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Question 1 / 20
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1
A, B, and C invested their capitals in the ratio 2:3:5. The durations for which A, B, and C invested were in the ratio 4:2:3. If C's share of profit is Rs. 1,47,000 greater than A's share, what is B's share of profit?
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Solution: Step 1: Determine the profit sharing ratio using the formula: Profit Ratio = (Capital Ratio * Time Ratio). Capital ratio A:B:C = 2:3:5. Time ratio A:B:C = 4:2:3. Profit Ratio A:B:C = (2*4) : (3*2) : (5*3) = 8:6:15. Step 2: Calculate the difference in profit shares between C and A in terms of ratio units. Difference in units = C's units - A's units = 15 - 8 = 7 units. Step 3: Relate this difference to the given monetary value. Given difference = Rs. 1,47,000. So, 7 units = Rs. 1,47,000. Step 4: Calculate the value of 1 ratio unit. 1 unit = 1,47,000 / 7 = Rs. 21,000. Step 5: Calculate B's share of profit. B's share = 6 units. B's profit = 6 * 21,000 = Rs. 1,26,000.
2
In 2020, the income ratio of A to B was 5:4. A's income ratio from 2020 to 2021 was 4:5, and B's income ratio from 2020 to 2021 was 2:3. If their combined income in 2021 was Rs. 7,05,600, what was B's income in 2021?
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Solution: Step 1: Represent incomes in 2020 based on their ratio: Let A's 2020 income be 5k, and B's 2020 income be 4k. Step 2: Relate A's 2020 and 2021 incomes using the ratio 4:5: If 4 parts of A's income correspond to 5k (2020 income), then 1 part is 5k/4. A's 2021 income = 5 parts * (5k/4) = 25k/4. Step 3: Relate B's 2020 and 2021 incomes using the ratio 2:3: If 2 parts of B's income correspond to 4k (2020 income), then 1 part is 4k/2 = 2k. B's 2021 income = 3 parts * (2k) = 6k. Step 4: Formulate an equation using the total income in 2021: A's 2021 income + B's 2021 income = 7,05,600 25k/4 + 6k = 7,05,600 (25k + 24k) / 4 = 7,05,600 49k / 4 = 7,05,600. Step 5: Solve for k: 49k = 7,05,600 * 4 49k = 2,822,400 k = 2,822,400 / 49 = 57600. Step 6: Calculate B's income in 2021: B's 2021 income = 6k = 6 * 57600 = Rs. 3,45,600.
3
Express the ratio 4^3.5 : 2^5 in its simplest form.
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Solution: Step 1: Rewrite the base of the first term so it matches the second term. Since 4 = 2², then 4^3.5 = (2²)^3.5. Step 2: Apply the exponent rule (a^m)^n = a^(m*n): (2²)^3.5 = 2^(2 * 3.5) = 2^7. Step 3: The ratio becomes 2^7 : 2^5. Step 4: Divide both sides of the ratio by the lowest common power, 2^5: * 2^7 / 2^5 = 2^(7-5) = 2² = 4 * 2^5 / 2^5 = 1 Step 5: The simplified ratio is 4 : 1.
4
Given that the ratio of x to y is 5:2, determine the ratio of (8x + 9y) to (8x + 2y).
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Solution: Step 1: From the given ratio `x : y = 5 : 2`, we can assume `x = 5k` and `y = 2k`. Since we are finding a ratio, the common factor 'k' will cancel out, so we can directly substitute `x = 5` and `y = 2`. Step 2: Substitute these assumed values into the expression `(8x + 9y) : (8x + 2y)`. - Calculate the first part: `8x + 9y = 8(5) + 9(2) = 40 + 18 = 58`. - Calculate the second part: `8x + 2y = 8(5) + 2(2) = 40 + 4 = 44`. Step 3: Form the new ratio and simplify it. - The new ratio is 58 : 44. - Divide both sides by their greatest common divisor, which is 2. - 58 / 2 = 29 - 44 / 2 = 22 - The simplified ratio is 29 : 22.
