Question 1

Depreciation applicable to an equipment is 20%. The value of the equipment 3 years from now will be less by

Accepted Answer

Concept & Formula: This problem asks for the total effective percentage decrease over 3 years. This requires applying the successive percentage change concept on an assumed base value. For an annual depreciation rate R, the remaining value after n years is calculated as: Final Value = Initial Value × (1 - (R/100))^n Step-by-Step Solution: • Assume Initial Value: Let the current value of the equipment be 100. • Depreciation Rate: 20% per year. • Fractional Multiplier: 100% - 20% = 80%. As a fraction, this is (80/100) = (4/5) or 0.8. • Calculate Final Value after 3 years: Final Value = 100 × 0.8 × 0.8 × 0.8 Final Value = 100 × 0.512 = 51.2 • Calculate the Total Depreciation: The question asks how much the value will be *less by*. Total Reduction = Initial Value - Final Value Total Reduction = 100 - 51.2 = 48.8 • Since our base was 100, this reduction is directly 48.8%. Exam Strategy & Shortcut: Effective Percentage Formula (for 3 successive changes): Alternatively, remember the cube of the multiplier. A 20% decrease means the value is multiplied by 0.8 each year. 0.8^3 = 0.512. This means 51.2% of the value remains. The depreciation is simply 100% - 51.2% = 48.8%. Knowing your cubes (like 8^3 = 512) makes this a 5-second mental math problem. Common Pitfall: The single most common mistake is choosing 51.2%. Students correctly calculate the final remaining value (51.2%) but fail to read the exact wording of the question: "will be less by". You must subtract the remaining value from the original 100% to find the actual depreciation amount. Final Answer: Therefore, the correct answer is 48.8%.

Question 2

Ashish started a business with an initial investment of ₹ 500000. In the first year, he incurred a loss of 4%. However, during the second year, he earned a profit of 5% which in the third year rose to 10%. Calculate his net profit for the entire per…

Accepted Answer

Concept & Formula: This requires calculating the final amount after three successive percentage changes (one loss, two profits), and then finding the difference between the final amount and the initial investment to determine the net profit. Final Amount = Initial × (1 - (L_1/100)) × (1 + (P_1/100)) × (1 + (P_2/100)) Net Profit = Final Amount - Initial Amount Step-by-Step Solution: • Given: Initial investment = ₹ 500000. Year 1: 4% loss (multiplier 0.96). Year 2: 5% profit (multiplier 1.05). Year 3: 10% profit (multiplier 1.10). • Set up the final amount equation: Final Amount = 500000 × (96/100) × (105/100) × (110/100) • Simplify by cancelling zeros: There are 5 zeros in 500000 and 1 zero in 110. That's 6 zeros in the numerator. The denominator has 100 × 100 × 100 = 1000000 (6 zeros). They cancel out perfectly! • Calculate the product: Final Amount = 5 × 96 × 105 × 11 Final Amount = 480 × 105 × 11 Final Amount = 50400 × 11 = 554400 • Calculate Net Profit: Net Profit = 554400 - 500000 = 54400 Exam Strategy & Shortcut: Effective Percentage Method: You can combine the percentage changes successively using the formula x + y + (xy/100). 1. Combine Year 1 (-4%) and Year 2 (+5%): -4 + 5 + ((-4 × 5)/100) = 1 - 0.20 = 0.8% (Net 0.8% profit after 2 years) 2. Combine this result (+0.8%) with Year 3 (+10%): 0.8 + 10 + ((0.8 × 10)/100) = 10.8 + 0.08 = 10.88% The total net profit is exactly 10.88% of the initial investment. 10.88% of 500000 = 10.88 × 5000 = 54400 This eliminates heavy multiplication and delivers the answer directly. Common Pitfall: A frequent mistake is applying the 10% profit in the third year to the base 5% profit of the second year (meaning only a 15% total profit), rather than applying a 10% compound growth onto the actual final accrued capital of the second year. Always use successive multipliers. Final Answer: Therefore, the correct answer is ₹ 54400.

Question 3

The production of a company has ups and downs every year. The production increases for two consecutive years consistently by 15% and in the third year it decreases by 10%. Again in the next two years it increases by 15% each year and decreases by 10…

