Question 1

Convert a repeating–decimal mixed number to a fraction The intended value is 10.46(7) = 10.467777… (only the 7 repeats). Express this as a simplified fraction.

Accepted Answer

Introduction / Context: This item converts a decimal with a non-repeating part followed by a single repeating digit into a fraction. The options indicate a denominator of 900, which matches a pattern where there are two non-repeating decimal digits and a repeating block of length one. Given Data / Assumptions: Target number: 10.46(7) = 10.467777… (only the digit 7 repeats). Non-repeating part after decimal: “46” (two digits). Repeating block: “7” (one digit). Goal: obtain the simplest fraction. Concept / Approach: Separate the integer part and the fractional part with repetition. Let y = 0.46(7). For two non-repeating digits and one repeating digit, a standard shortcut is y = (467 − 46) / 900. Then add the integer part 10 to this fraction. Step-by-Step Solution: Let y = 0.46(7) = 0.467777…Compute numerator: 467 − 46 = 421.Denominator: since there are two non-repeating digits and one repeating, use 900 = 9 × 100.Thus y = 421/900.Add the integer part 10: 10 + 421/900 = (9000 + 421)/900 = 9421/900. Verification / Alternative check: Compute 9421 ÷ 900 ≈ 10.467777…, consistent with the intended decimal 10.46(7). Why Other Options Are Wrong: 9422/900, 9435/900, and 9437/900 represent nearby values with slight numerator shifts; 8521/900 corresponds to 9.46(7), a different integer part, hence not matching the choices linked to this item. Common Pitfalls: Forgetting to subtract the non-repeating part from the combined block; using 990 instead of 900 when the non-repeating part has length two and the repeating block has length one. Final Answer: 9421/900

Question 2

Evaluate the quotient of recurring decimals Compute (0.142857) ÷ (0.285714) exactly and choose the correct value.

Accepted Answer

Introduction / Context: Many repeating decimals correspond to simple fractions. Recognizing 0.142857 and 0.285714 as classic recurring forms linked to sevenths makes the division straightforward without long computation. Given Data / Assumptions: 0.142857 = 1/7 (recurring cycle of 142857). 0.285714 = 2/7 (the same cycle, shifted). We must compute (1/7) ÷ (2/7). Concept / Approach: Division of fractions a/b ÷ c/d equals (a/b) * (d/c). Here, (1/7) ÷ (2/7) = (1/7) * (7/2) = 1/2. Identifying the recurring-decimal–fraction mapping saves time. Step-by-Step Solution: 0.142857 = 1/7; 0.285714 = 2/7.Compute the quotient: (1/7) ÷ (2/7) = (1/7) * (7/2) = 1/2. Verification / Alternative check: Decimal check: 0.142857 ÷ 0.285714 ≈ 0.5. Since both are precise recurring forms of sevenths, the exact result is 1/2. Why Other Options Are Wrong: 1/3 and 2 do not match the straightforward fraction division; 10 is much too large; 3/2 inverts the intended ratio. Common Pitfalls: Forgetting that both decimals are exact fractional forms of sevenths, leading to unnecessary long division. Final Answer: 1/2

Question 3

Convert the repeating decimal 0.252525… into a fraction in lowest terms Identify the correct simplified fraction for the recurring block 25.

Accepted Answer

Introduction / Context: Purely recurring decimals with a block of r digits convert to fractions whose denominator is r copies of 9. For 0.252525…, the repeating block is “25” with length r = 2. Given Data / Assumptions: Number: x = 0.(25) = 0.252525… Repeating length r = 2. We want the simplified fraction. Concept / Approach: For x = 0.(AB) where AB is two digits, x = AB/99. Here AB = 25, hence x = 25/99 directly. This is a standard result from shifting the decimal by powers of 10 and subtracting to eliminate the repeat. Step-by-Step Solution: Let x = 0.252525…100x = 25.252525…Subtract: 100x − x = 25.2525… − 0.2525… = 25So 99x = 25 ⇒ x = 25/99. Verification / Alternative check: Compute 25 ÷ 99 ≈ 0.252525…, confirming the correct repeating block length and digits. Why Other Options Are Wrong: 25/90 and 25/999, 25/9999 use incorrect denominators for a two-digit repeating block; 5/18 = 0.277… represents a single-digit repeating 7, not “25”. Common Pitfalls: Using 9, 99, 999 indiscriminately without matching the block length; forgetting that the decimal is purely repeating with no non-repeating prefix. Final Answer: 25/99

Question 4

Evaluate the sum of unit fractions with patterned denominators Compute S = 1/3 + 1/15 + 1/35 + 1/63 + 1/99 and express it as a simplified fraction.

