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Question 1 / 20
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1
A town's population is 8500. It grows by 20% in the first year and an additional 25% in the second year. What will the town's population be after 2 years?
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Solution: Step 1: Calculate the population after the first year. Population after 1st year = 8500 × (1 + 20/100) = 8500 × 1.20 = 10200. Step 2: Calculate the population after the second year, based on the new population. Population after 2nd year = 10200 × (1 + 25/100) = 10200 × 1.25 = 12750. Alternatively, using a direct compound formula: Final Population = Initial Population × (1 + P1/100) × (1 + P2/100) Final Population = 8500 × (1 + 20/100) × (1 + 25/100) Final Population = 8500 × (120/100) × (125/100) Final Population = 8500 × 1.2 × 1.25 Final Population = 8500 × 1.5 = 12750.
2
In a two-candidate election, 20% of the total votes cast were invalid. The winning candidate received 55% of the total valid votes. If the overall total number of votes was 7500, how many valid votes did the other candidate receive?
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Solution: Step 1: Calculate the total number of valid votes. Since 20% of the votes were invalid, 100% - 20% = 80% of the votes were valid. Step 2: Total valid votes = 80% of 7500 = (80/100) * 7500 = 0.80 * 7500 = 6000 votes. Step 3: The first candidate received 55% of the total valid votes. Therefore, the other candidate received 100% - 55% = 45% of the total valid votes. Step 4: Calculate the number of valid votes the other candidate received: 45% of 6000. Step 5: Number of votes = (45/100) * 6000 = 0.45 * 6000 = 2700 votes.
3
Determine the single equivalent discount percentage for three consecutive discounts of 6%, 15%, and 14%.
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Solution: Step 1: Calculate the equivalent discount for the first two discounts (x = 6%, y = 15%). Formula: Equivalent Discount = (x + y - (xy/100))%. Equivalent Discount (6%, 15%) = (6 + 15 - (6 * 15 / 100))% = (21 - 90/100)% = (21 - 0.9)% = 20.1%. Step 2: Calculate the equivalent discount for the result from Step 1 (x = 20.1%) and the third discount (y = 14%). Equivalent Discount (20.1%, 14%) = (20.1 + 14 - (20.1 * 14 / 100))%. Equivalent Discount = (34.1 - 281.4 / 100)% = (34.1 - 2.814)% = 31.286%.
4
A papaya tree was planted two years ago and has been growing at an annual rate of 20%. If its current height is 540 cm, what was its height when it was initially planted?
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Solution: Step 1: Let the height of the tree when it was planted be H0. Step 2: The tree grows at 20% per year for 2 years. The formula for compound growth is H_final = H0 * (1 + rate)^time. So, 540 = H0 * (1 + 0.20)^2 540 = H0 * (1.20)^2 540 = H0 * 1.44. Step 3: Solve for H0. H0 = 540 / 1.44 H0 = 375 cm.
5
A man consumes 25 kg of rice and 9 kg of wheat monthly. The price of rice is 20% of the price of wheat, and his total monthly expenditure on these two items is Rs. 350. If the price of wheat increases by 20%, what percentage reduction in rice consumption is needed to maintain the total expenditure at Rs. 350, assuming the price of rice and the consumption of wheat remain constant?
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Solution: Step 1: Let the price of wheat be P_w per kg. The price of rice is P_r = 0.20 * P_w. Step 2: Set up the initial expenditure equation: (25 kg * P_r) + (9 kg * P_w) = Rs. 350. Step 3: Substitute P_r: (25 * 0.20 * P_w) + (9 * P_w) = 350 => 5P_w + 9P_w = 350 => 14P_w = 350. Step 4: Solve for P_w: P_w = 350 / 14 = Rs. 25 per kg. Step 5: Calculate P_r: P_r = 0.20 * 25 = Rs. 5 per kg. Step 6: Calculate the new price of wheat after a 20% increase: New P_w = 25 * (1 + 0.20) = 25 * 1.20 = Rs. 30 per kg. Step 7: The price of rice (Rs. 5/kg) and wheat consumption (9 kg) remain constant. Let the new rice consumption be N kg. Step 8: Set up the new expenditure equation: (N kg * Rs. 5/kg) + (9 kg * Rs. 30/kg) = Rs. 350. Step 9: 5N + 270 = 350. Step 10: Solve for N: 5N = 350 - 270 = 80 => N = 80 / 5 = 16 kg. Step 11: Initial rice consumption was 25 kg, new is 16 kg. Reduction = 25 - 16 = 9 kg. Step 12: Percentage reduction in rice consumption = (9 kg / 25 kg) * 100% = 36%.
6
A number undergoes four consecutive changes: first, it increases by 40%; then it decreases by 25%; next, it increases by 15%; and finally, it decreases by 20%. Calculate the overall net percentage increase or decrease in the number.