Accepted Answer

Concept & Formula: This problem tests the application of successive percentage changes over a specified timeline. When a value undergoes multiple percentage changes sequentially, we use the compound growth multiplier formula: Final Value = Initial Value × (1 \pm (R_1/100)) × (1 \pm (R_2/100)) × \dots Step-by-Step Solution: • Determine the Timeline: Starting from the end of 2008 (Base Year / Year 0). • Year 1 (End of 2009): Increase of 15% • Year 2 (End of 2010): Increase of 15% • Year 3 (End of 2011): Decrease of 10% • Year 4 (End of 2012): Increase of 15% Note: The pattern repeats, but we only need to calculate up to 2012, which is 4 years of changes. • Set up the Multipliers: • +15% \rightarrow 1.15 • -10% \rightarrow 0.90 • Calculate the Net Effect: Net Multiplier = 1.15 × 1.15 × 0.90 × 1.15 • Execute the Multiplication: 1.15 × 1.15 = 1.3225 1.3225 × 0.90 = 1.19025 1.19025 × 1.15 = 1.3687875 • Convert Multiplier to Percentage: A multiplier of 1.3687875 represents an overall increase of approximately 36.88%. Rounding to the nearest whole number given in the options, we get a 37% increase. Exam Strategy & Shortcut: Effective Percentage Approximation: Use the successive percentage formula: A + B + (AB/100) 1. Combine the first two 15% increases: 15 + 15 + (15 × 15/100) = 30 + 2.25 = 32.25% 2. Combine this 32.25% increase with the 10% decrease: 32.25 - 10 - (32.25 × 10/100) = 22.25 - 3.225 = 19.025% 3. Combine this 19.025% increase with the final 15% increase: 19.025 + 15 + (19.025 × 15/100) = 34.025 + 2.85375 = 36.87875% Approximating this yields 37%. Breaking it into pairs avoids dealing with large four-term decimal multiplications. Common Pitfall: The most common error is miscounting the number of years between 2008 and 2012. Students often calculate for 5 years instead of 4, incorrectly adding the final -10% drop to their sequence, which yields a drastically lower and incorrect answer. Always map out the year-to-year timeline explicitly. Final Answer: Therefore, the correct answer is 37% increase.

Question 4

The present population of a country estimated to be 10 crores is expected to increase to 13.31 crores during the next three years. The uniform rate of growth is

Accepted Answer

Concept & Formula: This problem requires finding the unknown uniform annual growth rate when the initial value, final value, and time period are given. This is a direct application of the compound interest formula: A = P (1 + (R/100))^n Where A is the final population, P is the initial population, R is the rate of growth, and n is the number of years. Step-by-Step Solution: • Given: Initial Population (P) = 10 crores Final Population (A) = 13.31 crores Time (n) = 3 years • Set up the equation: 13.31 = 10 × (1 + (R/100))^3 • Isolate the growth factor: (13.31/10) = (1 + (R/100))^3 1.331 = (1 + (R/100))^3 • Solve for the rate (R): Recognize that 1.331 is the cube of 1.1. (Since 11^3 = 1331, shifting the decimal point gives 1.1^3 = 1.331). (1.1)^3 = (1 + (R/100))^3 Taking the cube root of both sides: 1.1 = 1 + (R/100) 0.1 = (R/100) R = 0.1 × 100 = 10% Exam Strategy & Shortcut: Cube Root Recognition: In compound growth problems spanning 3 years, the ratio of the Final Value to the Initial Value will almost always form a perfect cube. Ratio = (13.31/10) = (1331/1000). Take the cube root immediately: \sqrt[3]{(1331/1000)} = (11/10). The fraction (11/10) means an increase of (1/10) per year. (1/10) expressed as a percentage is exactly 10%. This avoids all formal algebra. Common Pitfall: A common mistake is attempting to use the simple interest formula: calculating the total increase (3.31 crores) and dividing it by 3 years to get an average annual increase (1.103 crores), which is roughly 11%. Population growth is strictly a compounding phenomenon, not linear. Final Answer: Therefore, the correct answer is 10%.

Question 5

The population of a colony was 3600 three years back. It is 4800 right now. What will be the population three years down the line, if the rate of growth of population has been constant over the years and has been compounding annually?

Accepted Answer

Concept & Logic: This problem relies on the principle of Constant Compounding Multipliers. When a quantity grows at a constant compound rate over equal time intervals, it gets multiplied by a constant factor for each of those intervals. If the population grows from P_1 to P_2 in t years, the multiplying factor is (P_2/P_1). Over the next t years, it will multiply by the exact same factor. Step-by-step Solution: • Given: • Population 3 years ago = 3600 • Current population = 4800 • Time interval = 3 years • Calculation: First, find the growth multiplying factor over a 3-year period. Factor = \frac{Current Population}{Past Population} Factor = (4800/3600) = (4/3) • This means every 3 years, the population becomes (4/3) times its previous value. • To find the population 3 years from now, multiply the current population by this same factor: Future Population = 4800 × (4/3) Future Population = 1600 × 4 = 6400 Exam Strategy & Shortcut: Recognize the symmetry in the time intervals: 3 years back to now, and now to 3 years future. When intervals are equal, the values form a Geometric Progression (G.P.). Let the populations be A, B, C. (B/A) = (C/B) \implies C = (B^2/A) C = (4800 × 4800/3600) = 4800 × (4/3) = 6400 This G.P. shortcut bypasses solving for the actual percentage rate R, saving crucial minutes. Common Pitfall: The most time-consuming mistake is trying to calculate the exact annual percentage rate R by solving 3600(1 + (R/100))^3 = 4800. Since you need to find the cube root of (4/3), this calculation becomes messy and completely unnecessary when you realize the time intervals are identical blocks of 3 years. Final Answer: Therefore, the correct answer is 6400.