Accepted Answer

Introduction / Context: This problem revisits the classic technique of summing unit fractions whose denominators follow a pattern (often products of consecutive integers or related factors). Quick decimal estimation can guide you to the right option, but a structured approach confirms the exact fraction. Given Data / Assumptions: S = 1/3 + 1/15 + 1/35 + 1/63 + 1/99. Each term is a positive unit fraction. We seek the exact simplified sum. Concept / Approach: As a rough check, compute decimal approximations: 1/3 ≈ 0.3333; 1/15 ≈ 0.0667; 1/35 ≈ 0.0286; 1/63 ≈ 0.0159; 1/99 ≈ 0.0101. The total is about 0.4546, which equals 5/11 exactly (since 5/11 = 0.4545… recurring). This strongly indicates 5/11 as the precise result. Step-by-Step Solution: Compute a quick sum: 0.3333 + 0.0667 = 0.4000.Add 1/35 ≈ 0.0286 ⇒ 0.4286.Add 1/63 ≈ 0.0159 ⇒ 0.4445.Add 1/99 ≈ 0.0101 ⇒ 0.4546 ≈ 5/11. Verification / Alternative check: Cross-verify with exact arithmetic by bringing to a common denominator (for example, using prime factorization to build the LCM); the result will reduce to 5/11. Why Other Options Are Wrong: 10/11 = 0.909… is much too large; 9/11 and 7/11 do not match the approximate total; 1/2 = 0.5 is slightly larger than the sum. Common Pitfalls: Arithmetic slips in decimal addition; assuming an oversimplified common denominator without reducing; not recognizing the recurring decimal 0.4545… as 5/11. Final Answer: 5/11

Question 5

Evaluate a product/quotient of decimals efficiently Compute the exact value of 2 × [(3.6 × 0.48 × 2.50) / (0.12 × 0.09 × 0.5)].

Accepted Answer

Introduction / Context: Large-looking decimal expressions often simplify cleanly when grouped strategically. This problem rewards rearranging factors and canceling early to avoid handling many zeros and decimal places at once. Given Data / Assumptions: Expression: E = 2 × [(3.6 × 0.48 × 2.50) / (0.12 × 0.09 × 0.5)]. All numbers are finite decimals; exact arithmetic is expected. Concept / Approach: Multiply the numerators and denominators in convenient pairs. Use the fact that 2.50 = 2.5 and 0.5 = 1/2 to simplify. Grouping as (3.6 × 0.48) first is neat because 36 × 48 has a well-known product pattern. Step-by-Step Solution: Compute numerator: 3.6 × 0.48 = 1.728.Then 1.728 × 2.5 = 4.32.Compute denominator: 0.12 × 0.09 = 0.0108.Then 0.0108 × 0.5 = 0.0054.Inside bracket: 4.32 / 0.0054 = 800.Multiply by the outside 2: E = 2 × 800 = 1600. Verification / Alternative check: Scale all numbers by powers of 10 to clear decimals (e.g., multiply top and bottom by 1000) and then cancel common factors. You will reach the same exact integer result 1600. Why Other Options Are Wrong: 800 is the bracketed value before multiplying by 2; 500, 900, 600 come from partial or misgrouped computations. Common Pitfalls: Forgetting the outer factor 2; mishandling decimal places when multiplying or dividing; rounding mid-calculation instead of keeping exact values. Final Answer: 1600

Question 6

Find a fraction from an operation description A fraction f is multiplied by itself and then divided by its reciprocal. The result equals 18 26/27. Determine the fraction f in simplest form.

