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1
The total number of boys in a school is 16% greater than the total number of girls. What is the ratio of the total number of boys to the total number of girls in the school?
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Solution: Step 1: Let the total number of girls in the school be G. Step 2: The total number of boys (B) is 16% more than the girls. * B = G + (16/100)G = G + 0.16G = 1.16G. Step 3: Form the ratio of boys to girls: B : G. * B / G = 1.16G / G = 1.16. Step 4: Convert the decimal 1.16 to a fraction: 116/100. Step 5: Simplify the fraction/ratio by dividing both the numerator and denominator by their greatest common divisor, which is 4. * 116 / 4 = 29 * 100 / 4 = 25 * So, the ratio is 29/25. Step 6: The respective ratio of boys to girls is 29:25.
2
In an examination, 5% of applicants were ineligible. Of the eligible candidates, 85% were from the general category. If 4275 eligible candidates belonged to other categories, how many candidates applied for the examination in total?
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Solution: Step 1: Let the total number of applicants be 'A'. Step 2: Calculate the percentage of eligible candidates. Eligible candidates = 100% - 5% (ineligible) = 95% of A = 0.95A Step 3: Calculate the percentage of eligible candidates belonging to 'other categories'. Candidates from other categories = 100% - 85% (general category) = 15% of eligible candidates. Step 4: Form an equation using the given number of candidates from other categories. 15% of the eligible candidates = 4275 15% of (0.95A) = 4275 (15 / 100) * 0.95A = 4275 0.15 * 0.95A = 4275 0.1425A = 4275 Step 5: Solve for 'A'. A = 4275 / 0.1425 A = 30000 The total number of candidates who applied for the examinations was 30,000.
3
Rajeev purchases goods valued at Rs. 6650. He receives a 6% rebate on this amount. After applying the rebate, a 10% sales tax is levied on the reduced price. What is the total amount Rajeev must pay for the goods?
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Solution: Step 1: Original price of goods = Rs. 6650. Step 2: Calculate rebate amount: 6% of Rs. 6650 = (6/100) * 6650 = Rs. 399. Step 3: Price after rebate = Original price - Rebate amount = 6650 - 399 = Rs. 6251. Step 4: Calculate sales tax amount: 10% of price after rebate = 10% of Rs. 6251 = (10/100) * 6251 = Rs. 625.10. Step 5: Final amount to pay = Price after rebate + Sales tax amount. Step 6: Final amount = 6251 + 625.10 = Rs. 6876.10.
4
A hotel provides a 12% discount for booking two or more rooms and an extra 5% discount for payments made with any SBI card. Rakesh booked two rooms for one day, each costing Rs. 1,500 per day. If he paid with an SBI Silver Card upon checkout, what was the total amount he had to pay?
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Solution: Step 1: Calculate the total initial cost without discounts. Cost per room = Rs. 1,500. Number of rooms = 2. Total initial cost = 2 × 1500 = Rs. 3,000. Step 2: Apply the first discount (12% for booking 2 or more rooms). Amount after 12% discount = 3000 × (1 - 12/100) = 3000 × 0.88 = Rs. 2,640. Step 3: Apply the second discount (5% for SBI card payment) on the discounted amount. Amount after 5% additional discount = 2640 × (1 - 5/100) = 2640 × 0.95. Amount = 2640 × 0.95 = Rs. 2,508. Rakesh had to pay Rs. 2,508.
5
Rs. 9800 is invested, partly in a 9% stock at Rs. 75 and partly in a 10% stock at Rs. 80, such that both investments yield an equal amount of income. How much was invested in the 9% stock?
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Solution: Step 1: Let the investment in the 9% stock be Rs. X. Then, the investment in the 10% stock is Rs. (9800 - X). Step 2: Calculate the income from the 9% stock. For every Rs. 75 invested, the income is Rs. 9. Income from X = X * (9/75). Step 3: Calculate the income from the 10% stock. For every Rs. 80 invested, the income is Rs. 10. Income from (9800 - X) = (9800 - X) * (10/80). Step 4: Since the incomes are equal, set up the equation: X * (9/75) = (9800 - X) * (10/80). Step 5: Simplify the fractions: X * (3/25) = (9800 - X) * (1/8). Step 6: Multiply both sides by 200 (LCM of 25 and 8) to clear denominators: 8 * 3X = 25 * (9800 - X). Step 7: Solve for X: 24X = 245000 - 25X => 49X = 245000 => X = 245000 / 49 = Rs. 5000. Step 8: The investment in the 9% stock is Rs. 5000.
6
Given that 5/9 of a first number equals 25% of a second number, and the second number is 1/4 of a third number, calculate 30% of the first number if the third number is 2960.
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Solution: Step 1: Determine the value of the second number. It is 1/4 of the third number. Second number = (1/4) * 2960 = 740. Step 2: Set up the relationship between the first and second number: 5/9 of the first number = 25% of the second number. 25% of 740 = (25/100) * 740 = (1/4) * 740 = 185. Step 3: Now we have: (5/9) * First number = 185. Step 4: Solve for the First number: First number = 185 * (9/5). First number = 37 * 9 = 333. Step 5: Calculate 30% of the first number. 30% of 333 = (30/100) * 333. = 0.30 * 333. Step 6: Perform the multiplication: 0.30 * 333 = 99.9.
