Question 1

$47.7 	imes 12.4 	imes 8.6 = $x$

Accepted Answer

Concept & Formula The problem requires basic decimal multiplication. The key insight is to count the total number of decimal places in the factors to determine the correct decimal placement in the final product. Step-by-Step Solution • First, ignore the decimals and multiply the numbers as integers: 477 × 124 × 86. • Let's group them: 477 × 124 = 59148. • Now multiply by 86: 59148 × 86 = 5086728. • Count the total decimal places in the original problem: 47.7 (1 place), 12.4 (1 place), and 8.6 (1 place). Total = 3 decimal places. • Place the decimal point 3 positions from the right in the product: 5086.728. Exam Strategy & Shortcut Use digital root and unit digit analysis to save time. Unit digit of the integers: 7 × 4 = 28 (ends in 8). 8 × 6 = 48 (ends in 8). The exact answer must end in 8. This eliminates option (d). Next, use approximation: 50 × 12 × 9 = 5400. The answer must be reasonably close to 5400. Doing a slightly closer rough calculation (48 × 12 \approx 576; 576 × 8.5 \approx 4896) points clearly to option (a). Common Pitfall Students often misplace the decimal point by adding digits incorrectly or by messing up the large integer multiplication. Always double-check by estimating the magnitude (50 × 12 × 8 = 4800) to ensure your decimal is in the right spot. Final Answer Therefore, the correct answer is 5086.728.

Question 2

$32.4 	imes 11.5 	imes 8.5 = $x$

Accepted Answer

Concept & Formula This is a continuous decimal multiplication problem. When multiplying decimals, multiply the numbers as if they were integers, then place the decimal point so the product has the same number of decimal places as the sum of decimal places in the factors. Step-by-Step Solution • Treat the numbers as integers: 324, 115, and 85. • Multiply the first two: 324 × 115 = 37260. • Multiply the result by the third: 37260 × 85 = 3167100. • Count the total decimal places in the given expression: 1 + 1 + 1 = 3 decimal places. • Apply the decimal point to the product: 3167.100. • Drop the trailing zeros: 3167.1. Exam Strategy & Shortcut Calculate the unit digit of the non-zero integer components. Look at 32.4 × 11.5 = 372.6. The trailing zero was dropped. Next multiply 372.6 × 8.5. The last digit before the decimal will end based on 6 × 5 = 30, meaning it ends in a single decimal place `.1` (from the integer math 3726 × 85). Looking at the options, (b) and (c) end in .1. Estimating 30 × 11 × 8 = 2640 helps confirm magnitude, pushing us toward (c). Common Pitfall Forgetting to drop trailing zeros after placing the decimal point, which leads to confusion when matching the calculated answer (3167.100) with the given simplified options (3167.1). Final Answer Therefore, the correct answer is 3167.1.

Question 3

$(0.05 	imes 5 - 0.005 	imes 5)$ equals

Accepted Answer

Concept & Formula This problem requires basic arithmetic operations following the order of operations (BODMAS/PEMDAS). Alternatively, you can use the distributive property of multiplication to factor out common terms and simplify the calculation: a × c - b × c = c(a - b) Step-by-Step Solution Method 1: Direct Calculation (BODMAS) • Perform the multiplications first: 0.05 × 5 = 0.25 0.005 × 5 = 0.025 • Subtract the second product from the first: 0.25 - 0.025 = 0.225 Method 2: Factoring • Both terms are multiplied by 5, so factor it out: 5 × (0.05 - 0.005) • Perform the subtraction inside the parentheses: 5 × 0.045 • Multiply by 5: 0.225 Exam Strategy & Shortcut Using direct calculation is usually faster for simple multiples like 5. Think of money or simple fractions: 5 × 5 cents is 25 cents (0.25). 5 × 5 tenths of a cent is 2.5 cents (0.025). Aligning the decimals mentally: 0.250 - 0.025 leaves 0.225. Common Pitfall Misaligning the decimal points during subtraction is the primary error here. Students often calculate 0.25 - 0.025 in their heads and mistakenly get 0.0225 or 0.250, forgetting to append a zero (0.250) to align the places properly before subtracting. Final Answer Therefore, the correct answer is 0.225.