Accepted Answer

Introduction / Context: This problem encodes an algebraic equation in words. Translating the description into symbolic form reveals an elegant simplification. It assesses comfort with reciprocals, mixed numbers, and cube roots of rational numbers. Given Data / Assumptions: A fraction f is multiplied by itself (forming f^2). The result is divided by the reciprocal of the same fraction (1/f). The final value is 18 26/27, which is a mixed number. We must find f in lowest terms. Concept / Approach: Dividing by the reciprocal is equivalent to multiplying by the original fraction: f^2 ÷ (1/f) = f^2 * f = f^3. Hence f^3 equals the given mixed number converted to an improper fraction. Take the rational cube root to obtain f. Step-by-Step Solution: Convert the mixed number: 18 26/27 = (18*27 + 26)/27 = (486 + 26)/27 = 512/27.Set up the equation: f^3 = 512/27.Recognize 512 = 8^3 and 27 = 3^3.Therefore f = cube_root(512/27) = 8/3.Reduce if necessary (already simplest). So f = 8/3. Verification / Alternative check: Check: f^3 = (8/3)^3 = 512/27, which equals 18 26/27, matching the given result. The reverse operation (multiply by itself and divide by reciprocal) indeed returns f^3. Why Other Options Are Wrong: 8/27 is far too small and (8/27)^3 = 512/19683; 11/3, 22/3, 32/3 all cube to values much larger than 512/27. Only 8/3 produces exactly 512/27. Common Pitfalls: Misreading “divided by the reciprocal” as dividing by the fraction itself (which would give f), or mishandling the mixed number to improper fraction conversion. Final Answer: 8/3

Question 7

In a school’s annual sports meet, 1/4 of the boys and 3/8 of the girls participated. What fractional part of the total student population (boys + girls) took part in the event?

Accepted Answer

Introduction / Context: In problems that ask for a fraction or percentage of a combined population, you must know either the actual counts or at least the ratio between the groups. Here, different fractions of boys and girls participated. Without knowing how many boys there are relative to girls, the overall (combined) fraction cannot be uniquely determined. Given Data / Assumptions: Participating boys = 1/4 of all boys. Participating girls = 3/8 of all girls. Total students = boys + girls. No ratio or count of boys to girls is given. Concept / Approach: Overall participation fraction = (participating boys + participating girls) / (boys + girls). This simplifies to ( (1/4)*B + (3/8)*G ) / (B + G ). The value depends on the ratio B:G. Without B:G, there is no single numeric answer. Step-by-Step Solution: Let B = number of boys, G = number of girls.Participating = (1/4)B + (3/8)G.Total = B + G.Overall fraction = [ (1/4)B + (3/8)G ] / (B + G ).This expression varies with B:G; e.g., if B = G, fraction = (1/4 + 3/8) / 2 = (5/8) / 2 = 5/16 = 31.25%.If B = 2G, fraction = [ (1/4)*(2G) + (3/8)G ] / (3G) = [ (1/2)G + (3/8)G ] / (3G) = (7/8) / 3 = 7/24 ≈ 29.17%. Verification / Alternative check: Try different B:G ratios (1:1, 2:1, 1:2). Each gives a different combined percentage, confirming non-uniqueness. Why Other Options Are Wrong: 32%: Only true for specific B:G, not universally.20%: Also depends on B:G; not guaranteed.36%: Same issue—no unique ratio provided.24%: Arbitrary without B:G information. Common Pitfalls: Assuming B = G without being told; treating weighted fractions as simple averages; ignoring the dependence on the population mix. Final Answer: data inadequate

Question 8

If the fraction a/b is positive, which of the following statements must always be true?

Accepted Answer

Introduction / Context: This question tests sign analysis for rational numbers. A fraction a/b is positive when its numerator and denominator have the same sign: both positive or both negative. Recognizing the necessary and sufficient condition prevents incorrect assumptions about individual signs. Given Data / Assumptions: a/b > 0. b ≠ 0 (division by zero is undefined). Concept / Approach: For any real numbers a and b (b ≠ 0), a/b > 0 if and only if a and b have the same sign. Algebraically, that means the product ab is positive, i.e., ab > 0. This holds regardless of the individual signs of a or b as long as both are not zero simultaneously and b ≠ 0. Step-by-Step Solution: Case 1: a > 0 and b > 0 → a/b > 0 and ab > 0.Case 2: a < 0 and b < 0 → a/b > 0 and ab > 0.Both cases share ab > 0 as the must-true condition. Verification / Alternative check: Counterexamples show why individual sign claims fail: If a = -2 and b = -3, then a/b = 2/3 (positive), yet a > 0 and b > 0 are both false. However, ab = 6 > 0 remains true. Why Other Options Are Wrong: a > 0: Could be negative if b is also negative.b > 0: Could be negative if a is also negative.a + b > 0: Sum can be any sign even if a/b is positive.a and b are both even: Parity is irrelevant to sign. Common Pitfalls: Assuming positivity forces both numerator and denominator individually positive; ignoring the valid both-negative case. Final Answer: ab > 0

Question 9

5/12 of an amount equals 3 3/4 of ₹100. Find the amount.

