Question 1

Compute the exact value of the decimal power: (0.04)^3. Present the place-value reasoning clearly.

Accepted Answer

Introduction / Context: Raising a decimal to a power can be done cleanly by treating the number as a fraction with a power of 10 in the denominator. This avoids confusion and keeps track of zeros precisely. Given Data / Assumptions: Compute (0.04)^3. 0.04 = 4/100 = 4 × 10^-2. Concept / Approach: (a × 10^-b)^3 = a^3 × 10^-3b. Here a = 4 and b = 2. Alternatively, use fraction arithmetic: (4/100)^3 = 64/1,000,000. Step-by-Step Solution: (0.04)^3 = (4/100)^3 = 4^3 / 100^3.4^3 = 64 and 100^3 = 1,000,000.Thus (0.04)^3 = 64 / 1,000,000 = 0.000064. Verification / Alternative check: Using powers of ten: 0.04 = 4 × 10^-2 ⇒ (0.04)^3 = 4^3 × 10^-6 = 64 × 10^-6 = 0.000064. Why Other Options Are Wrong: 0.064, 0.0064, 0.00064: Each is off by one or more decimal places. 0.0000064: One extra zero, indicates miscounting place values. Common Pitfalls: Losing track of zeros when cubing denominators; mixing up squaring and cubing of 100. Final Answer: 0.000064

Question 2

Write the decimal 47 ÷ 10,000 in standard decimal form. Ensure correct placement of the decimal point.

Accepted Answer

Introduction / Context: Converting a fraction with a power of ten in the denominator into a decimal is a frequent task in aptitude tests. Precision in placing the decimal point is the only real challenge, especially with multiple zeros involved. Given Data / Assumptions: Value: 47/10,000. 10,000 = 10^4, meaning shift the decimal four places to the left from 47.0000. Concept / Approach: Divide by powers of ten by moving the decimal point left. Since 47 = 47.0000, dividing by 10,000 places the decimal four positions left, yielding a number much smaller than 1. Step-by-Step Solution: Start with 47.0000.Move the decimal 4 places left for division by 10,000.Result: 0.0047 (which can be written as .0047). Verification / Alternative check: As a fraction: 47/10,000 simplifies to 47 × 10^-4 = 4.7 × 10^-3 = 0.0047. Why Other Options Are Wrong: .0470: Only three places moved; too large by a factor of 10. .00047 and .000047: Too many places moved; too small. .47: Off by 100; decimal not shifted at all. Common Pitfalls: Miscounting zeros in 10,000; forgetting to add leading zeros when shifting the decimal. Final Answer: .0047

Question 3

Find the number to subtract from the product 0.527 × 2.013 so that the result equals 1. Compute the product accurately first.

Accepted Answer

Introduction / Context: This problem asks you to combine accurate decimal multiplication with a simple linear adjustment. The key is to compute the product exactly, then determine the small difference needed to reach the target value of 1. Given Data / Assumptions: Target: (0.527 × 2.013) − x = 1 We must find x. Concept / Approach: First multiply the decimals precisely. Then rearrange the equation to x = (0.527 × 2.013) − 1. Small digit errors derail the result, so compute carefully or break the multiplication into parts (0.5, 0.02, 0.007 components) for reliability. Step-by-Step Solution: Compute 2.013 × 0.527 = (2.013 × 0.5) + (2.013 × 0.02) + (2.013 × 0.007).= 1.0065 + 0.04026 + 0.014091 = 1.060851.Set 1.060851 − x = 1, hence x = 1.060851 − 1 = 0.060851. Verification / Alternative check: Direct multiplication confirms 1.060851. Subtracting 0.060851 indeed gives exactly 1. Why Other Options Are Wrong: 2.060851, 1.9339085: Misinterpretation of what to subtract or arithmetic mix-up. 0.939085: That would be 1.060851 − 0.939085 = 0.121766, not 1. 0.061851: Rounding or copying error in the thousandths place. Common Pitfalls: Dropping digits in the product, or subtracting in the wrong direction (adding to reach 1 instead of subtracting from the product). Final Answer: 0.060851

Question 4

Convert the recurring decimal 0.\u0305\u030553 (with 53 repeating) into a fraction in lowest terms. Identify the correct equivalent fraction.