7
Calculate the value of: 125% of 860 + 75% of 480.
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Solution: Step 1: Calculate '125% of 860'. 125% can be written as 1.25 or 5/4. 1.25 * 860 = 1075. Step 2: Calculate '75% of 480'. 75% can be written as 0.75 or 3/4. 0.75 * 480 = 360. Step 3: Add the results from Step 1 and Step 2: 1075 + 360 = 1435. Step 4: The value is 1435.
8
Gauri made purchases worth Rs. 25 at a stationer's. Out of this total, 30 paise was incurred as sales tax on taxable purchases. If the tax rate was 6%, what was the cost of the items that were tax-free?
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Solution: Step 1: Convert the sales tax from paise to Rupees: 30 paise = Rs. 0.30. Step 2: Let the cost of the taxable purchases (before tax) be 'x' Rupees. Step 3: The sales tax paid (Rs. 0.30) represents 6% of the taxable purchases. Step 4: Formulate the equation: 6% of x = 0.30. Step 5: (6/100) * x = 0.30. Step 6: Solve for x: x = (0.30 * 100) / 6 = 30 / 6 = 5. Step 7: So, the cost of taxable purchases was Rs. 5. Step 8: The total amount paid by Gauri was Rs. 25. Step 9: The combined cost of taxable items (including tax) is Rs. 5 (cost of items) + Rs. 0.30 (tax) = Rs. 5.30. Step 10: To find the cost of tax-free items, subtract the total cost of taxable items from the total amount paid: Cost of tax-free items = Rs. 25 - Rs. 5.30 = Rs. 19.70.
9
Person X's income is 25% higher than Person Y's. What percentage of X's income is Y's income?
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Solution: Step 1: Let Y's income = 100 units Step 2: X's income = 100 + 25% * 100 = 125 units Step 3: Y's income as percentage of X's = (100 / 125) * 100 = 80%
10
A city's population is 35,000. If the number of men increases by 6% and the number of women by 4%, the total population becomes 36,760. What was the initial number of women?
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Solution: Step 1: Let the initial number of men be M and the initial number of women be W. We know the total initial population: M + W = 35000 (Equation 1). Step 2: Calculate the total increase in population: Total increase = New Population - Old Population = 36760 - 35000 = 1760. Step 3: Express the increase in terms of men and women: Increase in men = 6% of M = 0.06M. Increase in women = 4% of W = 0.04W. The sum of these increases equals the total population increase: 0.06M + 0.04W = 1760 (Equation 2). Step 4: From Equation 1, express M in terms of W: M = 35000 - W. Substitute this into Equation 2: 0.06(35000 - W) + 0.04W = 1760 2100 - 0.06W + 0.04W = 1760 2100 - 0.02W = 1760 Step 5: Solve for W: 0.02W = 2100 - 1760 0.02W = 340 W = 340 / 0.02 W = 17000. Thus, the initial number of women was 17,000.
11
If 50% of (x - y) equals 30% of (x + y), what percentage of x is y?
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Solution: Step 1: Write down the given equation in mathematical form. 50% of (x - y) = 30% of (x + y) (50/100)(x - y) = (30/100)(x + y) Step 2: Simplify the equation. (1/2)(x - y) = (3/10)(x + y) Step 3: Eliminate the denominators by multiplying both sides by 10. 5(x - y) = 3(x + y) Step 4: Expand both sides of the equation. 5x - 5y = 3x + 3y Step 5: Rearrange the terms to group x terms and y terms. 5x - 3x = 3y + 5y 2x = 8y Step 6: Express y in terms of x. y = 2x/8 = x/4. Step 7: Calculate what percentage of x is y. (y/x) × 100% = ( (x/4) / x ) × 100% = (1/4) × 100% = 25%.
12
A town's population grows by 4% annually. If the initial population was 50000, what will its population be after two years?
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Solution: Step 1: Initial population = 50000. Step 2: Annual growth rate = 4% = 0.04. Step 3: Number of years = 2. Step 4: Using the compound growth formula: Final Population = Initial Population * (1 + Rate)^Number of Years. Step 5: Population after 2 years = 50000 * (1 + 0.04)^2. Step 6: Population after 2 years = 50000 * (1.04)^2 = 50000 * 1.0816. Step 7: Population after 2 years = 54080. Step 8: The population after two years will be 54080.
13
A number is first decreased by 10% and subsequently increased by 10%. The resulting number is 50 less than the original number. What was the original number?
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Solution: Step 1: Let the original number be X. Step 2: Calculate the number after it is decreased by 10%. Number after 10% decrease = X × (1 - 10/100) = X × 0.9 = 0.9X. Step 3: Calculate the number after it is then increased by 10%. Final number = (0.9X) × (1 + 10/100) = 0.9X × 1.1 = 0.99X. Step 4: Set up an equation based on the given difference from the original number. The final number (0.99X) is 50 less than the original number (X). X - 0.99X = 50. Step 5: Solve for X. 0.01X = 50. X = 50 / 0.01. X = 5000. Alternatively, using the net percentage change formula for successive increase/decrease by the same percentage (p): Net change = -(p^2/100)%. Net change = -(10^2/100)% = -(100/100)% = -1%. So, the final number is 1% less than the original number. This 1% reduction corresponds to 50. 1% of Original Number = 50. (1/100) × Original Number = 50. Original Number = 50 × 100 = 5000.