Question 4

$4.4 	imes 5.8 	imes 11.5 - 141.27 = $x$

Accepted Answer

Concept & Strategy When multiplying multiple decimals, group terms strategically to create easier intermediate numbers. Look for pairings that result in whole numbers or drop trailing decimals (like multiplying an even number by a number ending in .5). Step-by-Step Solution • Group the first and third terms to make the multiplication easier: 4.4 × 11.5 • Break 4.4 into (4 + 0.4): 4 × 11.5 = 46 0.4 × 11.5 = 4.6 Sum = 46 + 4.6 = 50.6 • Now multiply this result by the remaining term (5.8): 50.6 × 5.8 Treat as integers: 506 × 58 = 29348. Restore two decimal places: 293.48. • Finally, subtract the given value from this product: 293.48 - 141.27 152.21 Exam Strategy & Shortcut Use aggressive approximation to find the correct range instantly. Round the terms: 4.5 × 6 × 11 = 297. Subtract the final term: 297 - 141 = 156. The actual answer must be close to 156. Option (c) is 152.21, which is the only option in this immediate neighborhood. To confirm without full calculation, check the last digit: Product unit digit: 4 × 8 × 5 = 160 (ends in 0, but wait, the decimal shifts). 50.6 × 5.8 \rightarrow 6 × 8 = 48. The product ends in .x8. Subtraction: x.x8 - x.x7 = x.x1. Options (b) and (c) end in 1, but our approximation (156) heavily favors (c). Common Pitfall Multiplying strictly left-to-right (4.4 × 5.8 = 25.52, then 25.52 × 11.5) creates unnecessarily large and complex intermediate decimal multiplication, wasting time and increasing the risk of arithmetic mistakes. Final Answer Therefore, the correct answer is 152.21.

Question 5

$4.036$ divided by $0.04$ gives

Accepted Answer

Concept & Strategy: To divide a decimal by another decimal, the most reliable strategy is to eliminate the decimal points by multiplying both the numerator and the denominator by the same power of 10. (a/b) = (a × 10^n/b × 10^n) Step-by-Step Solution: • Write the division as a fraction: (4.036/0.04) • Multiply both the numerator and the denominator by 1000 to remove the decimals completely (or multiply by 100 to make the denominator a whole number): (4.036 × 100/0.04 × 100) = (403.6/4) • Perform standard division: 403.6 \div 4 = 100.9 Exam Strategy & Shortcut: Instead of writing out the full fraction, simply shift the decimal point in both numbers to the right by two places (the number of decimal places in the divisor 0.04). This immediately changes the problem from 4.036 \div 0.04 to 403.6 \div 4. Dividing 400 by 4 is 100, and 3.6 by 4 is 0.9, yielding 100.9 instantly. Common Pitfall: Students often misplace the decimal point during direct division, incorrectly answering 10.09 or 1.009. Always convert the divisor (the number you are dividing by) into a whole number first to avoid alignment errors. Final Answer: Therefore, the correct answer is 100.9.

Question 6

$\frac{1}{0.04}$ is equal to

Accepted Answer

Concept & Formula: When dividing by a decimal, convert the decimal into a fraction or multiply both the numerator and denominator by a power of 10 to clear the decimal point. 0.04 = (4/100) Step-by-Step Solution: • Convert the denominator from a decimal to a fraction: 0.04 = (4/100) = (1/25) • Substitute this back into the original expression: (1/\frac{1){25}} • When dividing by a fraction, multiply by its reciprocal: 1 × 25 = 25 Exam Strategy & Shortcut: Memorize the fraction equivalents of common decimals. Since 0.04 is exactly 4%, or (1/25), dividing 1 by 0.04 is simply asking "how many 4 cents are in a dollar?" The answer is instantly 25. Alternatively, multiply top and bottom by 100 to get (100/4) = 25. Common Pitfall: A frequent mistake is moving the decimal incorrectly and calculating (10/4) = 2.5 or answering (1/40). Always count the number of decimal places carefully—two decimal places require multiplying by 100. Final Answer: Therefore, the correct answer is 25.