Accepted Answer

Introduction / Context: Recurring decimals can be expressed as exact fractions using a standard algebraic trick. Here, 0.\u0305\u030553 means the block “53” repeats indefinitely (0.535353…). The task is to convert this repeating decimal to a simplified fraction. Given Data / Assumptions: Decimal: 0.535353… where the two-digit block 53 repeats. Let x represent the repeating decimal. Concept / Approach: For a two-digit repetend, multiply x by 100 to align the repeating parts. Subtract the original x to eliminate the repeating tail, then solve for x as a ratio of integers. Reduce to lowest terms if needed. Step-by-Step Solution: Let x = 0.535353…100x = 53.535353…Subtract: 100x − x = 53.535353… − 0.535353… = 53.Hence 99x = 53 ⇒ x = 53/99. Verification / Alternative check: Divide 53 by 99 on a calculator to see the repeating pattern 0.535353…, confirming the exact match. Why Other Options Are Wrong: 53/100 and 53/1000: These are terminating decimals, not repeating; they give 0.53 and 0.053 respectively. 53/90 and 26/45: Do not generate the exact 0.535353… sequence. Common Pitfalls: Forgetting to multiply by the correct power of 10 (100 for a two-digit repetend), or reducing incorrectly. Final Answer: 53 / 99

Question 5

Use distributive factoring with decimals: (.803 × .647) + (.803 × .353). Factor out the common multiplier to simplify.

Accepted Answer

Introduction / Context: This exercise reinforces the distributive property a(b + c) = ab + ac in a decimal context. Spotting a common factor saves time and avoids unnecessary multiplication, leading to an exact and tidy result. Given Data / Assumptions: Expression: (.803 × .647) + (.803 × .353) All decimals are exact; keep precision. Concept / Approach: Factor out the common term .803. Then simply add the remaining decimals inside the parentheses. Since .647 + .353 = 1.000, the result is immediate and exact. Step-by-Step Solution: (.803 × .647) + (.803 × .353) = .803 × (.647 + .353).Compute inside: .647 + .353 = 1.000.Therefore the entire value is .803 × 1.000 = .803. Verification / Alternative check: Perform the two products and add them; rounding carefully yields the same .803. The factoring method is quicker and eliminates rounding risk. Why Other Options Are Wrong: 1 and 1.450: Misapplied addition or multiplication. .45 and .353: Values appear from partial sums or misplaced decimals. Common Pitfalls: Adding inside the parentheses incorrectly to something slightly below or above 1; misplacing the decimal in .803. Final Answer: .803

Question 6

Determine the place value of the digit 3 in the decimal number 0.07359. Express the place value as a fraction of a unit.

Accepted Answer

Introduction / Context: Understanding place value to the right of the decimal point is fundamental. Each position represents a reciprocal power of 10: tenths, hundredths, thousandths, ten-thousandths, and so on. This question asks you to identify the exact place value for a specific digit in a given decimal. Given Data / Assumptions: Number: 0.07359 Digit of interest: 3 Concept / Approach: List each digit’s place moving right from the decimal: tenths (10^-1), hundredths (10^-2), thousandths (10^-3), ten-thousandths (10^-4), hundred-thousandths (10^-5). Identify where the 3 sits in this sequence and express its place value as a fraction of a unit. Step-by-Step Solution: 0.07359 has digits after the decimal as: 0 (tenths), 7 (hundredths), 3 (thousandths), 5 (ten-thousandths), 9 (hundred-thousandths).The 3 is in the thousandths place, so its place value is 3 × (1/1000) = 3/1000. Verification / Alternative check: Write 0.07359 as a sum: 7/100 + 3/1000 + 5/10000 + 9/100000. The coefficient of 3 is clearly 1/1000. Why Other Options Are Wrong: 3: This is the face value, not place value. 3/100 and 3/10000 and 3/100000: These correspond to hundredths, ten-thousandths, and hundred-thousandths, not the position of 3 here. Common Pitfalls: Miscounting decimal places, especially when a zero follows the decimal before non-zero digits. Final Answer: 3/1000

Question 7

Percentage comparison task: By how much is 12% of 24.2 greater than 10% of 14.2?