Accepted Answer

Introduction / Context: This restatement clarifies the same computation: turning a mixed fraction into an improper fraction and solving a simple linear equation for the unknown total amount. Given Data / Assumptions: (5/12) * X = (3 3/4) * 100. Concept / Approach: Compute the right-hand side and multiply by the reciprocal of 5/12 to isolate X. Keep arithmetic exact to avoid rounding errors in currency problems. Step-by-Step Solution: 3 3/4 = 15/4; (15/4) * 100 = 375.(5/12) * X = 375 → X = 375 * (12/5).Compute: 375 * 12 = 4500; 4500 / 5 = 900. Verification / Alternative check: (5/12) of 900 is 375, which matches (3 3/4) of ₹100, so the solution is consistent. Why Other Options Are Wrong: All other listed amounts do not yield 375 when multiplied by 5/12. Common Pitfalls: Misreading 3 3/4; arithmetic errors when multiplying by reciprocals; attempting unnecessary decimal conversions before finishing fraction arithmetic. Final Answer: ₹ 900

Question 10

Convert the terminating decimal 0.2341 into a fraction p/q in lowest terms, where p and q are integers and q ≠ 0.

Accepted Answer

Introduction / Context: Terminating decimals convert cleanly to fractions by placing the digits over an appropriate power of 10 and then reducing if possible. Recognizing when a decimal terminates is the key first step. Given Data / Assumptions: Decimal: 0.2341. It has four digits after the decimal point. Concept / Approach: A decimal with four places equals an integer divided by 10^4. Therefore, 0.2341 = 2341/10000. If the numerator and denominator share a common factor, reduce to lowest terms; otherwise, the fraction is already simplified. Step-by-Step Solution: Write 0.2341 as 2341/10000 (since there are four decimal places).Check simplification: 2341 is not divisible by 2 or 5, and quick divisibility checks for small primes (3, 7, 11, 13) fail.Thus, the fraction 2341/10000 is already in lowest terms. Verification / Alternative check: Convert back to decimal: 2341/10000 = 0.2341 exactly, confirming correctness. Why Other Options Are Wrong: 2341/9990, 2339/9990, 781/3330, 11705/50000: These correspond to non-terminating (repeating) or mismatched decimals and do not equal 0.2341 exactly. Common Pitfalls: Confusing terminating decimals with repeating ones; using 999… denominators unnecessarily; attempting to reduce by 2 or 5 when the numerator is not divisible by these factors. Final Answer: 2341/10000

Question 11

Compute the highest common factor (H.C.F.) of the decimals 0.54, 1.8, and 7.2.

Accepted Answer

Introduction / Context: Finding the highest common factor (H.C.F.) of decimal numbers is easier if we remove decimals by scaling to integers, compute the integer H.C.F., and then scale back. This standard technique avoids repeated decimal operations. Given Data / Assumptions: Numbers: 0.54, 1.8, 7.2. All are finite decimals with at most one or two decimal places. Concept / Approach: Multiply each number by 100 to clear decimals uniformly: 0.54 → 54, 1.8 → 180, 7.2 → 720. Compute H.C.F.(54, 180, 720), then divide by 100 to return to the original scale. Step-by-Step Solution: Scale: 0.54, 1.8, 7.2 → 54, 180, 720.H.C.F.(54, 180) = 18.H.C.F.(18, 720) = 18 (since 720 ÷ 18 = 40).Scale back by 100: 18 / 100 = 0.18. Verification / Alternative check: Divide each original number by 0.18: 0.54/0.18 = 3, 1.8/0.18 = 10, 7.2/0.18 = 40. All results are integers, so 0.18 is indeed a common factor. No larger decimal works because the corresponding integer H.C.F. is 18. Why Other Options Are Wrong: 1.8 and 18: Too large when scaled back; do not divide 0.54 exactly..018: Too small; not the highest.0.9: Does not divide 0.54 exactly into an integer. Common Pitfalls: Forgetting to scale all numbers by the same power of 10; taking common factor but not the highest; mismanaging decimal places when scaling back. Final Answer: 0.18

Question 12

Arrange the fractions 16/19, 11/14, and 17/22 in ascending (increasing) order.

