📘 Quiz

Solution: Step 1: Calculate investment ratio: 42000:60000:80000 = 21:30:40 Step 2: Total ratio parts = 21 + 30 + 40 = 91 Step 3: Second partner's ratio = 30/91 of total profit Step 4: Profit share = (30/91) * 18200 = Rs.6000

Solution: Step 1: Let the original shares of A, B, and C be A, B, and C respectively. The total sum is A + B + C = 1087. Step 2: The amounts after diminishing the shares are (A - 10), (B - 12), and (C - 15). Step 3: Calculate the total sum of these diminished amounts: (A - 10) + (B - 12) + (C - 15) = (A + B + C) - (10 + 12 + 15) = 1087 - 37 = 1050. Step 4: These diminished amounts are in the ratio 5:7:9. Let the common multiplier for the ratio be k. A - 10 = 5k B - 12 = 7k C - 15 = 9k Step 5: The sum of these ratio parts is 5k + 7k + 9k = 21k. Step 6: Equate the sum of ratio parts to the total diminished amount: 21k = 1050 k = 1050 / 21 = 50. Step 7: Calculate B's original share: B - 12 = 7k = 7 * 50 = 350. B = 350 + 12 = Rs. 362.

Solution: Step 1: Let the original value = x Step 2: Set up equation: x - (2/5)x = 45 Step 3: Simplify: (3/5)x = 45 Step 4: Solve for x: x = 45 * (5/3) Step 5: Calculate: x = 75

Solution: Step 1: Let the number to be subtracted be x. Step 2: The new numbers are (7 - x), (9 - x), (11 - x), and (15 - x). Step 3: For these numbers to be in proportion, the ratio of the first two must equal the ratio of the last two: (7 - x) / (9 - x) = (11 - x) / (15 - x). Step 4: Cross-multiply the equation: (7 - x)(15 - x) = (11 - x)(9 - x). Step 5: Expand both sides: Left side: 7*15 - 7x - 15x + x² = 105 - 22x + x² Right side: 11*9 - 11x - 9x + x² = 99 - 20x + x² Step 6: Set the expanded expressions equal: 105 - 22x + x² = 99 - 20x + x². Step 7: Subtract x² from both sides: 105 - 22x = 99 - 20x. Step 8: Rearrange to solve for x: 105 - 99 = 22x - 20x => 6 = 2x => x = 3.

Solution: Step 1: The total duration of the business is one year, which is 12 months. Step 2: A's investment is Rs. 85,000 for the full 12 months. A's effective investment = 85000 * 12. Step 3: Let 'x' be the number of months B remained in the business. B's investment is Rs. 42,500. B's effective investment = 42500 * x. Step 4: The profits are divided in the ratio 3:1 (A's share : B's share). Step 5: Set up the ratio of effective investments equal to the ratio of profits: (A's effective investment) / (B's effective investment) = (A's profit share) / (B's profit share) (85000 * 12) / (42500 * x) = 3 / 1. Step 6: Simplify the equation: (85000 / 42500) * (12 / x) = 3 2 * (12 / x) = 3 24 / x = 3. Step 7: Solve for x: x = 24 / 3 x = 8. Step 8: B joined for a period of 8 months.

Solution: Step 1: Let the numbers be 2X and 9X Step 2: Then their H.C.F. is X, so X = 19 Step 3: Therefore, the numbers are (2*19 and 9*19) i.e. 38 and 171

Solution: Step 1: Identify the given set of numbers: {-3, -2, -1, 0, 1, 2, 3}. Step 2: Determine the total number of elements in the set. There are 7 numbers. Step 3: Understand the condition |X| < 2. This means -2 < X < 2. Step 4: Identify the numbers from the given set that satisfy the condition -2 < X < 2. These numbers are {-1, 0, 1}. Step 5: Count the number of favorable outcomes: There are 3 such numbers. Step 6: Calculate the probability: (Favorable outcomes) / (Total outcomes) = 3 / 7.