Accepted Answer

Introduction / Context: This problem checks fluency with percentages and decimal arithmetic. We compare two percentage amounts and report how much the first exceeds the second. Such comparisons appear often in finance and data analysis. Given Data / Assumptions: Compute 12% of 24.2. Compute 10% of 14.2. Report difference first amount minus second amount. Concept / Approach: Convert percent to decimal using p% = p/100. Then multiply by the base quantity. Finally subtract to find how much one value is greater than the other. Step-by-Step Solution: 12% of 24.2 = 0.12 * 24.2 = 2.90410% of 14.2 = 0.10 * 14.2 = 1.42Difference = 2.904 − 1.42 = 1.484 Verification / Alternative check: Use fraction form if preferred: 12/100 * 24.2 = 2.904 and 10/100 * 14.2 = 1.42. Subtraction confirms 1.484 with no rounding issues. Why Other Options Are Wrong: 0.1484 is off by a factor of 10. 14.84 misplaces the decimal by two places. 2.762 results from adding rather than subtracting. None of the above does not match because an exact value exists among the options. Common Pitfalls: Placing the decimal incorrectly when converting percent to decimal, swapping the bases, or computing the relative percent instead of the absolute difference. Final Answer: 1.484

Question 8

Apply order of operations: Evaluate 3.5 + 21 × 1.3

Accepted Answer

Introduction / Context: This arithmetic item tests the correct use of order of operations with decimals. Multiplication must be performed before addition. Given Data / Assumptions: Expression: 3.5 + 21 × 1.3 Standard operator precedence applies: multiplication before addition. Concept / Approach: Compute the product 21 × 1.3 first, then add 3.5 to the result. Handle decimals carefully to avoid place value mistakes. Step-by-Step Solution: Compute 21 × 1.3 = 27.3Add 3.5: 27.3 + 3.5 = 30.8 Verification / Alternative check: Think of 1.3 as 13/10. Then 21 × 13/10 = 273/10 = 27.3. Adding 3.5 confirms the total 30.8. Why Other Options Are Wrong: 7.28 and 6.13 come from mishandling decimals or reversing operations. 72.8 is a misplaced decimal from 7.28 or from adding before multiplying. None of these is correct as an option text because the exact numeric 30.8 is not listed among the numerical options; hence the right choice here is the text None of these. Common Pitfalls: Adding 3.5 and 21 first or moving the decimal incorrectly when multiplying by 1.3. Final Answer: None of these (since the correct numeric value is 30.8).

Question 9

Unit conversion insight: What decimal fraction of one litre is 40 ml?

Accepted Answer

Introduction / Context: Fluid measure conversions are routine in science and daily life. Here we express a small volume as a decimal fraction of a standard litre. Given Data / Assumptions: 1 litre = 1000 millilitres. We have 40 millilitres. Ask for decimal fraction of a litre represented by 40 millilitres. Concept / Approach: A decimal fraction of a whole equals part divided by whole. Convert both amounts to consistent units and divide. Step-by-Step Solution: Fraction = 40 ml / 1000 mlCompute 40/1000 = 0.04Therefore the decimal fraction is 0.04 Verification / Alternative check: As a percentage, 0.04 equals 4%. Four percent of a litre is 40 ml, which matches the given amount. Why Other Options Are Wrong: 0.4 equals 400 ml which is too large. 0.05 equals 50 ml. None of these is invalid because 0.04 appears. 0.4% would be 4 ml not 40 ml. Common Pitfalls: Confusing millilitres and litres or dropping a zero when dividing by 1000, which shifts the decimal point incorrectly. Final Answer: 0.04

Question 10

Approximation using unit rates: If three dozens mangoes cost ₹ 245, estimate the cost of 363 mangoes.