Question 6

Given that carbon-14 decays at a constant rate in such a way that it reduces to 50% in 5568 years, find the age of an old wooden piece in which the carbon is only 12.5% of the original.

Accepted Answer

Concept & Logic: This is a classic application of Half-Life Decay. The half-life is the time required for a quantity to reduce to half (50%) of its initial value. In radioactive decay, the amount remaining after n half-lives is given by: Remaining Amount = Initial Amount × ((1/2))^n Step-by-step Solution: • Given: • Half-life (t_{1/2}) = 5568 years. • Remaining carbon-14 = 12.5% of the original. • Deduction: We need to find how many half-lives it takes to reach 12.5%. Let's track the decay sequentially starting from 100%: • After 1 half-life: 100% \rightarrow 50% • After 2 half-lives: 50% \rightarrow 25% • After 3 half-lives: 25% \rightarrow 12.5% • It takes exactly 3 half-lives to reduce the carbon to 12.5%. • Calculation: Total Age = Number of half-lives × Length of one half-life Total Age = 3 × 5568 Total Age = 16704 years Exam Strategy & Shortcut: Instead of stepping through percentages manually, convert the remaining percentage directly into a power of (1/2): 12.5% = (12.5/100) = (1/8) = ((1/2))^3 The exponent 3 immediately tells you that 3 half-lives have passed. Multiply 3 × 5568 = 16704. Common Pitfall: A frequent arithmetic mistake occurs when multiplying 3 × 5568 under pressure. Instead of full multiplication, use the unit digit method to quickly eliminate options. 3 × 8 = 24, so the answer must end in 4. Looking at the options, only (c) 16704 ends in a 4. You don't even need to finish the multiplication! Final Answer: Therefore, the correct answer is 16704 years.

Question 7

A building worth ₹ 1,33,100 is constructed on land worth ₹ 72,900. After how many years will the value of both be the same if land appreciates at 10% p.a. and building depreciates at 10% p.a.?

Accepted Answer

Concept & Equation: This problem combines Compound Appreciation and Compound Depreciation. We need to track the future value of two separate assets changing at different rates (one positive, one negative) and find the point in time (n years) when their values are equal. • Formula for Appreciation (Land): A_1 = P_1(1 + (R_1/100))^n • Formula for Depreciation (Building): A_2 = P_2(1 - (R_2/100))^n We set A_1 = A_2 and solve for n. Step-by-step Solution: • Given: • Initial Land Value (P_1) = ₹ 72,900 • Land Appreciation Rate (R_1) = +10% p.a. • Initial Building Value (P_2) = ₹ 1,33,100 • Building Depreciation Rate (R_2) = -10% p.a. • Setting up the Equation: • Future value of Land = 72900 × (1 + (10/100))^n = 72900 × (1.1)^n • Future value of Building = 133100 × (1 - (10/100))^n = 133100 × (0.9)^n • Since their values become equal: 72900 × (1.1)^n = 133100 × (0.9)^n • Calculation: • Rearrange to group the exponential terms on one side and constants on the other: ((1.1)^n/(0.9)^n) = (133100/72900) • Simplify both sides: ((1.1/0.9))^n = (1331/729) • Multiply the numerator and denominator of the fraction inside the parenthesis by 10: ((11/9))^n = (1331/729) • Recognize the cubes of the numbers. You should know that 11^3 = 1331 and 9^3 = 729: ((11/9))^n = ((11/9))^3 • Equating the exponents: n = 3 Exam Strategy & Shortcut: In competitive exams, numbers like 1331 and 729 are massive hints. They are highly recognizable perfect cubes (11^3 and 9^3). When you see +10% and -10%, you are dealing with multipliers of (11/10) and (9/10). The ratio of their starting values is (133100/72900), which instantly reduces to (1331/729). Because this ratio perfectly matches the cubes of the rate multipliers ((11^3/9^3)), you can instantly conclude that n = 3 without writing out the full algebraic equation. Common Pitfall: A common trap is accidentally swapping the rates (e.g., applying depreciation to the land and appreciation to the building). Always double-check which asset is increasing and which is decreasing. Buildings generally depreciate due to wear and tear, while land appreciates due to scarcity, which aligns with logical real-world principles. Final Answer: Therefore, the correct answer is 3.

Question 8

The population of a town increases $4\%$ annually but is decreased by emigration annually to the extent of $(1/2)\%$. What will be the increase percent in 3 years?