Solution: Step 1: Let initial number of individuals = x Step 2: Initial supply lasts 42 days for x individuals Step 3: After 10 days, (x * 32) supply remains Step 4: (x + 200) individuals consume remaining supply in 24 days Step 5: Set up proportion: (x * 32) / (x + 200) = 24 Step 6: Cross-multiply: x * 32 = 24(x + 200) Step 7: Expand: 32x = 24x + 4800 Step 8: Simplify: 8x = 4800 Step 9: Solve: x = 600 individuals

Solution: Step 1: Write the given equation: 5.5a = 0.65b Step 2: Convert decimals to fractions: (55/10)a = (65/100)b Step 3: Simplify the fractions: (11/2)a = (13/20)b Step 4: Rearrange to find a/b: a/b = (13/20) / (11/2) Step 5: Perform the division: a/b = (13/20) * (2/11) = 26 / 220 Step 6: Reduce the fraction to its simplest form: a/b = 13 / 110 Step 7: Express as a ratio: a : b = 13 : 110

Solution: Step 1: Write down the given ratios: A:B = 2:3 and B:C = 4:5. Step 2: Identify the common term, which is B. The values for B are 3 and 4. Step 3: Find the Least Common Multiple (LCM) of 3 and 4, which is 12. Step 4: Adjust the first ratio (A:B) so that B becomes 12. Multiply both parts of 2:3 by 4: A:B = (2*4) : (3*4) = 8:12. Step 5: Adjust the second ratio (B:C) so that B becomes 12. Multiply both parts of 4:5 by 3: B:C = (4*3) : (5*3) = 12:15. Step 6: Now that B has the same value in both adjusted ratios, combine them: A:B:C = 8:12:15.

Solution: Step 1: Convert the quantity of 49 dozens into individual mangoes. 1 dozen = 12 mangoes. 49 dozens = 49 × 12 = 588 mangoes. Step 2: Set up a direct proportion between the number of mangoes and their price, as the rate is constant: (Quantity1 / Price1) = (Quantity2 / Price2). Step 3: Plug in the given values: Quantity1 = 357 mangoes, Price1 = Rs. 1517.25. Quantity2 = 588 mangoes, Price2 = x (unknown). Step 4: Substitute these values into the proportion: 357 / 1517.25 = 588 / x. Step 5: Solve for x: x = (588 × 1517.25) / 357. Step 6: Perform the calculation: x = 892182 / 357 ≈ 2499.109. Step 7: Since the approximate price is requested, round the value to the nearest whole number: x ≈ Rs. 2499. Given the options, Rs. 2500 is the closest approximation.

Solution: Step 1: Let the two numbers be 3x and 5x, where x is a common factor. Step 2: Since 3 and 5 are coprime, the LCM of 3x and 5x is (3 * 5 * x) = 15x. Step 3: We are given that the LCM is 225. Set up the equation: 15x = 225. Step 4: Solve for x: x = 225 / 15 = 15. Step 5: The smaller number is 3x. Substitute x = 15: Smaller number = 3 * 15 = 45.

Solution: Step 1: Understand the objective: Determine R's share of profit. Profit in a partnership is generally distributed in the ratio of each partner's (Capital * Time). Step 2: Analyze Statement I: * Q's investment = Rs. 80,000. The duration for which Q invested (total business duration) is unknown. R's investment is unknown. * Statement I alone is insufficient. Step 3: Analyze Statement II: * R joined after 3 months. R's investment amount is unknown. The total business duration is unknown. * Statement II alone is insufficient. Step 4: Analyze Statement III: * P joined after 4 months with a capital of Rs. 1,20,000. P's share of profit = Rs. 6000. The total business duration is unknown. The total profit is unknown. * Statement III alone is insufficient. Step 5: Combine Statements I, II, and III. * Let the total duration of the business be 'T' months. * Q's (Capital * Time) = 80,000 * T. * R joined after 3 months, so R's investment duration = (T - 3) months. R's capital (let's denote it as C_R) is still unknown. * P joined after 4 months, so P's investment duration = (T - 4) months. P's capital = 1,20,000. * P's (Capital * Time) = 1,20,000 * (T - 4). * We know P's profit share is Rs. 6000. To find R's share, we need: 1. The total profit. 2. R's capital (C_R). * Even with all three statements, R's investment amount (C_R) is not provided. Without R's capital, we cannot calculate R's share of profit. Step 6: Conclude that even with all three statements (I, II, and III) combined, the data is not sufficient to answer the question.