5
In a school, 5/12 of the students are girls, and the remaining are boys. Among the boys, 4/7 are under 14 years of age, and among the girls, 2/5 are 14 years or above 14 years of age. If the number of students under 14 years of age is 1120, what is the total number of students in the school?
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Solution: Step 1: Represent the total number of students with a common multiple. Let the total students be 12 units (based on the first fraction 5/12). Step 2: Determine the number of girls and boys. Girls = (5/12) * 12 units = 5 units. Boys = 12 units - 5 units = 7 units. Step 3: Calculate the number of students under 14 years of age. Boys under 14 years = (4/7) * 7 units = 4 units. Girls who are 14 years or above = (2/5) * 5 units = 2 units. Therefore, Girls under 14 years = Total girls - Girls 14+ years = 5 units - 2 units = 3 units. Step 4: Find the total number of students under 14 years of age in terms of units. Total students under 14 years = (Boys under 14) + (Girls under 14) Total students under 14 years = 4 units + 3 units = 7 units. Step 5: Relate the calculated units to the given number of students. 7 units = 1120 students. 1 unit = 1120 / 7 = 160 students. Step 6: Calculate the total number of students in the school. Total students = 12 units = 12 * 160 = 1920.
6
A total of Rs. 1250 is to be distributed among A, B, C, and D. The combined share of B and D is 14/11 of the combined share of A and C. D's share is half of A's share. C's share is 1.2 times A's share. What are the individual shares of A, B, C, and D?
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Solution: Step 1: Set up initial equations based on the problem statement. Total sum: A + B + C + D = 1250 (Equation 1) Combined share ratio: (B + D) / (A + C) = 14 / 11 (Equation 2) Share of D in terms of A: D = A / 2 (Equation 3) Share of C in terms of A: C = 1.2A = (6/5)A (Equation 4) Step 2: Use Equation 2 with total sum. (A + C) + (B + D) = 1250. Given (B+D) / (A+C) = 14/11, we can say A+C is 11 parts and B+D is 14 parts, making total 25 parts. 25 parts = 1250 => 1 part = 50. Therefore, A + C = 11 * 50 = 550 and B + D = 14 * 50 = 700. Step 3: Substitute Equation 3 and Equation 4 into the sum of A and C: A + (6/5)A = 550 (5A + 6A) / 5 = 550 11A / 5 = 550 11A = 2750 A = 2750 / 11 = 250 Step 4: Calculate shares of C and D using A's share: C = (6/5) * 250 = 6 * 50 = 300 D = A / 2 = 250 / 2 = 125 Step 5: Calculate B's share using the total sum: B = 1250 - (A + C + D) B = 1250 - (250 + 300 + 125) B = 1250 - 675 = 575 Step 6: State the shares of A, B, C, and D: A = Rs. 250, B = Rs. 575, C = Rs. 300, D = Rs. 125.
7
Given A:B = 7:9 and B:C = 5:4, find the combined ratio A:B:C.
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Solution: Step 1: List the given ratios: A : B = 7 : 9 B : C = 5 : 4 Step 2: To combine these ratios, make the common term 'B' consistent. Find the LCM of the 'B' values (9 and 5). LCM(9, 5) = 45 Step 3: Adjust each ratio so that the 'B' value becomes 45: For A : B = 7 : 9, multiply both parts by (45/9 = 5): A : B = (7 * 5) : (9 * 5) = 35 : 45 For B : C = 5 : 4, multiply both parts by (45/5 = 9): B : C = (5 * 9) : (4 * 9) = 45 : 36 Step 4: Now that 'B' is consistent in both adjusted ratios, combine them: A : B : C = 35 : 45 : 36
8
A box contains nine light bulbs, with 4 of them being defective. If four bulbs are chosen randomly from the box, what is the probability that all four selected bulbs are defective?