Accepted Answer

Concept & Formula: This problem is about finding the Net Compound Growth Rate. When there are simultaneous increases and decreases happening constantly per period, we first find the net rate of change per period, and then apply the standard compound interest formula over the given time. The net effective annual growth rate is the difference between the increase rate and the decrease rate. Step-by-step Solution: • Given: • Annual increase rate = 4% • Annual decrease (emigration) rate = (1/2)% = 0.5% • Time period (n) = 3 years • Calculation: • First, calculate the net annual growth rate: Net Rate (R) = 4% - 0.5% = 3.5% • The population after 3 years will be: Final Population = P(1 + (3.5/100))^3 • Let the initial population P be 100 to make percentage calculations easier. Final Population = 100 × (1 + (7/200))^3 = 100 × ((207/200))^3 • Calculating this explicitly takes time, so let's use the successive percentage change formula or approximation. The formula for successive changes of x%, y%, and z% is complex. Instead, let's approximate using simple interest for a rough estimate, then add a bit for compounding. • Simple interest over 3 years: 3.5% × 3 = 10.5% • Because it's compound growth, the actual increase will be slightly higher than 10.5%. • Let's look at the options: (a) 9.8, (b) 10, (c) 10.5, (d) 10.8. • Since it must be greater than 10.5%, the only logical answer is 10.8. Exam Strategy & Shortcut: Use the Effective Percentage Change Approximation. For a small rate R over n years, the effective change is roughly nR + a small compounding factor. Here, R = 3.5% and n = 3. 3 × 3.5% = 10.5%. Because it compounds, the actual value must be strictly greater than 10.5%. Looking at the options, 10.8% is the only viable choice. This avoids calculating 1.035^3 entirely. Common Pitfall: The most common mistake is simply multiplying the net rate by the number of years (3.5% × 3 = 10.5%) and selecting option (c), forgetting that population growth is a compounding process, not a simple linear process. Final Answer: Therefore, the correct answer is 10.8.

Question 9

The current birth rate per thousand is 32, whereas the corresponding death rate is 11 per thousand. The net growth rate in terms of population increase in percent is given by

Accepted Answer

Concept & Logic: This question involves converting Rates per Thousand into Percentage Rates (per hundred). The net growth is the difference between births and deaths. The crucial step is understanding the base unit. "Per thousand" means out of every 1000 people. "Percent" means out of every 100 people. Step-by-step Solution: • Given: • Birth rate = 32 per thousand • Death rate = 11 per thousand • Calculation: • First, find the net growth rate per thousand: Net Growth per thousand = Birth rate - Death rate Net Growth per thousand = 32 - 11 = 21 • This means the population grows by 21 people for every 1000 people. • To find the percentage (per 100), we divide this rate by 10: Net Growth Percent = (21/1000) × 100 Net Growth Percent = (21/10) = 2.1% Exam Strategy & Shortcut: Understand the direct relationship between "per mille" (per thousand) and "per cent" (per hundred). Since 1000 is ten times 100, to convert a rate per thousand to a percent, simply divide by 10. Net rate = 32 - 11 = 21. Percent = (21/10) = 2.1%. Common Pitfall: The trap here is the decimal placement. Students might see the net difference of 21 and hastily select option (d) 21% without converting "per thousand" to "percent". Always pay close attention to the units requested in the final question. Final Answer: Therefore, the correct answer is 2.1%.

Question 10

Mr. Jones' total annual gross salary, which was ₹ 10 lakhs per year in 2007, was reduced by 10% in 2008. In 2007 his family expenditure for food items was 40% of the total annual gross salary. The prices of average food items increased by 5% between…

Accepted Answer

Concept & Logic: This problem is an exercise in tracking Independent Percentage Changes on different base values. We need to calculate the new salary and the new food expenditure separately based on their respective changes, and then find the new ratio between them. New Percentage = ( \frac{New Food Expenditure}{New Total Salary} ) × 100 Step-by-step Solution: • Given Information (2007): • Total Salary in 2007 = ₹ 10,00,000 (10 Lakhs) • Food expenditure in 2007 = 40% of Total Salary • Calculating 2007 Values: • Food Expenditure (2007) = 40% of 10,00,000 = 0.4 × 10,00,000 = ₹ 4,00,000 • Calculating 2008 Changes: • New Salary (2008): Salary reduced by 10%. Salary (2008) = 10,00,000 × (1 - (10/100)) = 10,00,000 × 0.9 = ₹ 9,00,000 • New Food Expenditure (2008): Food prices increased by 5%, and consumption is the same, so the total expenditure increases by 5%. Food Exp (2008) = 4,00,000 × (1 + (5/100)) = 4,00,000 × 1.05 = ₹ 4,20,000 • Final Percentage Calculation: • We need the new food expenditure as a percentage of the new salary. Required % = ( (4,20,000/9,00,000) ) × 100 Required % = ( (42/90) ) × 100 Required % = ( (7/15) ) × 100 Required % = (700/15) = 46.66...% • Rounding to the nearest integer given in the options, we get 47%. Exam Strategy & Shortcut: Assume a Base of 100. The actual value of 10 Lakhs is a distractor and completely unnecessary. Let 2007 Salary = 100 units. 2007 Food = 40 units. 2008 Salary = 100 - 10% = 90 units. 2008 Food = 40 + 5% = 40 + 2 = 42 units. New Percentage = (42/90) × 100 = (420/9) = 46.66% \approx 47%. This method avoids large numbers and zeros, significantly speeding up calculation. Common Pitfall: A major time-waster is using the literal "10 Lakhs" figure throughout the calculations. While mathematically correct, it increases the risk of dropping a zero and slows down the process. Whenever asked for a final percentage, relative base values (like 100) are almost always faster and safer. Final Answer: Therefore, the correct answer is 47%.