Solution: Step 1: The total duration of the business is one year, which is 12 months. Step 2: A started the business, so A's investment of Rs. 3500 was for the full 12 months. A's effective investment = 3500 * 12. Step 3: B joined after 5 months, meaning B's investment was for 12 - 5 = 7 months. Step 4: Let B's capital contribution be Rs. x. B's effective investment = x * 7. Step 5: The profits are divided in the ratio 2:3 (A's share : B's share). Step 6: Set up the ratio of effective investments equal to the ratio of profits: (A's effective investment) / (B's effective investment) = (A's profit share) / (B's profit share) (3500 * 12) / (x * 7) = 2 / 3. Step 7: Cross-multiply to solve for x: 3 * (3500 * 12) = 2 * (x * 7) 3 * 42000 = 14x 126000 = 14x Step 8: Calculate x: x = 126000 / 14 = 9000. Step 9: B's capital contribution was Rs. 9000.

Solution: Step 1: Let the two numbers be 3x and 4x, given their ratio is 3:4. Step 2: The sum of their cubes is 5824. Set up the equation: (3x)³ + (4x)³ = 5824. Step 3: Calculate the cubes: 27x³ + 64x³ = 5824. Step 4: Combine the terms: 91x³ = 5824. Step 5: Solve for x³: x³ = 5824 / 91. Step 6: Perform the division: x³ = 64. Step 7: Take the cube root of both sides: x = ³√64. Step 8: x = 4. Step 9: Find the two numbers: First number = 3x = 3 × 4 = 12. Second number = 4x = 4 × 4 = 16. Step 10: Calculate the sum of the numbers: 12 + 16 = 28.

Solution: Step 1: The given ratio x : y = 7 : 3 means x/y = 7/3. For homogeneous expressions (where all terms have the same degree), we can directly substitute x = 7 and y = 3. Step 2: Substitute x=7 and y=3 into the numerator (xy + y^2): xy + y^2 = (7 × 3) + (3)^2 = 21 + 9 = 30. Step 3: Substitute x=7 and y=3 into the denominator (x^2 - y^2): x^2 - y^2 = (7)^2 - (3)^2 = 49 - 9 = 40. Step 4: Form the fraction and simplify: (xy + y^2) / (x^2 - y^2) = 30 / 40. Step 5: Reduce the fraction to its simplest form: 30 / 40 = 3 / 4.

Solution: Step 1: Let the tin contain x bottles initially. Step 2: 4/5 of x - 6 + 4 = 3/4 of x Step 3: 4x/5 - 2 = 3x/4 Step 4: Multiply by 20 to clear fractions: 16x - 40 = 15x Step 5: Rearrange: 16x - 15x = 40 Step 6: Simplify: x = 40 Step 7: Verify: 4/5 of 40 = 32, 3/4 of 40 = 30 Step 8: Difference of 2 bottles corresponds to 6 - 4 = 2 bottles Step 9: Therefore, initial bottles = 32

Solution: Step 1: Let P's capital be C_P, Q's capital be C_Q, and R's capital be C_R. Step 2: The given relationship is 4C_P = 6C_Q = 10C_R. Step 3: To find the ratio C_P : C_Q : C_R, we can find the LCM of the coefficients (4, 6, 10), which is 60. Step 4: Express each capital in terms of a common constant 'k' such that 4C_P = 6C_Q = 10C_R = k. C_P = k/4 C_Q = k/6 C_R = k/10 Step 5: The ratio of capitals is C_P : C_Q : C_R = 1/4 : 1/6 : 1/10. Step 6: Multiply each term by the LCM (60) to get whole numbers: C_P : C_Q : C_R = (1/4 * 60) : (1/6 * 60) : (1/10 * 60) = 15 : 10 : 6. Step 7: The profit-sharing ratio is the same as the capital ratio: 15 : 10 : 6. Step 8: The sum of the ratio parts is 15 + 10 + 6 = 31. Step 9: R's share of the total profit (Rs. 4,650) is calculated as (R's ratio part / Total ratio parts) * Total Profit. Step 10: R's share = (6 / 31) * 4650. Step 11: Calculate R's share: 6 * (4650 / 31) = 6 * 150 = Rs. 900.

Solution: Step 1: Let the price of the refrigerator be 5x and the price of the television set be 3x, based on the given ratio 5:3. Step 2: The problem states that the refrigerator costs Rs. 5500 more than the television set. This difference can be expressed as 5x - 3x. Step 3: Formulate the equation: 5x - 3x = 5500. Step 4: Simplify the equation: 2x = 5500. Step 5: Solve for x: x = 5500 / 2 = 2750. Step 6: Calculate the price of the refrigerator: Price of refrigerator = 5x = 5 * 2750 = Rs. 13750.