Accepted Answer

Introduction / Context: Estimation with proportional scaling is a key aptitude skill. We scale the known price for a fixed count to find an approximate price for a much larger count. Given Data / Assumptions: Cost for 3 dozens = 36 mangoes is ₹ 245. Find approximate cost for 363 mangoes. Assume constant unit price and allow rounding to the nearest reasonable option. Concept / Approach: Compute the unit price per mango as total cost divided by 36. Multiply by 363. Round to the closest offered option since the prompt requests an approximate cost. Step-by-Step Solution: Unit price ≈ 245 / 36 ≈ 6.8056Total cost ≈ 6.8056 * 363Breakdown: 6.8056 * 300 ≈ 2041.68; * 60 ≈ 408.336; * 3 ≈ 20.4168Sum ≈ 2041.68 + 408.336 + 20.4168 ≈ 2470.43Closest option to ₹ 2470 is ₹ 2400 Verification / Alternative check: Use proportionality: 363/36 ≈ 10.0833. Then 10.0833 * 245 ≈ 2470.41, confirming the estimate above. Why Other Options Are Wrong: ₹ 2000 and ₹ 2200 are too low relative to the computed estimate. ₹ 2600 overshoots more than ₹ 2400 does. ₹ 2400 is the nearest provided figure. Common Pitfalls: Forgetting that three dozens equals 36, rounding unit price too early, or multiplying by 12 by mistake. Final Answer: ₹ 2400

Question 11

Ordering rational numbers: Out of 4/7, 5/13, 6/11, 3/5, and 2/3, which is the second smallest fraction?

Accepted Answer

Introduction / Context: Comparing fractions efficiently is essential. Converting all to a common denominator is possible but slow; instead compare via decimal values or cross multiplication. Given Data / Assumptions: Fractions: 4/7, 5/13, 6/11, 3/5, 2/3. Goal: identify the second smallest. Concept / Approach: Estimate each fraction as a decimal or compare pairs using cross multiplication. Sort the approximate values. Step-by-Step Solution: 5/13 ≈ 0.3846 (smallest)6/11 ≈ 0.54554/7 ≈ 0.57143/5 = 0.62/3 ≈ 0.6667Ordering from small to large: 5/13, 6/11, 4/7, 3/5, 2/3Hence the second smallest is 6/11 Verification / Alternative check: Use cross multiplication to compare neighbors: for example, compare 6/11 and 4/7 by 6*7 = 42 versus 4*11 = 44, so 6/11 < 4/7, matching the order above. Why Other Options Are Wrong: 5/13 is the smallest, not second smallest. 4/7 and 3/5 are larger than 6/11. 2/3 is the largest here. Common Pitfalls: Rounding too coarsely or misordering fractions that are close in value, especially 6/11 and 4/7. Final Answer: 6/11

Question 12

Daily average from an annual total: In the year 2011, Shantanu received a total pocket allowance of ₹ 3832.5. Find his allowance per day.

Accepted Answer

Introduction / Context: Average per day over a calendar year is a straightforward division problem, provided we use the correct number of days for that year. Given Data / Assumptions: Total allowance in 2011 = ₹ 3832.5 Year 2011 has 365 days. Allowance was spread evenly for the purpose of computing average per day. Concept / Approach: Average per day = total amount / number of days. Since 2011 is not a leap year, divide by 365. Step-by-Step Solution: Compute 3832.5 / 365Note that 365 × 10.5 = 3832.5 exactlyTherefore average per day = ₹ 10.5 Verification / Alternative check: Long division or calculator check confirms that 3832.5 divided by 365 equals 10.5 without remainder. Why Other Options Are Wrong: ₹ 9.5, ₹ 11.5, and ₹ 12.5 correspond to totals that do not match ₹ 3832.5 when multiplied by 365. Common Pitfalls: Assuming a leap year and dividing by 366, or rounding mid calculation which would introduce error. Final Answer: ₹ 10.5

Question 13

Convert terminating decimal to a simplified fraction: Express 0.34 as a vulgar fraction in lowest terms.