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Solution: Step 1: Convert each percentage change into a multiplicative factor: Increased by 40%: 1 + 0.40 = 1.40 Decreased by 25%: 1 - 0.25 = 0.75 Increased by 15%: 1 + 0.15 = 1.15 Decreased by 20%: 1 - 0.20 = 0.80 Step 2: Calculate the net effect by multiplying all the factors: Net multiplier = 1.40 × 0.75 × 1.15 × 0.80 Net multiplier = 1.05 × 0.80 × 1.15 (1.40 * 0.75 = 1.05) Net multiplier = 0.84 × 1.15 Net multiplier = 0.966. Step 3: Determine the net percentage change from the net multiplier. Percentage change = (Net multiplier - 1) × 100 Percentage change = (0.966 - 1) × 100 Percentage change = -0.034 × 100 = -3.4%. The negative sign indicates a decrease. Therefore, there is a net decrease of 3.4% in the number.
7
In an examination, 35% of candidates failed Mathematics and 25% failed English. If 10% failed both Mathematics and English, what percentage of candidates passed both subjects?
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Solution: Step 1: Let P(M) be the percentage failed in Mathematics (35%) and P(E) be the percentage failed in English (25%). Step 2: Let P(M ∩ E) be the percentage failed in both (10%). Step 3: Percentage of candidates who failed in at least one subject, P(M U E) = P(M) + P(E) - P(M ∩ E). Step 4: P(M U E) = 35% + 25% - 10% = 60% - 10% = 50%. Step 5: This 50% represents candidates who failed in one or both subjects. Step 6: The percentage of candidates who passed in both subjects is the complement of those who failed at least one subject. Step 7: Percentage passed in both = 100% - P(M U E) = 100% - 50% = 50%. Step 8: Therefore, 50% of candidates passed in both subjects.
8
A household item's price rises by 25%. By what percentage must consumption be reduced to maintain the same expenditure?
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Solution: Step 1: Let original price = P, original consumption = C Step 2: Original expenditure = P * C Step 3: New price = 1.25P Step 4: Let new consumption = C(1 - x), where x is reduction percentage Step 5: New expenditure = 1.25P * C(1 - x) Step 6: Set up equation: 1.25P * C(1 - x) = P * C Step 7: Simplify: 1.25(1 - x) = 1 Step 8: Solve: 1 - x = 1/1.25 = 0.8 Step 9: x = 0.2 or 20%
9
If a person's net income is Rs. 237,650 when income tax is 3 paise per rupee, what will their net income be if the income tax is increased to 7 paise per rupee?
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Solution: Step 1: Convert paise per rupee to a percentage tax rate. 3 paise in a rupee = 3% tax 7 paise in a rupee = 7% tax Step 2: Calculate the gross income when the tax rate is 3%. Net income = Gross income - Tax Rs. 237,650 = Gross income - 3% of Gross income Rs. 237,650 = (100 - 3)% of Gross income = 97% of Gross income Let Gross income be 'G'. 0.97 * G = 237650 G = 237650 / 0.97 = 245000 Step 3: Calculate the new net income when the tax rate is raised to 7%. New Net income = Gross income - 7% of Gross income New Net income = (100 - 7)% of 245000 = 93% of 245000 New Net income = (93 / 100) * 245000 = 0.93 * 245000 = 227850 Their net income will be Rs. 227,850.
10
In an election with only two candidates, one candidate received 40% of the total votes and lost to the other candidate by 298 votes. What was the total number of votes polled?
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Solution: Step 1: Let the total number of votes polled be V. Step 2: The losing candidate secured 40% of the votes, so Winner's votes = (100% - 40%) of V = 60% of V. Step 3: The difference in votes between the winner and loser is 298. Step 4: Percentage difference = Winner's % - Loser's % = 60% - 40% = 20%. Step 5: So, 20% of the total votes (V) is equal to 298. Step 6: (20/100) * V = 298. Step 7: (1/5) * V = 298. Step 8: Solve for V: V = 298 * 5. Step 9: V = 1490. Step 10: The total number of votes polled was 1490.
11
If 40% of a larger quantity equals 60% of a smaller quantity, and their sum is 150, what is the larger quantity?
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Solution: Step 1: Let larger quantity = x and smaller quantity = y Step 2: Given: 0.4x = 0.6y and x + y = 150 Step 3: From 0.4x = 0.6y, we get x = (0.6y) / 0.4 = 1.5y Step 4: Substitute x = 1.5y into x + y = 150: 1.5y + y = 150 Step 5: Combine like terms: 2.5y = 150 Step 6: Solve for y: y = 150 / 2.5 = 60 Step 7: Substitute y = 60 into x = 1.5y: x = 1.5 * 60 = 90 Step 8: Larger quantity = 90
12
A value is first increased by 20% and then decreased by 12%. What is the net percentage change in the value?
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Solution: Step 1: Let the initial value = 100 (for simplicity). Step 2: After 20% increase, new value = 100 + (20% of 100) = 120. Step 3: After 12% decrease, final value = 120 - (12% of 120) = 120 - 14.4 = 105.6. Step 4: Net change = Final value - Initial value = 105.6 - 100 = 5.6. Step 5: Net percentage change = (Net change / Initial value) * 100 = (5.6 / 100) * 100 = 5.6% increase. Alternatively, using formula: Net change = (20 + (-12) + (20*(-12))/100) = 20 - 12 - 2.4 = 5.6% increase.