Accepted Answer

Introduction / Context: Ordering fractions requires comparing their sizes. You can convert to decimals or use cross-multiplication to avoid approximate arithmetic. The goal is to find the correct increasing order among the three given fractions. Given Data / Assumptions: Fractions: 16/19, 11/14, 17/22. We want ascending order (smallest to largest). Concept / Approach: Two common methods: (1) Convert each fraction to a decimal; (2) Cross-multiply for pairwise comparisons without computing decimals. We will use decimals for clarity and then confirm with cross-multiplication. Step-by-Step Solution: Compute approximate decimals: 16/19 ≈ 0.8421, 11/14 ≈ 0.7857, 17/22 ≈ 0.7727.Order by value: 0.7727 < 0.7857 < 0.8421.Thus, ascending order: 17/22, 11/14, 16/19.Cross-check (pairwise): Compare 17/22 vs 11/14: 17*14 = 238, 11*22 = 242 → 238 < 242 → 17/22 < 11/14 (consistent).Compare 11/14 vs 16/19: 11*19 = 209, 16*14 = 224 → 209 < 224 → 11/14 < 16/19 (consistent). Verification / Alternative check: Both decimal and cross-multiplication methods agree on the same order, ensuring correctness. Why Other Options Are Wrong: Any order placing 16/19 before either fraction or 11/14 after 16/19 contradicts the computed comparisons. Common Pitfalls: Rounding too aggressively when using decimals; misreading ascending vs descending; arithmetic errors in cross-multiplication. Final Answer: 17/22, 11/14, 16/19

Question 13

Evaluate the expression (.35 × .0015) / (.25 × .07) and express the result as a percentage.

Accepted Answer

Introduction / Context: This problem focuses on decimal multiplication and division followed by converting the result to a percentage. Precision with place values is crucial to avoid order-of-magnitude mistakes. Given Data / Assumptions: Expression: (.35 × .0015) / (.25 × .07). Final output required as a percentage. Concept / Approach: Compute numerator and denominator separately, then divide. To convert a decimal to a percentage, multiply by 100 and append the % symbol. Step-by-Step Solution: Numerator: .35 × .0015 = 0.000525.Denominator: .25 × .07 = 0.0175.Quotient: 0.000525 / 0.0175 = 0.03.Percentage: 0.03 × 100 = 3 %. Verification / Alternative check: Scale by powers of 10 to remove decimals: (35 × 15) / (25 × 7) × (10^-2 × 10^-4) / (10^-2 × 10^-2) = (525 / 175) × 10^-2 = 3 × 10^-2 = 0.03, which confirms 3 % after conversion. Why Other Options Are Wrong: 0.3 % and 30 %: Off by factors of 10 arising from incorrect decimal place handling.None of these and 0.03 %: Do not match the correct 3 % computation. Common Pitfalls: Misplacing decimal points; forgetting to multiply by 100 to convert a decimal to a percentage; rounding prematurely. Final Answer: 3 %

Question 14

Compute the value of √[ (0.324 × 0.081 × 4.624) / (1.5625 × 0.0289 × 72.9 × 64 ) ].

Accepted Answer

Introduction / Context: This evaluation problem blends decimal arithmetic with a square root. Recognizing familiar squares among decimals (like 1.5625 = 1.25^2, 0.0289 = 0.17^2, 64 = 8^2) can simplify the process, but careful computation also works. Given Data / Assumptions: Expression inside the square root: (0.324 × 0.081 × 4.624) / (1.5625 × 0.0289 × 72.9 × 64). All positive values, so the principal square root applies. Concept / Approach: Multiply the numerators and denominators separately; then take the square root. When possible, group terms as squares to reduce effort. Keep track of place values to avoid scaling errors. Step-by-Step Solution: Compute numerator: 0.324 × 0.081 × 4.624 = 0.121352256 (rounded at the end if needed).Compute denominator: 1.5625 × 0.0289 × 72.9 × 64 = 210.681 (approx.).Form the ratio: ≈ 0.121352256 / 210.681 ≈ 0.000576.Square root: √0.000576 = 0.024. Verification / Alternative check: Note 0.0289 = 0.17^2, 1.5625 = 1.25^2, and 64 = 8^2. If the remaining product reduces to 0.000576, the square root must be 0.024, confirming the result without a calculator by recognizing 24^2 = 576 and adjusting for decimal places. Why Other Options Are Wrong: 24 and 2.4: Off by multiple powers of 10; they ignore small decimal outcomes.None of these and 0.24: Do not match the correctly computed 0.024. Common Pitfalls: Premature rounding before taking the square root; mishandling decimal places; not simplifying with perfect square recognitions that speed up checking. Final Answer: 0.024