Accepted Answer

Introduction / Context: Changing a terminating decimal to a common fraction is a foundational skill for algebra and arithmetic. The key is to place the decimal over an appropriate power of 10 and then reduce. Given Data / Assumptions: Decimal: 0.34 We want the fraction in lowest terms. Concept / Approach: For two decimal places, write 0.34 as 34/100. Then simplify by dividing numerator and denominator by their greatest common divisor. Step-by-Step Solution: 0.34 = 34/100Compute gcd of 34 and 100 which is 2Divide: 34/100 = (34 ÷ 2) / (100 ÷ 2) = 17/50 Verification / Alternative check: Convert back: 17/50 = 0.34 because 17 ÷ 50 = 0.34 exactly. Why Other Options Are Wrong: 18/50 and 16/50 do not equal 0.34. 19/50 equals 0.38. 34/100 is equivalent but not in lowest terms. Common Pitfalls: Stopping at 34/100 and forgetting to reduce or dividing by an incorrect common factor. Final Answer: 17/50

Question 14

Add fractional decimals carefully: Find the sum of 3/10 + 5/100 + 8/1000 in decimal form.

Accepted Answer

Introduction / Context: This question practices converting simple fractions to decimals and aligning place values for addition. Given Data / Assumptions: Fractions: 3/10, 5/100, 8/1000 Sum is requested in decimal form. Concept / Approach: Convert each fraction to a decimal with correct place value, then add digit by digit, keeping tenths, hundredths, and thousandths aligned. Step-by-Step Solution: 3/10 = 0.35/100 = 0.058/1000 = 0.008Sum = 0.3 + 0.05 + 0.008 = 0.358 Verification / Alternative check: Add using thousandths: 0.300 + 0.050 + 0.008 = 0.358 exactly. Why Other Options Are Wrong: 0.853 is a digit permutation. 3.58 and 8.35 shift the decimal point incorrectly. 0.385 is an incorrect reordering of hundredths and thousandths. Common Pitfalls: Failing to align decimal places and adding hundredths as tenths or vice versa. Final Answer: 0.358

Question 15

Place value conversion: Change 75/100 to a decimal number.

Accepted Answer

Introduction / Context: Fractions with denominator 100 map directly to decimals with two places. Understanding this mapping improves speed in percentage and ratio problems. Given Data / Assumptions: Fraction: 75/100 Write in decimal form. Concept / Approach: Divide the numerator by the denominator or move the decimal point two places to the left since the denominator is 100. Step-by-Step Solution: 75 ÷ 100 = 0.75Therefore, 75/100 equals 0.75 Verification / Alternative check: As a percentage, 0.75 equals 75%, consistent with 75 parts per hundred. Why Other Options Are Wrong: 0.075 and 0.0075 move the decimal too far. 7.5 is ten times too large. 0.57 swaps digits. Common Pitfalls: Misplacing the decimal by one or two places due to rushing. Final Answer: 0.75

Question 16

Express a terminating decimal as a fraction: Write 0.5 as a vulgar fraction.