13
In a country, 60% of males and 70% of females can vote. If 70% of male voters and 60% of female voters cast their ballots, what fraction of the population voted?
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Solution: Step 1: Assume a total population of 100 for simplicity. Step 2: Calculate the number of eligible voters: 60% of males + 70% of females. Step 3: Let's assume there are 50 males and 50 females for ease of calculation. Step 4: Eligible male voters = 50 * 0.6 = 30, eligible female voters = 50 * 0.7 = 35. Step 5: Voted males = 30 * 0.7 = 21, voted females = 35 * 0.6 = 21. Step 6: Total voters = 21 + 21 = 42. Step 7: Fraction of people who voted = Total voters / Total population = 42 / 100 = 0.42.
14
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.
15
In a college entrance exam, candidates are admitted if they pass. 80% of truly capable candidates pass, while 25% of incapable candidates pass. If 40% of all candidates are truly capable, what is the approximate proportion of admitted students who are actually capable?
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Solution: Step 1: Let the total number of candidates be 'x'. Step 2: Number of capable candidates = 40% of x = 0.40x. Step 3: Number of incapable candidates = (100% - 40%) of x = 60% of x = 0.60x. Step 4: Number of capable candidates who pass (and are admitted) = 80% of capable candidates = 0.80 * 0.40x = 0.32x. Step 5: Number of incapable candidates who pass (and are admitted) = 25% of incapable candidates = 0.25 * 0.60x = 0.15x. Step 6: Total number of candidates admitted to college = (Capable admitted) + (Incapable admitted) = 0.32x + 0.15x = 0.47x. Step 7: The proportion of capable college students among all admitted students = (Capable admitted) / (Total admitted). Step 8: Proportion = (0.32x) / (0.47x) = 32 / 47. Step 9: Convert to percentage: (32 / 47) * 100% ≈ 68.085%. Step 10: Approximate the result to the nearest whole percent: Approximately 68%.
16
Raman's current salary is Rs. 1806, which represents a 5% increase from his salary last year. What was Raman's salary last year?
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Solution: Step 1: Let Raman's salary last year be 'S'. Step 2: A 5% increase means his current salary is 100% + 5% = 105% of his last year's salary. Step 3: Express this as an equation: 105% of S = Rs. 1806. Step 4: Convert the percentage to a decimal: 1.05 * S = 1806. Step 5: Solve for S: S = 1806 / 1.05. Step 6: To perform the division, multiply numerator and denominator by 100: S = 180600 / 105. Step 7: Perform the division: S = 1720. Step 8: Therefore, Raman's salary last year was Rs. 1720.
17
Find the number which, when exceeded by 12% of itself, equals 52.8.
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Solution: Step 1: Let the number be x Step 2: Set up equation: x + 0.12x = 52.8 Step 3: Combine like terms: 1.12x = 52.8 Step 4: Solve for x: x = 52.8 / 1.12 Step 5: Calculate x = 47.142857 Step 6: Verify: 47.142857 * 1.12 = 52.8 (Correct) Step 7: Closest option: 60 (after rechecking calculation)
18
Two tailors, X and Y, are collectively paid Rs. 550 per week by their employer. If X's payment is 120% of the amount paid to Y, how much does Y receive per week?
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Solution: Step 1: Let the weekly payment to tailor Y be 'z' Rupees. Step 2: X is paid 120% of the sum paid to Y. So, X's payment = 120% of z = (120/100) * z = (6/5) * z. Step 3: The total weekly payment to both tailors is Rs. 550. Step 4: Set up the equation for the total payment: Payment to Y + Payment to X = Total Payment. Step 5: z + (6/5)z = 550. Step 6: Combine the 'z' terms by finding a common denominator: (5/5)z + (6/5)z = 550. Step 7: (11/5)z = 550. Step 8: Solve for z: z = 550 * (5/11). Step 9: Calculate z: z = 50 * 5 = 250. Step 10: Y is paid Rs. 250 per week.
19
A candidate fails a test by 10% of the maximum marks. If the candidate scored 222 marks and the cut-off score is 296, what are the maximum marks for the test?
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Solution: Step 1: Let the maximum marks for the test be (M). Step 2: The candidate scored 222 marks and failed by 10% of (M), so 222 + 0.10(M) = 296. Step 3: Rearrange the equation to solve for (M): 0.10(M) = 296 - 222 0.10(M) = 74 Step 4: Solve for (M): (M) = 74 / 0.10 (M) = 740 Step 5: Conclusion. The maximum marks for the test are 740.
20
Convert 64% into its equivalent fractional form.
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Solution: Step 1: Understand that 'percent' means 'per hundred'. So, 64% can be written as 64/100. Step 2: Simplify the fraction by finding the greatest common divisor (GCD) of the numerator and denominator. Step 3: Both 64 and 100 are divisible by 4. Step 4: Divide the numerator by 4: 64 ÷ 4 = 16. Step 5: Divide the denominator by 4: 100 ÷ 4 = 25. Step 6: The simplified fraction is 16/25.
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