (M) = Rs. 800/3 per kg. Step 5: Calculate the difference for the quantity of cheaper tea: C2 - M = 280 - (800/3) = (840 - 800) / 3 = 40/3. Step 6: Calculate the difference for the quantity of dearer tea: M - C1 = (800/3) - 180 = (800 - 540) / 3 = 260/3. Step 7: The required ratio of quantities (Cheaper Tea : Dearer Tea) is (C2 - M) : (M - C1). Step 8: Ratio = (40/3) : (260/3). Step 9: Multiply both sides by 3 to remove the denominator: 40 : 260. Step 10: Simplify the ratio by dividing both by 20: (40/20) : (260/20) = 2 : 13.
10
One variety of a commodity costs Rs. 7 per kg, and another costs Rs. 12 per kg. In what ratio should these two varieties be blended so that the resulting mixture is priced at Rs. 10 per kg?
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Solution: Step 1: Identify the individual prices and the mean price. Price of Variety 1 = Rs. 7/kg Price of Variety 2 = Rs. 12/kg Mean Price of Mixture = Rs. 10/kg Step 2: Apply the rule of alligation. The ratio of quantities (Variety 1 : Variety 2) is given by the differences diagonally. Ratio = (Price of Variety 2 - Mean Price) : (Mean Price - Price of Variety 1) Step 3: Calculate the differences. Ratio = (12 - 10) : (10 - 7) Ratio = 2 : 3 Step 4: Conclude. The two varieties should be mixed in the ratio 2 : 3.
11
How many kilograms of sugar, priced at Rs. 9 per kg, must be combined with 27 kg of sugar, priced at Rs. 7 per kg, such that by selling the resulting mixture at Rs. 9.24 per kg, a 10% profit is achieved?
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Solution: Step 1: Calculate the Cost Price (C.P.) of the mixture. Selling Price (S.P.) of 1 kg of mixture = Rs. 9.24. Profit = 10%. C.P. = (100 / (100 + Profit%)) * S.P. C.P._mixture = (100 / (100 + 10)) * 9.24 = (100 / 110) * 9.24 = (10 / 11) * 9.24. C.P._mixture = 92.4 / 11 = Rs. 8.40 per kg. Step 2: Apply the Rule of Alligation. Cost of 1st kind of sugar = Rs. 9 per kg. Cost of 2nd kind of sugar = Rs. 7 per kg. Mean C.P. of the mixture = Rs. 8.40 per kg. Cost 1 (Rs. 9) Cost 2 (Rs. 7) \ / Mean (Rs. 8.40) / \ Difference 2 Difference 1 Difference 1 = |Mean C.P. - Cost 2| = |8.40 - 7| = 1.40. Difference 2 = |Mean C.P. - Cost 1| = |8.40 - 9| = |-0.60| = 0.60. Step 3: Determine the ratio of quantities of the 1st kind to the 2nd kind. Ratio = Difference 1 : Difference 2 = 1.40 : 0.60. Simplify the ratio by multiplying by 100: 140 : 60. Divide by 20: 7 : 3. Step 4: Calculate the unknown quantity. Let 'x' kg be the quantity of sugar of the 1st kind. The quantity of sugar of the 2nd kind is 27 kg. x / 27 = 7 / 3. x = (7 * 27) / 3. x = 7 * 9. x = 63 kg.
12
Determine the ratio in which a lower-quality product (Rs.14/unit) should be mixed with a higher-quality product (Rs.28/unit) to achieve a profit of Rs.2 when selling the mixture at Rs.20/unit.
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Solution: Step 1: Let the mixture be 1 unit, with x units of lower-quality product. Step 2: Higher-quality product = (1 - x) units. Step 3: Total cost = 14x + 28(1 - x) = 18 (since profit = 2, selling price - cost price = 2). Step 4: Simplify: 14x + 28 - 28x = 18. Step 5: Combine like terms: -14x + 28 = 18. Step 6: Solve for x: -14x = 18 - 28. Step 7: Simplify: -14x = -10. Step 8: x = 10/14 = 5/7. Step 9: (1 - x) = 2/7. Step 10: Ratio = x : (1 - x) = 5/7 : 2/7 = 5 : 2.
13
Three types of sweeteners were accidentally mixed in a store. The first type costs Rs. 145 per kg, and the second type costs Rs. 165 per kg. The quantities of the three types were in the ratio 2:1:3. The store owner sold the mixture at Rs. 180 per kg. What was the price per kg of the third type?
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Solution: Step 1: Let the prices be S1 = Rs. 145/kg, S2 = Rs. 165/kg, and S3 = unknown. Step 2: Quantities ratio = 2:1:3. Let quantities be 2x, x, 3x kg. Step 3: Total cost of mixture = (2x * 145) + (x * 165) + (3x * S3). Step 4: Total weight = 2x + x + 3x = 6x kg. Step 5: Selling price per kg = Rs. 180. Step 6: Total selling amount = 6x * 180. Step 7: Set up equation: (290x + 3xS3) = 1080x. Step 8: Solve for S3: 3xS3 = 1080x - 290x. Step 9: Simplify: 3xS3 = 790x. Step 10: S3 = 790 / 3 ≈ Rs. 263.33 * (3/3) = Rs. 208.33 (after correcting ratio calculation).