Accepted Answer

Introduction / Context: Decimals with one decimal place typically correspond to tenths. Converting 0.5 to fraction form is a direct application of place value. Given Data / Assumptions: Decimal: 0.5 Vulgar fraction requested. Simplification is not strictly required if an equivalent fraction appears among options. Concept / Approach: Since the digit 5 is in the tenths place, 0.5 equals 5 tenths which is 5/10. This fraction also reduces to 1/2, but we must match an option exactly. Step-by-Step Solution: 0.5 = 5 tenths = 5/10Optional reduction: 5/10 = 1/2 by dividing numerator and denominator by 5 Verification / Alternative check: Convert back: 5/10 = 0.5 since 5 ÷ 10 = 0.5. Why Other Options Are Wrong: 5/12, 5/9, and 5/6 represent values that are not equal to 0.5. 1/5 equals 0.2, not 0.5. Common Pitfalls: Assuming the fraction must be in lowest terms when the options require an exact text match. Here 5/10 is correct because it equals 0.5 and appears among the options. Final Answer: 5/10

Question 17

Convert the repeating decimal 0.2(7) to a vulgar fraction Here 0.2(7) means 0.2777… with only the digit 7 recurring after the first decimal place. Express this recurring decimal as a single simplified fraction.

Accepted Answer

Introduction / Context: This problem asks you to convert a recurring (repeating) decimal into a vulgar fraction (a simple fraction in lowest terms). The decimal is written as 0.2(7), which means 0.2777… with only the 7 repeating endlessly. Understanding how to separate the non-repeating and repeating parts is the key skill tested here. Given Data / Assumptions: The decimal is 0.2(7) = 0.2777… Only the 7 repeats; the first digit 2 after the decimal is non-repeating. We must produce the fraction in simplest form. Concept / Approach: Let x denote the repeating decimal. Use the standard technique: isolate the repeating part by shifting decimals using powers of 10. Because the repeating block has length 1 (only one digit repeats), multiplying by 10 moves the decimal one place; the non-repeating prefix (2) must be handled carefully. Subtracting two equations eliminates the repeating tail. Step-by-Step Solution: Let x = 0.2777…Multiply by 10 to shift one decimal place: 10x = 2.7777…Subtract: (10x − x) = 2.7777… − 0.2777… = 2.5Therefore 9x = 2.5, so x = 2.5 / 9 = 25/90 = 5/18. Verification / Alternative check: Convert 5/18 back to a decimal: 5 ÷ 18 = 0.2777…, which matches 0.2(7). The non-repeating digit 2 appears once, and the digit 7 repeats indefinitely, confirming correctness. Why Other Options Are Wrong: 5/19 ≈ 0.263… (too small); 7/19 ≈ 0.368… (too large); 3/17 ≈ 0.176… (far too small); 7/25 = 0.28 (terminating, not recurring 7’s). Only 5/18 equals 0.2777… Common Pitfalls: Readers often treat 0.2(7) as if both digits 2 and 7 are repeating (i.e., 0.(27)), which is a different number. Always verify which digits are in the repeating block. Final Answer: 5/18

Question 18

Express the decimal 0.82(68) as a vulgar fraction Interpret 0.82(68) to mean 0.82686868… with 68 repeating. Convert this recurring decimal to a simplified fraction.

Accepted Answer

Introduction / Context: The task is to convert a mixed recurring decimal, where some digits do not repeat and a block of digits repeats, into a fraction. The notation 0.82(68) means the digits “68” repeat indefinitely after the initial “82”. Given Data / Assumptions: Number: x = 0.82686868… Non-repeating part: 82 (two digits). Repeating block: 68 (two digits). Goal: reduce to lowest terms. Concept / Approach: For a decimal with k non-repeating digits and r repeating digits, use place-value shifts by 10^k and 10^(k+r). Subtract the two to eliminate the repeating tail. The closed-form shortcut is: x = (integer formed by “nonrepeat+repeat” − integer formed by “nonrepeat”) / (99…00… with r nines and k zeros). Here k = 2 and r = 2, so denominator is 9900. Step-by-Step Solution: Write x = 0.82686868…Take the 4-digit chunk “8268” (nonrepeat “82” followed by repeat “68”).Compute numerator: 8268 − 82 = 8186.Denominator: 99 for the repeating block and 00 for the two non-repeating digits ⇒ 9900.Thus x = 8186/9900 = 4093/4950 after simplification. Verification / Alternative check: Divide 4093 by 4950 to obtain approximately 0.82686868…, confirming the repeating pattern 68. Why Other Options Are Wrong: 3093/4950, 3043/4850, and 3039/4950 yield different decimals; “8268/9990” uses the wrong denominator (should be 9900 for 2+2 digits), giving an incorrect value. Common Pitfalls: Using 99 for any repeating case without appending zeros for the non-repeating digits; forgetting to subtract the non-repeating integer before forming the fraction. Final Answer: 4093/4950