Question 7

$6425 \div 125 	imes 8 = x$

Accepted Answer

Concept & Logic: Follow the standard BODMAS/PEMDAS sequence, evaluating from left to right for multiplication and division. However, knowing fraction equivalents for large numbers (like 125) can drastically reduce calculation time. Step-by-Step Solution: • Given: 6425 \div 125 × 8 = x • Calculation: Process the division first. (6425/125) = 51.4 • Multiply the quotient by 8: 51.4 × 8 = 411.2 Exam Strategy & Shortcut: This problem is engineered for a specific shortcut. Notice that 125 = (1000/8). Substitute this into the expression: x = 6425 \div ((1000/8)) × 8 Dividing by a fraction is the same as multiplying by its reciprocal: x = 6425 × ((8/1000)) × 8 x = (6425 × 64/1000) Alternatively, a faster mental shortcut: 6425 \div 125 is the same as multiplying 6425 × 8 and dividing by 1000. 6425 × 8 = 51400. 51400 \div 1000 = 51.4. Finally, multiply by 8: 51.4 × 8 = 411.2. Common Pitfall: Multiplying 125 × 8 first to get 1000, and then doing 6425 \div 1000 = 6.425. Division and multiplication have the same precedence and must be executed left-to-right unless parentheses dictate otherwise. Final Answer: Therefore, the correct answer is 411.2.

Question 8

$(0.75 	imes 4.4 	imes 2.4) \div 0.6 = x$

Accepted Answer

Concept & Strategy: For complex decimal multiplication and division, the most efficient approach is to convert decimals into whole numbers or fractions, and leverage cancellation before performing any actual multiplication. (a × b × c/d) Step-by-Step Solution: • Given: (0.75 × 4.4 × 2.4/0.6) • Calculation: Notice that both 2.4 and 0.6 have one decimal place. You can immediately divide them as whole numbers. (2.4/0.6) = (24/6) = 4 • Substitute this simplified value back into the expression: 0.75 × 4.4 × 4 • Use the associative property to multiply 0.75 and 4 first, since 0.75 is (3/4). 0.75 × 4 = 3 • Finally, multiply the result by 4.4: 3 × 4.4 = 13.2 Exam Strategy & Shortcut: Never multiply straight across when division is involved. Always look for the easiest cancellation path. Spotting that 2.4 \div 0.6 = 4 turns a messy decimal calculation into a trivial mental math problem: 75% of 4 is 3, and 3 × 4.4 is 13.2. This avoids large intermediate decimals entirely. Common Pitfall: A common error is multiplying (0.75 × 4.4 × 2.4) first. This leads to calculating 3.3 × 2.4 = 7.92, and then attempting to divide 7.92 by 0.6. This is time-consuming and highly prone to arithmetic mistakes. Always cancel out fractions first. Final Answer: Therefore, the correct answer is 13.2.

Question 9

$(0.05 	imes 6.25) \div 2.5 = x$

Accepted Answer

Concept & Logic: The order of operations (BODMAS) states we evaluate terms within brackets first. To streamline decimal division and multiplication, converting terms to whole numbers or simplifying expressions before executing the calculation minimizes errors. Step-by-Step Solution: • Given: (0.05 × 6.25) \div 2.5 = x • Calculation: You can evaluate the bracket first. 0.05 × 6.25 = 0.3125 • Next, divide the result by 2.5. To do this easily, shift the decimal point one place to the right for both numbers: (0.3125/2.5) = (3.125/25) • Perform the division: 3.125 \div 25 = 0.125 • Check the options. The calculated value 0.125 is not among (a), (b), (c), or (d). Exam Strategy & Shortcut: Instead of calculating 0.05 × 6.25 first, recognize that (a × b) \div c is the same as a × (b \div c). 0.05 × (6.25 \div 2.5) Mentally calculate 6.25 \div 2.5. It is exactly 2.5 (since 25^2 = 625). Now, just multiply 0.05 × 2.5. 5 × 25 = 125. Adjusting for three decimal places gives 0.125. This takes seconds and avoids ugly decimals. Common Pitfall: Misplacing the decimal point during multiplication (e.g., getting 3.125 instead of 0.3125 for the bracket), leading to an incorrect final quotient of 1.25 (option d). Always count total decimal places carefully. Final Answer: Therefore, the correct answer is None of these (0.125).