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Solution: Step 1: Determine the total number of bulbs: 9. Step 2: Identify the number of defective bulbs: 4. Step 3: Calculate the total number of ways to choose 4 bulbs from 9. This is C(9, 4) = (9 * 8 * 7 * 6) / (4 * 3 * 2 * 1) = 126. Step 4: Calculate the number of ways to choose 4 defective bulbs from the 4 available defective bulbs. This is C(4, 4) = 1. Step 5: Calculate the probability that all four bulbs chosen are defective: (Favorable outcomes) / (Total outcomes) = 1 / 126.
9
If 60% of quantity A is equal to 3/4 of quantity B, determine the ratio of A to B (A:B).
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Solution: Step 1: Write the given relationship as an equation: 60% of A = (3/4) of B. Step 2: Convert 60% to a fraction: 60/100 = 3/5. Step 3: Substitute the fraction into the equation: (3/5)A = (3/4)B. Step 4: To find the ratio A:B, divide both sides by B and multiply by the reciprocal of (3/5), which is 5/3. Step 5: A/B = (3/4) * (5/3). Step 6: Cancel out the common factor 3: A/B = 5/4. Step 7: Therefore, the ratio A : B is 5 : 4.
10
A total of Rs. 600 is distributed among A, B, and C. The distribution is such that an amount equal to (2/5 of A's share + Rs. 40), (2/7 of B's share + Rs. 20), and (9/17 of C's share + Rs. 10) are all equal. Determine A's share.
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Solution: Step 1: Let A's share be A, B's share be B, and C's share be C. The total amount is A + B + C = 600. Step 2: Set up the given equality: (2/5)A + 40 = (2/7)B + 20 = (9/17)C + 10. Step 3: Let k be the common equal value. (2/5)A + 40 = k => (2/5)A = k - 40 => A = (5/2)(k - 40) (2/7)B + 20 = k => (2/7)B = k - 20 => B = (7/2)(k - 20) (9/17)C + 10 = k => (9/17)C = k - 10 => C = (17/9)(k - 10) Step 4: Substitute A, B, C into the total sum equation: (5/2)(k - 40) + (7/2)(k - 20) + (17/9)(k - 10) = 600 Step 5: Expand and simplify: (5k/2 - 100) + (7k/2 - 70) + (17k/9 - 170/9) = 600 6k - 170 + 17k/9 - 170/9 = 600 6k + 17k/9 = 600 + 170 + 170/9 (54k + 17k)/9 = 770 + 170/9 71k/9 = (6930 + 170)/9 71k/9 = 7100/9 71k = 7100 k = 100 Step 6: Calculate A's share using the value of k: A = (5/2)(k - 40) = (5/2)(100 - 40) = (5/2)(60) = 5 * 30 = Rs. 150.
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The ratio of a school ground's length to its width is 5:2. If the width measures 40m, what is the length?
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Solution: Step 1: Let the length of the school ground be L and the width be W. The given ratio is L : W = 5 : 2. Step 2: This means L can be represented as 5x and W as 2x, where x is a common multiplier. Step 3: The width is given as 40m. So, set 2x equal to 40: 2x = 40 Step 4: Solve for x: x = 40 / 2 = 20. Step 5: Calculate the length L using the value of x: L = 5x = 5 * 20 = 100m.
12
A box contains 210 coins, consisting only of one-rupee and fifty-paise coins. If the ratio of their respective values is 13 : 11, how many one-rupee coins are there?