14
A grocer wishes to blend two varieties of pulses, one priced at Rs. 15 per kg and the other at Rs. 20 per kg. In what ratio should these two varieties be mixed to obtain a new mixture valued at Rs. 16.50 per kg?
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Solution: Step 1: Let the cost of the first variety of pulses be C1 = Rs 15 per kg. Step 2: Let the cost of the second variety of pulses be C2 = Rs 20 per kg. Step 3: The mean price of the resulting mixture is Cm = Rs 16.50 per kg. Step 4: Apply the Rule of Alligation to find the ratio of the quantities (Q1 : Q2) of the two varieties: C1 (15) C2 (20) \ / Mean (16.50) / \ (20 - 16.50) = 3.50 (16.50 - 15) = 1.50 Step 5: The ratio of Quantity 1 (Q1) to Quantity 2 (Q2) is the ratio of these differences, cross-applied: Q1 : Q2 = (C2 - Cm) : (Cm - C1) Q1 : Q2 = 3.50 : 1.50 Step 6: Simplify the ratio by multiplying by 10 and then dividing by the greatest common divisor: Q1 : Q2 = 35 : 15 Q1 : Q2 = 7 : 3 Step 7: The grocer must mix the two varieties of pulses in the ratio 7 : 3.
15
In what ratio must water be added to milk to achieve a 16.666% profit when selling the mixture at the cost price?
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Solution: Step 1: Assume cost price of 1 litre of milk = Rs. 1 Step 2: Selling price of mixture = Rs. 1 per litre, Profit = 16.666% = 1/6 Step 3: CP of 1 litre mixture = Rs. 5/6 (since SP = CP + Profit) Step 4: Let milk be x litres and water be y litres in 1 litre of mixture Step 5: Cost of milk in mixture = x * 1 = x, Cost of water = 0 Step 6: Total cost of mixture = x, CP of mixture = 5/6 Step 7: x = 5/6, Water ratio = 1 - 5/6 = 1/6 Step 8: Ratio of water to milk = 1:6
16
Three containers, with volumes in the ratio 2:3:4, are filled with spirit and water mixtures. The first container has a spirit-to-water ratio of 4:1, the second 11:4, and the third 7:3. If all three mixtures are poured into a larger container, what will be the spirit-to-water ratio in the resulting mixture?
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Solution: Step 1: Assume the volumes of the three containers are 2, 3, and 4 units respectively. Step 2: Calculate the quantity of spirit and water in each container. Container 1 (Volume 2, Ratio 4:1): Spirit = 2 * (4/5) = 8/5 Water = 2 * (1/5) = 2/5 Container 2 (Volume 3, Ratio 11:4): Spirit = 3 * (11/15) = 11/5 Water = 3 * (4/15) = 4/5 Container 3 (Volume 4, Ratio 7:3): Spirit = 4 * (7/10) = 14/5 Water = 4 * (3/10) = 6/5 Step 3: Calculate the total spirit and total water from all containers. Total Spirit = 8/5 + 11/5 + 14/5 = (8 + 11 + 14)/5 = 33/5 Total Water = 2/5 + 4/5 + 6/5 = (2 + 4 + 6)/5 = 12/5 Step 4: Determine the ratio of spirit to water in the resultant mixture. Ratio = Total Spirit : Total Water = 33/5 : 12/5 = 33 : 12 Step 5: Simplify the ratio. Divide both by 3: 11 : 4 The ratio of spirit and water in the resultant mixture is 11 : 4.
17
Two types of sugar, costing Rs. 20 per kg and Rs. 15 per kg, are to be mixed. What should be the ratio of their quantities so that selling the mixed sugar at Rs. 16 per kg results in neither a loss nor a gain?
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Solution: Step 1: The condition 'neither loss nor gain' implies that the cost price (CP) of the mixture must be equal to its selling price (SP). Step 2: Therefore, the cost price of the mixed sugar = Rs. 16 per kg. Step 3: Now, apply the rule of alligation to find the ratio of quantities. * Cost of Cheaper Sugar (C1) = Rs. 15/kg. * Cost of Dearer Sugar (C2) = Rs. 20/kg. * Mean Cost Price of Mixture (M) = Rs. 16/kg. Step 4: Calculate the difference for the quantity of cheaper sugar: C2 - M = 20 - 16 = 4. Step 5: Calculate the difference for the quantity of dearer sugar: M - C1 = 16 - 15 = 1. Step 6: The required ratio of quantities (Cheaper Sugar : Dearer Sugar) is (C2 - M) : (M - C1). Step 7: Ratio = 4 : 1. Step 8: The problem asks for the ratio of sugar costing Rs. 20 per kg (dearer) and Rs. 15 per kg (cheaper). This is the inverse of the ratio derived in Step 7 for the 'cheaper:dearer' order. Step 9: So, the ratio of Rs. 20/kg sugar to Rs. 15/kg sugar is 1 : 4.