Question 10

$5.5 - [6.5 - \{3.5 \div (6.5 - \overline{5.5 - 2.5})\}]$ is equal to

Accepted Answer

Concept & Strategy: This problem tests the comprehensive VBODMAS rule. The Vinculum (the overline bar) acts as the innermost parenthesis and must be resolved first. Next, resolve round brackets (), followed by curly braces \{\}, square brackets [], and finally the operations outside. Step-by-Step Solution: • Given: 5.5 - [6.5 - \{3.5 \div (6.5 - \overline{5.5 - 2.5})\}] • Calculation: Resolve the Vinculum (bar bracket) first. \overline{5.5 - 2.5} = 3.0 • Substitute and resolve the round brackets. (6.5 - 3.0) = 3.5 • Substitute and resolve the curly braces. \{3.5 \div 3.5\} = 1 • Substitute and resolve the square brackets. [6.5 - 1] = 5.5 • Final subtraction outside all brackets. 5.5 - 5.5 = 0 Exam Strategy & Shortcut: Do not rewrite the entire equation for every single step. Instead, simplify the nested brackets mentally or jot down running totals starting from the innermost bar. Track it like this: 5.5 - 2.5 \rightarrow 3 6.5 - 3 \rightarrow 3.5 3.5 \div 3.5 \rightarrow 1 6.5 - 1 \rightarrow 5.5 Final step: 5.5 - 5.5 \rightarrow 0. Working inwards-out visually saves critical time. Common Pitfall: Ignoring the vinculum completely and computing (6.5 - 5.5 - 2.5) straight across as if the bar didn't exist. This incorrectly yields (1 - 2.5) = -1.5, which completely derails the subsequent division and leads to a wrong answer. Final Answer: Therefore, the correct answer is 0.

Question 11

$.04 	imes x = .000016$

Accepted Answer

Concept & Formula: When dealing with very small decimals, tracking the number of decimal places or converting the numbers into fractions limits the risk of alignment errors during division. x = (a/b) Step-by-Step Solution: • Given: 0.04 × x = 0.000016 • Calculation: Isolate the unknown variable x. x = (0.000016/0.04) • Multiply both the numerator and denominator by 100 to clear the decimal point in the denominator (shifting the decimal two places to the right). x = (0.000016 × 100/0.04 × 100) x = (0.0016/4) • Divide 0.0016 by 4. 0.0016 \div 4 = 0.0004 Exam Strategy & Shortcut: Use the "decimal place subtraction" rule for multiplication. The target product (0.000016) has 6 decimal places. The given multiplier (0.04) has 2 decimal places. Therefore, the unknown factor must have 6 - 2 = 4 decimal places. Since 16 \div 4 = 4, the answer is simply a 4 placed with 4 decimal places, which writes exactly as 0.0004. Common Pitfall: Miscounting the zeros during standard long division. Under time pressure, it's easy to accidentally answer 0.04 or 0.004 by rushing the calculation. Relying strictly on counting decimal places entirely eliminates this risk. Final Answer: Therefore, the correct answer is 0.0004.

Question 12

$\frac{.009}{x} = .01$

Accepted Answer

Concept & Formula: This is a basic decimal division problem. The equation involves isolating the unknown variable x. Dividend \div Divisor = Quotient Step-by-Step Solution: • Given: (0.009/x) = 0.01 • Calculation: Rearrange the equation to solve for x: x = (0.009/0.01) Multiply the numerator and denominator by 1000 to remove decimals: x = (9/10) Convert back to a decimal: x = 0.9 Exam Strategy & Shortcut: When dividing decimals, focus on the shift of decimal places. You are dividing 0.009 (3 decimal places) by 0.01 (2 decimal places). The result will have 3 - 2 = 1 decimal place. 9 \div 1 = 9, with 1 decimal place makes it 0.9. This allows you to solve the equation mentally in seconds. Common Pitfall: The most common mistake is miscounting the decimal shifts, leading to 0.09 or 9. Always write out the powers of 10 if you are unsure, or cross-check by multiplying your answer by the quotient (0.9 × 0.01 = 0.009). Final Answer: Therefore, the correct answer is .9.