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Solution: Step 1: Understand the value and quantity relationship for each denomination. One-rupee coin: Value = 1 Rupee = 100 paise. Fifty-paise coin: Value = 0.5 Rupee = 50 paise. Step 2: Convert the given value ratio to a ratio of the number of coins. Let the value of one-rupee coins be 13V and the value of fifty-paise coins be 11V. Number of one-rupee coins = Value / (Value per coin) = 13V / 1 = 13V Number of fifty-paise coins = Value / (Value per coin) = 11V / 0.5 = 22V So, the ratio of the number of one-rupee coins to fifty-paise coins is 13V : 22V, which simplifies to 13 : 22. Step 3: Use the total number of coins to find the actual number of each type. Total coins = 210 Let the number of one-rupee coins be 13x and the number of fifty-paise coins be 22x. Total coins = 13x + 22x = 35x Step 4: Solve for x. 35x = 210 x = 210 / 35 x = 6 Step 5: Calculate the number of one-rupee coins. Number of one-rupee coins = 13x = 13 * 6 = 78
13
A total sum of money is distributed among A, B, C, and D in the ratio 3:4:9:10. If C's share is Rs. 2580 more than B's share, calculate the combined total amount received by A and D.
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Solution: Step 1: Let the shares of A, B, C, and D be 3x, 4x, 9x, and 10x respectively. Step 2: The problem states that C's share is Rs. 2580 more than B's share. Set up the equation based on this difference: 9x - 4x = 2580 5x = 2580 Step 3: Solve for x: x = 2580 / 5 = 516. Step 4: The question asks for the total amount A and D receive together. Calculate the combined ratio parts for A and D: Amount (A + D) = 3x + 10x = 13x. Step 5: Substitute the value of x to find the combined amount: Amount (A + D) = 13 * 516 = Rs. 6708.
14
The incomes of two individuals are in the ratio 5:3, and their expenditures are in the ratio 9:5. If their individual savings are Rs. 2600 and Rs. 1800, respectively, calculate their actual incomes.
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Solution: Step 1: Let the incomes of the two persons be 5x and 3x. Step 2: Let their expenditures be 9y and 5y. Step 3: For the first person, Income - Expenditure = Savings: 5x - 9y = 2600 (Equation 1). Step 4: For the second person, Income - Expenditure = Savings: 3x - 5y = 1800 (Equation 2). Step 5: Multiply Equation 1 by 3 and Equation 2 by 5 to eliminate 'x': (Equation 1 * 3): 15x - 27y = 7800 (Equation 3) (Equation 2 * 5): 15x - 25y = 9000 (Equation 4) Step 6: Subtract Equation 3 from Equation 4: (15x - 25y) - (15x - 27y) = 9000 - 7800 => 2y = 1200 => y = 600. Step 7: Substitute y = 600 into Equation 2: 3x - 5(600) = 1800 => 3x - 3000 = 1800 => 3x = 4800 => x = 1600. Step 8: Calculate the income of the first person: 5x = 5 * 1600 = Rs. 8000. Step 9: Calculate the income of the second person: 3x = 3 * 1600 = Rs. 4800. Step 10: The incomes are Rs. 8000 and Rs. 4800.
15
Express the ratio of 3 hours to 1 day in its simplest form.
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Solution: Step 1: Convert all quantities to the same unit. Since 1 day = 24 hours, convert 1 day to hours. Step 2: The ratio becomes 3 hours : 24 hours. Step 3: Simplify the ratio by dividing both numbers by their greatest common divisor, which is 3. 3/3 : 24/3. Step 4: The simplest form of the ratio is 1 : 8.
16
What is the probability of drawing four cards of the same suit when selecting four cards randomly from a standard deck of 52 playing cards?
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Solution: Step 1: Calculate the total number of ways to draw 4 cards from a standard deck of 52 cards. This is given by C(52, 4) = (52 * 51 * 50 * 49) / (4 * 3 * 2 * 1) = 270,725. Step 2: Understand that there are 4 suits (clubs, spades, hearts, diamonds), each with 13 cards. Step 3: Calculate the number of ways to draw 4 cards of a specific suit (e.g., all 4 clubs). This is C(13, 4) = (13 * 12 * 11 * 10) / (4 * 3 * 2 * 1) = 715. Step 4: Since there are 4 different suits, the number of ways to draw all four cards of the same suit is 4 times the number of ways to draw 4 cards of a specific suit. So, 4 * C(13, 4) = 4 * 715 = 2860. Step 5: Calculate the probability: (Favorable outcomes) / (Total outcomes) = 2860 / 270725. Step 6: Simplify the fraction: 2860/270725 = 44/4165.