Question 11

Two numbers are respectively 20% and 25% lower than a third number. By how much percentage is the second number lower than the first?

Accepted Answer

Concept & Strategy: When multiple values are defined in relation to a single unknown "third number," the most efficient strategy is to assign the third number a convenient base value, typically 100, to bypass complex algebra. Required % = \frac{Difference}{Base Value} × 100 Step-by-Step Solution: • Given: Two numbers are 20% and 25% lower than a third number. • Let the third number be exactly 100. • The first number is 20% lower: 100 - 20 = 80. • The second number is 25% lower: 100 - 25 = 75. • We must find the percentage by which the *second* number (75) is lower than the *first* number (80). • Absolute difference = 80 - 75 = 5. • The comparison base is the *first number* (80). • Required Percentage = (5/80) × 100. • Simplify the fraction: (1/16) × 100 = (100/16) = (25/4). • Convert to a mixed fraction: 6 (1/4)%. Exam Strategy & Shortcut: Always use 100 as the reference point for the independent variable in percentage chain problems. It directly translates percentages into hard numbers, making the secondary comparison ((5/80)) immediately obvious. Common Pitfall: Test-takers often spot the 5 unit difference between 80 and 75, and mistakenly select 5% as the answer because they subconsciously divide by 100 (the original base). The question explicitly shifts the base to the *first number* (80). Final Answer: Therefore, the correct answer is 6 (1/4)%.

Question 12

Two numbers are respectively $12 \frac{1}{2}\%$ and 25% more than a third number. The first number as a percentage of the second number is

Accepted Answer

Concept & Strategy: Like other successive comparison problems, defining the common reference point (the third number) as 100 simplifies the math. The goal is to find one number as a direct percentage of another, rather than finding a percentage increase/decrease. Percentage = \frac{Target Value}{Reference Value} × 100 Step-by-Step Solution: • Given: Number 1 is 12 (1/2)% more than Number 3. Number 2 is 25% more than Number 3. • Let the third number be 100. • The first number = 100 + 12.5 = 112.5. • The second number = 100 + 25 = 125. • The question asks for the first number *as a percentage of* the second number. • Required Percentage = (112.5/125) × 100. • To remove decimals, multiply numerator and denominator by 10: (1125/1250) × 100. • Simplify: (1125/1250) = (9/10). • (9/10) × 100 = 9 × 10 = 90. Exam Strategy & Shortcut: Recognize the fractional equivalents: 12.5% is (1/8) and 25% is (1/4) (or (2/8)). Let the third number be 8 to avoid decimals entirely. First number = 8 + 1 = 9. Second number = 8 + 2 = 10. First as a % of second = (9/10) = 90%. This ratio method is significantly faster. Common Pitfall: Students frequently misread "first number as a percentage of the second" and mistakenly calculate the *difference* between the two numbers before dividing, solving for how much *less* the first is than the second. Read the final prompt carefully. Final Answer: Therefore, the correct answer is 90.

Question 13

In a mixture of milk and water, the proportion of water by weight was 75%. If in the 60 gms mixture 15 gm of water was added, what would be the percentage of water?

Accepted Answer

Concept & Formula: This involves a Simple Mixture Addition. We must calculate the initial quantities of the components, add the new volume to the specific component (water), and then find the new percentage over the updated total weight. New Percentage = ( \frac{Initial Water + Added Water}{Initial Total + Added Mixture} ) × 100 Step-by-Step Solution: • Given: • Initial mixture weight = 60 gms • Initial proportion of water = 75% • Water added = 15 gm • Calculation: • Initial water quantity = 75% of 60 = (3/4) × 60 = 45 gms. • New water quantity after adding 15 gm = 45 + 15 = 60 gms. • New total weight of the mixture = 60 (initial) + 15 (added) = 75 gms. • New percentage of water = ((60/75)) × 100. • Simplifying the fraction: (60/75) = (4/5). • (4/5) × 100 = 80%. • The exact new percentage of water is 80%, which does not appear in options (a), (b), (c), or (d). Exam Strategy & Shortcut: Fractional Method: Calculate initial water rapidly: (3/4) × 60 = 45 gms. Add 15 gms water: new water = 60, new total = 75. Fraction = (60/75). Recognize immediately that both are divisible by 15. (4/5) = 80%. Instantly spot that 80% is missing from the numerical options, so 'None of these' is the correct choice. Don't second guess your calculation if the math is clean. Common Pitfall: A classic mistake is calculating the new water amount (60 gms) but forgetting to add the 15 gms to the *total mixture weight* denominator. This leads to an incorrect calculation of (60/60) = 100%, which is strategically placed as trap option (d). Final Answer: Therefore, the correct answer is None of these.