Question 13

If $\frac{144}{0.144} = \frac{14.4}{x}$, then the value of $x$ is

Accepted Answer

Concept & Logic: This problem tests proportional reasoning and the simplification of decimal fractions. The key is to simplify the known ratio before solving for the unknown. (a/b) = (c/d) \implies d = (b × c/a) Step-by-Step Solution: • Given: (144/0.144) = (14.4/x) • Calculation: First, simplify the left side of the equation. (144/0.144) = (144/\frac{144){1000}} = 144 × (1000/144) = 1000 Now, substitute this back into the equation: 1000 = (14.4/x) Rearrange to solve for x: x = (14.4/1000) Shift the decimal point three places to the left: x = 0.0144 Exam Strategy & Shortcut: Observe the powers of 10. The left side (144/0.144) is simply 10^3 = 1000. Therefore, x must be 14.4 shifted three decimal places smaller. Mentally shifting 14.4 left by three places instantly gives 0.0144. Common Pitfall: A frequent error is placing the decimal point incorrectly during division by 1000, leading to 0.144 or 1.44. Always remember that dividing by 10^n moves the decimal point n places to the left. Final Answer: Therefore, the correct answer is 0.0144.

Question 14

$40.04 \div 0.4 = x 	imes 0.05$

Accepted Answer

Concept & Logic: The equation requires simplifying a decimal division on one side and then isolating the variable x by dividing by a decimal fraction. (a/b) = c × d \implies c = (a/b × d) Step-by-Step Solution: • Calculation: First, evaluate the left side of the equation: (40.04/0.4) Multiply numerator and denominator by 10 to simplify: (400.4/4) = 100.1 Substitute this back into the main equation: 100.1 = x × 0.05 Solve for x: x = (100.1/0.05) Multiply numerator and denominator by 100: x = (10010/5) Perform the division: x = 2002 Exam Strategy & Shortcut: Instead of dividing step-by-step, you can combine the denominators. x = (40.04/0.4 × 0.05) Calculate the denominator product first: 0.4 × 0.05 = 0.02 So, x = (40.04/0.02) = (4004/2) = 2002. This method heavily reduces calculation steps and prevents decimal shifting errors. Common Pitfall: Students often mishandle the division by 0.05, multiplying instead or misplacing the decimal, incorrectly landing on 20.02 or 200.2. Always double-check decimal shifts when dividing by values less than 1. Final Answer: Therefore, the correct answer is 2002.

Question 15

$48 \div 75 	imes 84.5 \div 20 = x$

Accepted Answer

Concept & Strategy: According to the BODMAS/PEMDAS rule, division and multiplication have the same precedence and are evaluated from left to right. Converting divisions into multiplication by fractions simplifies the process. a \div b × c \div d = a × (1/b) × c × (1/d) Step-by-Step Solution: • Calculation: Convert the expression into fractions for easier cancellation: x = 48 × (1/75) × 84.5 × (1/20) Group the terms: x = (48/75) × (84.5/20) Simplify (48/75) by dividing numerator and denominator by 3: (16/25) Now substitute it back: x = (16/25) × (84.5/20) Simplify further by dividing 16 and 20 by 4: x = (4/25) × (84.5/5) Multiply the numerators and denominators: x = (338/125) To easily divide by 125, multiply numerator and denominator by 8: x = (338 × 8/1000) x = (2704/1000) x = 2.704 Exam Strategy & Shortcut: Approximation is key for verification. (48/75) is roughly (2/3) (or 0.64). (84.5/20) is roughly 4.2. 0.64 × 4.2 is approximately 2.68. Looking at the options, 2.704 is the only close answer, instantly eliminating the others without rigorous final calculations. Common Pitfall: Applying multiplication before division incorrectly (e.g., multiplying 75 by 84.5 first) because of misinterpreting the left-to-right rule. Always treat division as multiplying by the reciprocal to avoid order-of-operations errors. Final Answer: Therefore, the correct answer is 2.704.