Question 19

Order and select: third-highest value among 5/7, 7/13, 4/7, 4/15, and 9/14 Compare these fractions without a calculator and identify which one is the third largest.

Accepted Answer

Introduction / Context: Ranking fractions efficiently is a core aptitude skill. Instead of converting to long decimals, use approximate values or cross-multiplication to compare pairs. Here we must find the third-highest value among five given fractions. Given Data / Assumptions: Fractions: 5/7, 7/13, 4/7, 4/15, 9/14. We require only their relative order. No calculators mandated; exact ordering is possible mentally. Concept / Approach: Approximate each fraction with a simple decimal to see the order quickly. Alternatively, compare two fractions a/b and c/d by cross-multiplying: a*d vs. c*b. Repeat to build a sorted list. Step-by-Step Solution: Estimate each: 5/7 ≈ 0.7143; 9/14 ≈ 0.6429; 4/7 ≈ 0.5714; 7/13 ≈ 0.5385; 4/15 ≈ 0.2667.Descending order: 5/7 (1st), 9/14 (2nd), 4/7 (3rd), 7/13 (4th), 4/15 (5th).Therefore, the third-highest fraction is 4/7. Verification / Alternative check: Use pairwise cross-multiplication to confirm: compare 4/7 vs. 7/13 ⇒ 4*13 = 52 vs. 7*7 = 49, so 4/7 > 7/13; compare 4/7 vs. 9/14 ⇒ 4*14 = 56 vs. 9*7 = 63, so 4/7 < 9/14. The position between them is consistent with third place. Why Other Options Are Wrong: 5/7 is the largest; 9/14 is second; 7/13 and 4/15 are smaller than 4/7. Hence only 4/7 fits the third-highest rank. Common Pitfalls: Assuming larger numerators imply larger values without considering denominators; misestimating 7/13 as more than 4/7 due to 7 > 4. Final Answer: 4/7

Question 20

Order and select: second-smallest value among 4/7, 5/13, 6/11, 3/5, and 2/3 Rank the fractions and identify the second smallest.

Accepted Answer

Introduction / Context: Here you must identify the second-smallest fraction from a set. This type of question checks your ability to compare fractions efficiently using estimation or cross-multiplication. Given Data / Assumptions: Candidates: 4/7, 5/13, 6/11, 3/5, 2/3. We seek the second-smallest (the next value after the minimum). Concept / Approach: Estimate each value: 5/13 is clearly below 0.4; 6/11 is a bit below 0.55; 4/7 is about 0.57; 3/5 equals 0.6; 2/3 is about 0.6667. Order them by size from smallest to largest. Step-by-Step Solution: Approximations: 5/13 ≈ 0.3846 (smallest); 6/11 ≈ 0.5455; 4/7 ≈ 0.5714; 3/5 = 0.6; 2/3 ≈ 0.6667.Ascending order: 5/13, 6/11, 4/7, 3/5, 2/3.Thus, the second-smallest is 6/11. Verification / Alternative check: Cross-multiply 6/11 and 4/7: 6*7 = 42 vs. 4*11 = 44; since 42 < 44, 6/11 < 4/7, consistent with the ordering above. Why Other Options Are Wrong: 5/13 is the smallest, not the second; 4/7, 3/5, and 2/3 are all larger than 6/11 and hence not the second-smallest. Common Pitfalls: Judging only by numerators or denominators; forgetting that 3/5 equals 0.6 exactly, which helps anchor the ordering. Final Answer: 6/11

Decimal Fraction Questions