Question 14

The quantity of water (in ml) needed to reduce $9$ ml shaving lotion containing $50\%$ alcohol to a lotion containing $30\%$ alcohol, is

Accepted Answer

Concept & Logic: This is a classic dilution problem. By adding water, we are changing the total volume and the water content, but the absolute mass of the alcohol remains entirely unchanged. Initial Alcohol Mass = Final Alcohol Mass Step-by-Step Solution: • Given: • Initial total volume = 9 ml • Initial alcohol concentration = 50% • Target alcohol concentration = 30% • Calculation: • Find the absolute volume of alcohol in the original lotion: 50% of 9 ml = 4.5 ml. • Let the final total volume of the diluted lotion be V. • In the new lotion, the alcohol concentration must be 30%, but the absolute alcohol remains 4.5 ml. • Set up the equation: 30% of V = 4.5. • 0.30 × V = 4.5. • V = (4.5/0.30) = 15 ml. • The new total volume is 15 ml. The initial volume was 9 ml. • Quantity of water added = 15 - 9 = 6 ml. Exam Strategy & Shortcut: Percentage-Volume Inversion: The alcohol percentage drops from 50% to 30%, which is a ratio of 5:3. Because the alcohol volume is constant, the total mixture volume must be inversely proportional, which is a ratio of 3:5. If the initial volume (3 units) = 9 ml, then 1 unit = 3 ml. The final volume (5 units) = 15 ml. The added water is the difference (2 units) = 2 × 3 = 6 ml. Common Pitfall: Students frequently calculate the final total volume (15 ml) and accidentally select it as the answer if it's present in the options. Always reread the question to ensure you are answering what is asked—in this case, the *added* water, not the *final* volume. Final Answer: Therefore, the correct answer is 6.

Question 15

$6$ c.c. of a $20\%$ solution of alcohol in water is mixed with $4$ c.c. of a $60\%$ solution of alcohol in water. The alcoholic strength of the mixture is

Accepted Answer

Concept & Logic: This requires calculating the Weighted Average of two mixtures. To find the final strength, we must determine the total absolute amount of alcohol from both constituent solutions and divide it by their combined total volume. Final Percentage = ((P_1 × V_1) + (P_2 × V_2)/V_1 + V_2) × 100 Step-by-Step Solution: • Given: • Solution 1: Volume = 6 c.c., Concentration = 20% • Solution 2: Volume = 4 c.c., Concentration = 60% • Calculation: • Calculate absolute alcohol in Solution 1: 20% of 6 = 0.20 × 6 = 1.2 c.c. • Calculate absolute alcohol in Solution 2: 60% of 4 = 0.60 × 4 = 2.4 c.c. • Total absolute alcohol in the combined mixture = 1.2 + 2.4 = 3.6 c.c. • Total combined volume of the mixture = 6 + 4 = 10 c.c. • Calculate the final alcoholic strength: ((3.6/10)) × 100. • (3.6/10) = 0.36. • 0.36 × 100 = 36%. Exam Strategy & Shortcut: Alligation (Weighted Average Line): The weights (volumes) of the two mixtures are 6 c.c. and 4 c.c., giving a weight ratio of 6:4 or 3:2. The concentrations are 20% and 60%. The total distance between them is 60% - 20% = 40%. The final mixture concentration will divide this 40% gap in the *inverse* ratio of their volumes (2:3). Calculate the 2 parts (pulling away from the 20% side): (2/5) of 40% = 16%. Add this to the lower concentration: 20% + 16% = 36%. Common Pitfall: A very common trap is taking a simple unweighted average of the two percentages: (20 + 60/2) = 40%. This completely ignores the fact that there is more of the 20% solution (6 c.c.) than the 60% solution (4 c.c.), meaning the final answer must heavily skew closer to 20%. Trap option (d) catches students who make this mistake. Final Answer: Therefore, the correct answer is 36%.

Question 16

One type of liquid contains $20\%$ water and the second type of liquid contains $35\%$ of water. A glass is filled with $10$ parts of first liquid and $4$ parts of second liquid. The percentage of water in the new mixture in the glass is

Accepted Answer

Concept & Formula: This problem requires calculating the Weighted Average of two mixtures. Because the volumes (parts) of the two liquids are different, you cannot simply average their percentages. We must find the total absolute units of water and divide by the total units of mixture. Final Percentage = ((P_1 × V_1) + (P_2 × V_2)/V_1 + V_2) Step-by-Step Solution: • Given: • Liquid 1: Volume = 10 parts, Water = 20% • Liquid 2: Volume = 4 parts, Water = 35% • Calculation: • Calculate absolute water units from Liquid 1: 20% of 10 = 0.20 × 10 = 2 parts of water. • Calculate absolute water units from Liquid 2: 35% of 4 = 0.35 × 4 = 1.4 parts of water. • Total absolute water in the glass = 2 + 1.4 = 3.4 parts. • Total volume of liquid in the glass = 10 + 4 = 14 parts. • Calculate the final percentage: ( (3.4/14) ) × 100. • Simplify the fraction: (34/140) × 100 = (17/70) × 100 = (170/7)%. • Convert to a mixed fraction: 170 \div 7 = 24 with a remainder of 2, giving 24 (2/7)%. Exam Strategy & Shortcut: Alligation Method: The ratio of the parts is 10 : 4, which simplifies to 5 : 2. The total gap between the percentages (20% and 35%) is 15%. The final mixture will divide this 15% gap in the *inverse* ratio of their volumes (2 : 5). Calculate the shift from the 20% side (using the 2 parts): (2/7) of 15% = (30/7)% = 4 (2/7)%. Add this shift to the lower percentage: 20% + 4 (2/7)% = 24 (2/7)%. Solved without decimals! Common Pitfall: A very frequent mistake is calculating the simple, unweighted average: (20 + 35/2) = 27.5%. This incorrectly assumes the glass was filled with equal amounts (e.g., 7 parts and 7 parts) of each liquid. Always factor in the specific volume ratios. Final Answer: Therefore, the correct answer is 24 (2/7)%.