Question 16

$(5420 + 3312 + x) \div 600 = 25.93$

Accepted Answer

Concept & Logic: This is an algebraic equation requiring simplification using the BODMAS rule. The core concept is isolating the variable within the parentheses by reversing operations. Step-by-Step Solution: • Calculation: Start by eliminating the division by 600. Multiply both sides by 600: 5420 + 3312 + x = 25.93 × 600 Perform the multiplication on the right side: 25.93 × 600 = 2593 × 6 = 15558 Now, simplify the left side by adding the known numbers: 8732 + x = 15558 Isolate x by subtracting 8732 from both sides: x = 15558 - 8732 x = 6826 Exam Strategy & Shortcut: Use the Unit Digit Method to bypass full addition and subtraction. Right side: 25.93 × 600 results in an integer ending in 3 × 6 = 18 (unit digit 8). Left side: 5420 + 3312 gives a unit digit of 0 + 2 = 2. Equation for unit digits: 2 + x = 8. The unit digit of x must be 6. Looking at the options, both 6286 and 6826 end in 6. A quick estimation (15500 - 8700 \approx 6800) immediately points to 6826. Common Pitfall: A standard error is attempting to distribute the division (e.g., dividing 5420 by 600) prematurely, which complicates the arithmetic and introduces rounding errors. Always isolate the grouped term first. Final Answer: Therefore, the correct answer is 6826.

Question 17

$786 \div 24 	imes x = 6.55$

Accepted Answer

Concept & Formula: According to BODMAS rules, division and multiplication possess equal priority and must be executed sequentially from left to right. (a/b) × x = c \implies x = (c × b/a) Step-by-Step Solution: • Calculation: Express the equation as a fraction: (786/24) × x = 6.55 Simplify the fraction (786/24) by dividing both numerator and denominator by 6: 786 \div 6 = 131 24 \div 6 = 4 So, (131/4) × x = 6.55 Isolate x: x = (6.55 × 4/131) Calculate the numerator: 6.55 × 4 = 26.2 Solve for x: x = (26.2/131) Notice that 131 × 2 = 262. Therefore: x = (26.2/131) = 0.2 Exam Strategy & Shortcut: Look at the options and use unit logic. (786/24) \approx (800/25) = 32. The equation is roughly 32 × x = 6.55. x \approx (6.55/32) \approx (6.4/32) = 0.2. This approximation safely narrows down the answer to exactly 0.2 within moments. Common Pitfall: A common error is multiplying 24 × x first before dividing, which completely violates the left-to-right division-multiplication execution order rule. Always process operations from left to right. Final Answer: Therefore, the correct answer is 0.2.

Question 18

A maximum of how many pieces of 12.6 cm can be cut from a 857 cm long rod?

Accepted Answer

Concept & Formula: To find the maximum number of whole pieces that can be obtained, perform integer division (floor division) of the total length by the piece length. Number of pieces = \lfloor \frac{Total Length}{Piece Length} \rfloor Step-by-Step Solution: • Given: Total length of the rod = 857 cm Length of each piece = 12.6 cm • Calculation: Set up the division: Number of pieces = (857/12.6) Multiply both the numerator and denominator by 10 to clear the decimal: Number of pieces = (8570/126) Simplify the fraction by dividing by 2: (4285/63) Perform long division: 4285 \div 63 = 68.015... Since we can only cut whole pieces, truncate the decimal value to the nearest lower integer. Number of pieces = 68 Exam Strategy & Shortcut: Use fast multiplication and estimation. 12.6 × 60 = 756 Remaining length = 857 - 756 = 101 Now find how many times 12.6 goes into 101: 12.6 × 8 = 100.8 Total whole pieces = 60 + 8 = 68. This eliminates long division entirely. Common Pitfall: Students sometimes round up to the nearest integer if the decimal fraction is high. In cutting/partitioning problems, you can never round up because you cannot create a piece out of missing material. Final Answer: Therefore, the correct answer is 68.