17
Salaries of A, B, and C are in the proportion 2:3:5. If their salaries are increased by 15%, 10%, and 20% respectively, what will be the new ratio of their salaries?
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Solution: Step 1: Let the initial salaries of A, B, and C be 2k, 3k, and 5k, respectively, where 'k' is a constant. Step 2: Calculate A's new salary after a 15% increment: New salary of A = 2k * (1 + 15/100) = 2k * (115/100) = 2.3k. Step 3: Calculate B's new salary after a 10% increment: New salary of B = 3k * (1 + 10/100) = 3k * (110/100) = 3.3k. Step 4: Calculate C's new salary after a 20% increment: New salary of C = 5k * (1 + 20/100) = 5k * (120/100) = 6k. Step 5: Form the new ratio of their salaries: New Ratio = 2.3k : 3.3k : 6k. Step 6: Divide by 'k' and multiply by 10 to remove decimals and express the ratio in simplest integer form: New Ratio = 23 : 33 : 60. Step 7: The new ratio of their salaries is 23:33:60.
18
The current ages of Individual X and Individual Y are 59 and 37 years, respectively. What was the ratio of Individual Y's age to Individual X's age 13 years ago?
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Solution: Step 1: Individual X's age 13 years ago = 59 - 13 = 46 years Step 2: Individual Y's age 13 years ago = 37 - 13 = 24 years Step 3: Ratio of Y's age to X's age = 24:46 Step 4: Simplify the ratio: 12:23
19
The ratio of two numbers is 7:4, and their highest common factor (HCF) is 4. What is their lowest common multiple (LCM)?
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Solution: Step 1: Let the numbers be 7x and 4x. Step 2: HCF of the numbers = x = 4 Step 3: Numbers are 7x = 28 and 4x = 16 Step 4: LCM of 28 and 16 = 112
20
A, B, and C invested capital in a business with ratios of 1/2 : 1/3 : 1/5. After 4 months, A doubled his investment, and after 6 months, B halved his investment. If the total annual profit is Rs. 34650, what is the profit share for each partner?
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Solution: Step 1: Convert the fractional investment ratio to an integer ratio. Given ratio 1/2 : 1/3 : 1/5. Multiply by the LCM of 2, 3, 5 (which is 30) to clear fractions: (1/2)*30 : (1/3)*30 : (1/5)*30 = 15 : 10 : 6. Let their initial investments be 15x, 10x, and 6x respectively. Step 2: Calculate each partner's total effective capital for the year (12 months). A's investment: 15x for 4 months, then 2 * 15x = 30x for 12 - 4 = 8 months. A's effective capital = (15x * 4) + (30x * 8) = 60x + 240x = 300x. B's investment: 10x for 6 months, then 10x / 2 = 5x for 12 - 6 = 6 months. B's effective capital = (10x * 6) + (5x * 6) = 60x + 30x = 90x. C's investment: 6x for 12 months. C's effective capital = 6x * 12 = 72x. Step 3: Determine the profit sharing ratio (ratio of effective capitals). Ratio A : B : C = 300x : 90x : 72x. Divide by 6x: 50 : 15 : 12. Step 4: Calculate the total number of ratio parts. Total ratio parts = 50 + 15 + 12 = 77 parts. Step 5: Calculate the value of one ratio part. Total profit = Rs. 34650. Value of 1 part = 34650 / 77 = Rs. 450. Step 6: Calculate the individual profit share for A, B, and C. Profit of A = 50 parts * 450 = Rs. 22500. Profit of B = 15 parts * 450 = Rs. 6750. Profit of C = 12 parts * 450 = Rs. 5400.
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