Question 17

In some quantity of ghee, $60\%$ is pure ghee and $40\%$ is vanaspati. If $10$ kg of pure ghee is added, then the strength of vanaspati ghee becomes $20\%$. The original quantity was

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Concept & Logic: When only pure ghee is added to the mixture, the absolute quantity of the other ingredient (vanaspati) remains strictly constant. Setting up an equation based on this Constant Component is the fastest path to the solution. Initial Mass of Vanaspati = Final Mass of Vanaspati Step-by-Step Solution: • Given: • Initial pure ghee = 60%, meaning initial vanaspati = 40% • Added pure ghee = 10 kg • Final strength of vanaspati = 20% • Calculation: • Let the original quantity of the mixture be x kg. • The absolute mass of vanaspati in the original mixture is 40% of x = 0.40x. • After adding 10 kg of pure ghee, the new total quantity of the mixture becomes (x + 10) kg. • In this new mixture, the vanaspati constitutes 20%. So, its mass is 20% of (x + 10) = 0.20(x + 10). • Equating the constant vanaspati mass: 0.40x = 0.20(x + 10). • Divide both sides by 0.20: 2x = x + 10. • Subtract x from both sides: x = 10 kg. Exam Strategy & Shortcut: Inverse Ratio Shortcut: Focus entirely on the constant component (vanaspati). Its percentage drops from 40% to 20%, which is a ratio of 2 : 1. Because the absolute mass of vanaspati is constant, the total mixture weight must grow in the exact inverse ratio, which is 1 : 2. If the original weight is 1 unit and the new weight is 2 units, the difference (1 unit) represents the added ghee. We are given that 1 unit (added ghee) = 10 kg. Therefore, the original quantity (1 unit) must also be 10 kg. Common Pitfall: Students often waste time tracking the pure ghee: trying to solve 0.60x + 10 = 0.80(x + 10). While mathematically valid, it involves more complex arithmetic and terms on both sides of the equation, greatly increasing the chance of an algebra error compared to using the constant vanaspati. Final Answer: Therefore, the correct answer is 10 kg.

Question 18

85% and 92% alcoholic solutions are mixed to get 35 litres of an 89% alcoholic solution. How many litres of each solution are there in the new mixture?

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Concept & Strategy: When mixing two solutions of different concentrations to form a target concentration, the Rule of Alligation provides the exact ratio of the quantities required. It avoids complex linear equations by mapping the distances between the percentages. \frac{Quantity of Cheaper}{Quantity of Dearer} = \frac{Dearer Price - Mean Price}{Mean Price - Cheaper Price} Step-by-Step Solution: • Given: • Solution 1 (Lower) = 85% • Solution 2 (Higher) = 92% • Target Mixture (Mean) = 89% • Total Volume = 35 litres • Calculation / Deduction: • Set up the Alligation cross: • Place 85% on the top left and 92% on the top right. • Place 89% in the middle. • Calculate the diagonal differences: • Right diagonal (Difference between 92% and 89%) = 3. This goes under the 85% side. • Left diagonal (Difference between 89% and 85%) = 4. This goes under the 92% side. • The required ratio of Solution 1 to Solution 2 is 3 : 4. • The total number of ratio parts is 3 + 4 = 7 parts. • We know the total volume is 35 litres. So, 7 parts = 35 litres. • 1 part = 5 litres. • Volume of Solution 1 (3 parts) = 3 × 5 = 15 litres. • Volume of Solution 2 (4 parts) = 4 × 5 = 20 litres. Exam Strategy & Shortcut: The Alligation rule itself is the shortcut. However, to verify your answer even faster via Option Elimination: The target mixture (89%) is closer to 92% than it is to 85%. Therefore, the mixture must contain a larger quantity of the 92% solution. Looking at the options, only (a) and (c) have more of the second solution. A quick mental check of the 3:4 ratio confirms (c) is correct. Common Pitfall: Students often place the calculated ratio components under the wrong initial solutions (swapping the 3 and 4). Remember: the difference calculated from the right side value dictates the proportion of the left side substance, and vice versa. Final Answer: Therefore, the correct answer is 15 of the first and 20 of the second.