Question 15

Evaluate √(0.289 / 0.00121) and express the result as a simplified fraction.

Accepted Answer

Introduction / Context: This problem uses exact decimal-to-fraction conversion before taking a square root. Spotting perfect squares in the numerators and denominators greatly streamlines the solution. Given Data / Assumptions: Compute S = √(0.289 / 0.00121). All numbers are positive. Concept / Approach: Rewrite decimals as fractions with powers of 10 and then simplify using known squares. In particular, 0.289 = 289/1000 and 0.00121 = 121/100000. Step-by-Step Solution: 0.289 = 289/1000, 0.00121 = 121/100000.Ratio = (289/1000) / (121/100000) = (289 × 100000) / (121 × 1000) = 28900 / 121.Observe: 289 = 17^2 and 121 = 11^2, so 28900/121 = ( (17^2) × 100 ) / (11^2) = ( (170)^2 ) / (11^2 ).Therefore, √(28900/121) = 170/11. Verification / Alternative check: Compute decimal approximation: 170/11 ≈ 15.4545..., consistent with √(0.289 / 0.00121) since the ratio is about 238.842..., and the square root of a number near 238.8 is about 15.45. Why Other Options Are Wrong: 17/110 and 17/1100: Invert the scaling; results are far too small.17/11: Misses the factor of 10 inside the square root.10/11: Unrelated to the derived square-root structure. Common Pitfalls: Mistaking 0.289 for 0.0289; misplacing decimal powers when converting to fractions; taking the square root before simplifying to perfect squares. Final Answer: 170 / 11

Question 16

Given √4096 = 64, evaluate √40.96 + √0.4096 + √0.004096 + √0.00004096.

Accepted Answer

Introduction / Context: This problem exploits place-value scaling of a known square. If √4096 = 64, then moving the decimal point in 4096 simply scales the square root by powers of 10. This creates a quick path to compute several square roots without fresh squaring. Given Data / Assumptions: √4096 = 64. We need √40.96, √0.4096, √0.004096, and √0.00004096. Concept / Approach: If N = 4096, then for any k, √(N / 10^(2k)) = √N / 10^k because dividing inside the root by 10^(2k) divides outside the root by 10^k. Each given decimal equals 4096 divided by a power of 10. Step-by-Step Solution: 40.96 = 4096 / 100 → √40.96 = 64 / 10 = 6.4.0.4096 = 4096 / 10000 → √0.4096 = 64 / 100 = 0.64.0.004096 = 4096 / 1,000,000 → √0.004096 = 64 / 1000 = 0.064.0.00004096 = 4096 / 100,000,000 → √0.00004096 = 64 / 10000 = 0.0064.Sum = 6.4 + 0.64 + 0.064 + 0.0064 = 7.1104. Verification / Alternative check: Note the pattern: each successive term divides the previous square root by 10, ensuring the running total converges and is easy to add accurately. Why Other Options Are Wrong: 7.09, 7.1014, 7.12, 7.014: Each deviates from the exact decimal scaling and precise addition shown above. Common Pitfalls: Miscalculating powers of 10 under the square root; arithmetic slips when summing decimals with mixed place values; rounding prematurely instead of keeping exact tenths, hundredths, thousandths, and ten-thousandths. Final Answer: 7.1104

Question 17

Use the given square-root value to evaluate a related root: If √15 = 3.88, compute the numerical value of √(5/3). Show the relationship that links these two quantities and give the result to two decimal places.