Question 19

The price of commodity X increases by 40 paise every year, while the price of commodity Y increases by 15 paise every year. If in 2004, the price of commodity X was ₹ 4.20 and that of Y was ₹ 6.30, in which year commodity X will cost 40 paise more t…

Accepted Answer

Concept & Logic: This is a relative speed or linear equations problem. The price gap between the two commodities shrinks every year because Commodity X increases faster than Commodity Y. We need to find the number of years it takes for X to catch up and surpass Y by a specific margin. Target Year = Base Year + \frac{Initial Gap + Target Gap}{Relative Increase Rate} Step-by-Step Solution: • Given: Initial Price of X in 2004 = ₹ 4.20 Initial Price of Y in 2004 = ₹ 6.30 Annual increase of X = 0.40 ₹ (40 paise) Annual increase of Y = 0.15 ₹ (15 paise) Target difference = X is 40 paise (0.40 ₹) more than Y. • Calculation: Let n be the number of years after 2004. Price of X after n years = 4.20 + 0.40n Price of Y after n years = 6.30 + 0.15n Set up the equation based on the condition that X costs 0.40 ₹ more than Y: (4.20 + 0.40n) - (6.30 + 0.15n) = 0.40 Simplify the equation: 0.25n - 2.10 = 0.40 0.25n = 2.50 Solve for n: n = (2.50/0.25) n = 10 years Add 10 years to the base year: 2004 + 10 = 2014 Exam Strategy & Shortcut: Use the concept of relative speed. Commodity X catches up to Commodity Y at a relative rate of 40 - 15 = 25 paise per year. Current price gap = 6.30 - 4.20 = 2.10 ₹ (or 210 paise). Target state: X is 40 paise ahead. Total gap to cover = 210 paise (to tie) + 40 paise (to lead) = 250 paise. Time required = (250/25) = 10 years. 2004 + 10 = 2014. This skips building algebraic equations entirely. Common Pitfall: Students often solve for when the prices become equal (210 / 25 = 8.4 years) and forget to account for the additional 40 paise target margin required by the question. Final Answer: Therefore, the correct answer is 2014.

Question 20

A sum of ₹ 6.25 is made up of 80 coins which are either 10P or 5P. How many are there of each kind?

Accepted Answer

Concept & Strategy: This is a classic mixture problem that can be solved using simultaneous equations or the rule of alligation. The fastest approach is treating it as a weighted average problem (Alligation). Step-by-Step Solution: • Given: Total number of coins = 80 Total value = ₹ 6.25 = 625 Paise Denominations = 10 Paise and 5 Paise • Calculation (Using Equations): Let the number of 10P coins be x. Then, the number of 5P coins will be 80 - x. Set up the total value equation: 10x + 5(80 - x) = 625 Expand and simplify: 10x + 400 - 5x = 625 5x + 400 = 625 5x = 225 x = 45 So, there are 45 coins of 10P. Number of 5P coins = 80 - 45 = 35. Exam Strategy & Shortcut: Use the Rule of Alligation for lightning-fast solving. Assume ALL 80 coins are 10P \implies 80 × 10 = 800 Paise. Assume ALL 80 coins are 5P \implies 80 × 5 = 400 Paise. Actual mean value is 625 Paise. Apply alligation cross-subtraction: Ratio of 10P to 5P = (625 - 400) : (800 - 625) = 225 : 175 Simplify by dividing by 25: = 9 : 7 Total parts = 9 + 7 = 16. Number of 10P coins = (9/16) × 80 = 9 × 5 = 45. Number of 5P coins = (7/16) × 80 = 7 × 5 = 35. Common Pitfall: Mixing up the final order in the answer options. Since the question asks for "10P or 5P" in that order, the answer must be listed as (10P count, 5P count), which is 45, 35. Choosing 35, 45 is a tragic unforced error. Final Answer: Therefore, the correct answer is 45, 35.

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