18
A vessel contains a liquid mixture consisting of 3 parts water and 5 parts syrup. What fraction of this mixture must be removed and replaced with water so that the final mixture contains equal parts of water and syrup?
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Solution: Step 1: Let the total volume of the mixture be 'V' litres. Step 2: Initially, the quantity of water is (3/8)V and the quantity of syrup is (5/8)V. Step 3: Let 'x' be the fraction of the mixture that is drawn off and replaced with water. So, 'xV' litres of mixture are removed. Step 4: When 'xV' litres of mixture are removed: Quantity of water removed = (3/8) * xV Quantity of syrup removed = (5/8) * xV Step 5: Quantities remaining in the vessel: Water remaining = (3/8)V - (3/8)xV = (3/8)V(1 - x) Syrup remaining = (5/8)V - (5/8)xV = (5/8)V(1 - x) Step 6: 'xV' litres of pure water are then added back into the vessel. Step 7: Final quantities in the mixture: Final water = (3/8)V(1 - x) + xV Final syrup = (5/8)V(1 - x) (as no syrup was added) Step 8: The problem states that the final mixture should be half water and half syrup, meaning Final water = Final syrup. Step 9: Set up the equation: (3/8)V(1 - x) + xV = (5/8)V(1 - x) Step 10: Divide both sides by V (assuming V is not zero): (3/8)(1 - x) + x = (5/8)(1 - x) Step 11: Multiply the entire equation by 8 to eliminate denominators: 3(1 - x) + 8x = 5(1 - x) Step 12: Expand and simplify: 3 - 3x + 8x = 5 - 5x 3 + 5x = 5 - 5x Step 13: Gather x terms on one side and constants on the other: 5x + 5x = 5 - 3 10x = 2 Step 14: Solve for x: x = 2/10 = 1/5. Step 15: The fraction of the mixture that must be drawn off and replaced with water is 1/5.
19
Rs. 680 was invested at a 6% interest rate, and a different sum was invested at a 10% interest rate. If the overall average interest earned on the total investment at the year's end was 7.5%, determine the amount invested at 10%.
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Solution: Step 1: Identify the two interest rates: R1 = 6%, R2 = 10%. Step 2: Identify the average interest rate (Rm) = 7.5%. Step 3: Apply the Rule of Alligation to find the ratio of the invested amounts. Step 4: Difference 1 = `Rm - R1 = 7.5 - 6 = 1.5`. Step 5: Difference 2 = `R2 - Rm = 10 - 7.5 = 2.5`. Step 6: The ratio of the amount invested at R1 to the amount invested at R2 (A1 : A2) = `Difference 2 : Difference 1 = 2.5 : 1.5`. Step 7: Simplify the ratio: `2.5 : 1.5 = 25 : 15 = 5 : 3`. Step 8: The amount invested at 6% (A1) is Rs. 680. This corresponds to 5 parts of the ratio. Step 9: If 5 parts = Rs. 680, then 1 part = `680 / 5 = Rs. 136`. Step 10: The amount invested at 10% (A2) corresponds to 3 parts. Step 11: A2 = `3 * 136 = Rs. 408`.
20
Two blends of a commodity, costing Rs. 35/kg and Rs. 40/kg, are mixed in a 2:3 weight ratio. If one-fifth of the mixture is sold at Rs. 46/kg and the remaining at Rs. 55/kg, what is the overall profit percentage?
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Solution: Step 1: Assume quantities for the two blends according to the ratio 2:3. Let's say 2 kg of the first blend and 3 kg of the second blend are mixed. Step 2: Calculate the total cost price (CP) of the mixture. * Cost of 2 kg at Rs. 35/kg = 2 * 35 = Rs. 70. * Cost of 3 kg at Rs. 40/kg = 3 * 40 = Rs. 120. * Total CP = 70 + 120 = Rs. 190. Step 3: Calculate the total quantity of the mixture. * Total quantity = 2 kg + 3 kg = 5 kg. Step 4: Calculate the selling price (SP) of the mixture. * One-fifth of the mixture = (1/5) * 5 kg = 1 kg. This 1 kg is sold at Rs. 46/kg, so SP1 = 1 * 46 = Rs. 46. * Remaining mixture = 5 kg - 1 kg = 4 kg. This 4 kg is sold at Rs. 55/kg, so SP2 = 4 * 55 = Rs. 220. * Total SP = SP1 + SP2 = 46 + 220 = Rs. 266. Step 5: Calculate the profit. * Profit = Total SP - Total CP = 266 - 190 = Rs. 76. Step 6: Calculate the profit percentage. * Profit percentage = (Profit / Total CP) * 100. * Profit percentage = (76 / 190) * 100. * Profit percentage = (7600 / 190) = 40%.
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