14
A toy merchant offers a 25% discount on balls, each costing Rs. 32. To achieve a total rebate of Rs. 40, how many balls must be purchased?
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Solution: Step 1: Calculate the rebate amount on a single ball. Rebate per ball = 25% of Rs. 32 = (25 / 100) * 32 = Rs. 8 Step 2: Determine the number of balls required to get a total rebate of Rs. 40. Number of balls = Total desired rebate / Rebate per ball = Rs. 40 / Rs. 8 = 5 balls
15
Calculate the result of: 140% of 56 + 56% of 140.
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Solution: Step 1: Understand the property: X% of Y is equal to Y% of X. Therefore, 56% of 140 is the same as 140% of 56. Step 2: The expression can be rewritten as: (140% of 56) + (140% of 56) = 2 * (140% of 56). Step 3: Calculate 140% of 56: (140/100) * 56 = 1.4 * 56 = 78.4. Step 4: Multiply the result by 2: 2 * 78.4 = 156.8. Step 5: Alternatively, calculate each term separately: - 140% of 56 = 78.4. - 56% of 140 = 78.4. Step 6: Add the two results: 78.4 + 78.4 = 156.8. Step 7: The final answer is 156.8. Since this value is not among the given options, the correct choice is 'None of these'.
16
An institute's student body consists of 60% boys and the remaining are girls. 15% of the boys and 7.5% of the girls receive a full fee waiver. If 90 students in total receive this fee waiver, determine the total number of students eligible for a 50% fee concession, given that 50% of those who do not receive a full fee waiver are eligible for the half fee concession?
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Solution: Step 1: Assume total students = 100. Boys = 60, Girls = 40. Step 2: Boys getting fee waiver = 15% of 60 = 9. Step 3: Girls getting fee waiver = 7.5% of 40 = 3. Step 4: Total fee waiver students (assumed) = 9 + 3 = 12. Step 5: Given 90 students actually received fee waiver, so 12 units = 90 students. Therefore, 1 unit = 90/12 = 7.5 students. Step 6: Total number of students = 100 units * 7.5 = 750 students. Step 7: Students not getting fee waiver = 750 total - 90 fee waiver = 660 students. Step 8: Students eligible for 50% concession = 50% of those not getting fee waiver = 50% of 660 = 330 students.
17
If adding 8% of a number to itself results in 810, what is the original number?
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Solution: Step 1: Let the number = x Step 2: Equation: x + 0.08x = 810 Step 3: Combine like terms: 1.08x = 810 Step 4: Solve for x: x = 810 / 1.08 Step 5: Calculate: x = 750
18
If the price of a commodity increases by 50%, what fraction of its consumption must be reduced to maintain the original expenditure?
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Solution: Step 1: Assume the initial price of the commodity is 100 units. Step 2: With a 50% increase, the new price becomes 100 + 50 = 150 units. Step 3: To keep the expenditure constant (say, at 100 units), the consumption must be reduced by an amount that negates the price increase. Step 4: The required reduction in consumption (in terms of value relative to the new price) is 150 - 100 = 50 units. Step 5: Express this reduction as a fraction of the *new* price: 50 / 150 = 1/3. Thus, consumption must be reduced by 1/3.
19
In a school, 20% of the students are under 8 years old. There are 48 students who are exactly 8 years old. The number of students older than 8 years is two-thirds of the number of students who are 8 years old. Calculate the total number of students in the school.
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Solution: Step 1: Identify the number of students who are exactly 8 years old: 48 students. Step 2: Calculate the number of students above 8 years of age. This is (2/3) of the number of students of 8 years of age: (2/3) * 48 = 32 students. Step 3: Calculate the total number of students who are 8 years or older: 48 (exactly 8 years) + 32 (above 8 years) = 80 students. Step 4: The problem states that 20% of students are below 8 years of age. This implies that the remaining students (8 years or older) constitute 100% - 20% = 80% of the total student population. Step 5: Let the total number of students in the school be 'X'. So, 80% of X = 80. Step 6: Set up the equation: (80/100) * X = 80. Step 7: Solve for X: X = (80 * 100) / 80. Step 8: X = 100. Step 9: The total number of students in the school is 100.
20
An alloy is created by mixing 5 kg of metal A with 20 kg of metal B. What is the percentage of metal A in this alloy?
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Solution: Step 1: Calculate the total weight of the alloy. Total weight = Weight of metal A + Weight of metal B = 5 kg + 20 kg = 25 kg. Step 2: Calculate the percentage of metal A in the alloy. Percentage of metal A = (Weight of metal A / Total weight of alloy) * 100 Percentage of metal A = (5 / 25) * 100 = (1/5) * 100 = 20%.
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