Question 19

Two vessels contain equal quantities of $40\%$ alcohol. Sachin changed the concentration of the first vessel to $50\%$ by adding extra quantity of pure alcohol. Vivek changed the concentration of the second vessel to $50\%$ replacing a certain quant…

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Concept & Strategy: To adjust the concentration of a mixture, adding pure solute (Sachin's method) increases the total volume, while replacing part of the mixture with pure solute (Vivek's method) maintains a constant total volume. We can use basic ratios and the Alligation rule to find the exact quantities involved in both methods. Step-by-Step Solution: • Given: • Initial concentration in both vessels = 40% alcohol. • Final concentration in both vessels = 50% alcohol. • Let the initial volume of both vessels be 100 units. • Sachin's Calculation (Addition): • Initial alcohol = 40% of 100 = 40 units. • Let Sachin add x units of pure (100%) alcohol. • New total alcohol = 40 + x. New total volume = 100 + x. • Target concentration is 50% (which is (1/2)): • (40 + x/100 + x) = (1/2) • 80 + 2x = 100 + x \implies x = 20. • Sachin added 20 units. • Vivek's Calculation (Replacement): • Vivek replaces y units of the mixture. This is equivalent to mixing the remaining 40% solution with added pure 100% alcohol to get a 50% mixture. • Using Alligation: • Left side (remaining mix) = 40%. Right side (added pure) = 100%. Target (mean) = 50%. • Ratio of remaining volume to added volume = (100 - 50) : (50 - 40) = 50 : 10 = 5 : 1. • This means out of 6 total parts (5+1), 1 part was the replaced/added quantity. • Total volume = 100. Quantity replaced y = (1/6) × 100 = (50/3) units. • Comparison: • Sachin's quantity = 20. Vivek's quantity = (50/3). • Difference = 20 - (50/3) = (60 - 50/3) = (10/3). • Percentage by which Sachin's quantity is more than Vivek's = ( \frac{Difference}{Vivek's Quantity} ) × 100. • ( (10/3/50/3) ) × 100 = (10/50) × 100 = 20%. Exam Strategy & Shortcut: Assume initial volume is 60 (the LCM of the alligation parts 6 simplifies fractions). Sachin: (24+x/60+x) = (1/2) \implies 48 + 2x = 60 + x \implies x = 12. Vivek: Ratio 5:1, so replaced is (1/6) of 60 = 10. Sachin added 12, Vivek replaced 10. Sachin is 2 units more than Vivek (10). (2/10) = 20%. Choosing a smart assumed value eliminates complex fractions! Common Pitfall: Students frequently misread the final question and calculate the percentage based on Sachin's quantity as the denominator ((10/3/20) = 16.66%), matching trap option (c). The phrasing "more/less than that replaced by Vivek" strictly dictates that Vivek's quantity must be the denominator. Final Answer: Therefore, the correct answer is 20% more.

Question 20

An empty fuel tank of a car was filled with A type petrol. When the tank was half-empty, it was filled with B type petrol. Again when the tank was half-empty, it was filled with A type petrol. When the tank was half-empty again, it was filled with B…

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Concept & Logic: This problem requires tracking the sequential dilution and reinforcement of a mixture. Because the tank is always emptied by half and refilled to full, we can trace the remaining quantities of Type A petrol through each specific cycle of halving and adding. Step-by-Step Solution: • Given: • Let the full tank capacity be 100 units. • Calculation (Cycle by Cycle): • Initial State: Tank is filled with A. • Contents: 100 A. • Step 1: Tank becomes half-empty (50 units remain), filled with B (+50 B). • Contents: 50 A, 50 B. (Mixture is now 50% A). • Step 2: Tank becomes half-empty (halving the existing mixture leaves 25 A, 25 B), filled with A (+50 A). • Contents: 25 + 50 = 75 A, 25 B. (Mixture is now 75% A). • Step 3: Tank becomes half-empty (halving the existing mixture leaves 37.5 A, 12.5 B), filled with B (+50 B). • Contents: 37.5 A, 12.5 + 50 = 62.5 B. • The question asks for the final percentage of A type petrol. • Final amount of A = 37.5 units out of 100 units total = 37.5%. Exam Strategy & Shortcut: Fraction Tracking Method: Track only the fraction of Type A fuel at each stage. Start = 1. After Step 1 (Half A remains) = (1/2). After Step 2 (Half remains, plus full half added) = (1/2)((1/2)) + (1/2) = (1/4) + (2/4) = (3/4). After Step 3 (Half remains, no A added) = (1/2)((3/4)) = (3/8). Convert fraction to percentage: (3/8) = 37.5%. This avoids tracking both A and B entirely! Common Pitfall: A common mistake is losing track of which fuel type is being added during which step, or forgetting that "halving" the tank halves *both* fuel types proportionally. Proceed systematically step-by-step or use the fractional shortcut to avoid getting lost in the numbers. Final Answer: Therefore, the correct answer is 37.5%.

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