Accepted Answer

Introduction / Context: This question tests proportional reasoning with square roots. When a square root of a product is known (here √15), you can form related roots such as √(5/3) by using basic root properties. The goal is to connect √(5/3) to √15 cleanly and compute a decimal answer without a calculator-heavy approach. Given Data / Assumptions: Given: √15 = 3.88 (use this value as exact for the problem). Find: √(5/3). Standard property: √(a/b) = √a / √b for a, b > 0. Concept / Approach: Rewrite 5/3 as 15/9 so that its square root links directly to √15. Since √(15/9) = √15 / √9 and √9 = 3, we obtain a simple division of the provided value 3.88 by 3. This avoids approximations beyond the given data and keeps the computation short and reliable. Step-by-Step Solution: Write 5/3 as 15/9.Use the identity: √(a/b) = √a / √b.So √(5/3) = √(15/9) = √15 / √9.Compute with given values: √15 / √9 = 3.88 / 3 = 1.293333… ≈ 1.29. Verification / Alternative check: Square 1.29 as a quick check: 1.29^2 ≈ 1.6641, which is close to 5/3 ≈ 1.6667. The tiny difference comes from the rounded given √15. Why Other Options Are Wrong: 0.43: Much too small; would imply √(5/3) < 1. 1.63 and 1.89: Too large; they would correspond to squaring above the target 1.6667 baseline. Common Pitfalls: Forgetting to take √9 = 3 when converting 5/3 to 15/9. Rounding too early; compute 3.88 / 3 first, then round. Final Answer: 1.29

Question 18

Work carefully with radicals and decimals: Given √5 = 2.24, evaluate (3√5) / (2√5 − 0.48) by simplifying numerator and denominator step by step.

Accepted Answer

Introduction / Context: This problem combines radicals with decimals. The key is to simplify the numerator and the denominator separately using the provided value for √5, and then perform a clean division. Such steps reinforce operational fluency with expressions containing roots and constants. Given Data / Assumptions: √5 = 2.24 (use as given). Expression: (3√5) / (2√5 − 0.48). Concept / Approach: First compute the numerator 3√5 and the linear combination in the denominator 2√5 − 0.48 by substituting √5 = 2.24. Once both are numeric, divide to get the final value. Keep exact steps to avoid decimal slips. Step-by-Step Solution: Numerator: 3√5 = 3 × 2.24 = 6.72.Denominator: 2√5 − 0.48 = 2 × 2.24 − 0.48 = 4.48 − 0.48 = 4.00.Compute the quotient: 6.72 / 4.00 = 1.68. Verification / Alternative check: Because the denominator simplifies to a neat integer (4), mental division confirms 1.68 exactly without rounding. Why Other Options Are Wrong: 0.168: Off by a factor of 10. 16.8 and 168: Off by factors of 10 and 100; these ignore the small denominator correction. Common Pitfalls: Mistaking 2√5 − 0.48 as 2(√5 − 0.48) and distributing incorrectly. Arithmetic slip when subtracting 0.48 from 4.48. Final Answer: 1.68

Question 19

Use a known quotient to find a related one via scaling: Given 12276 ÷ 155 = 79.2, compute 122.76 ÷ 15.5 by relating both numerators and denominators to the known ratio.

Accepted Answer

Introduction / Context: When a quotient is already known, you can use proportional scaling to obtain another quotient rapidly. This avoids long division and leverages place-value reasoning with decimals—an essential skill in aptitude tests and quick calculations. Given Data / Assumptions: Known: 12276 ÷ 155 = 79.2. Find: 122.76 ÷ 15.5. Concept / Approach: Notice that 122.76 is 1/100 of 12276, and 15.5 is 1/10 of 155. Therefore, 122.76 ÷ 15.5 equals (12276/100) ÷ (155/10) = (12276 ÷ 155) × (10/100) = (12276 ÷ 155) × 0.1. This scaling logic preserves the ratio structure exactly. Step-by-Step Solution: Start with the known value: 12276 ÷ 155 = 79.2.Adjust numerator and denominator: 122.76 ÷ 15.5 = (12276/100) ÷ (155/10).Compute the scaling: = (12276 ÷ 155) × (10/100) = 79.2 × 0.1.Hence the result is 7.92. Verification / Alternative check: Shift decimals equally: 122.76 ÷ 15.5 = 1227.6 ÷ 155 ≈ 7.92 (exact with given data). Why Other Options Are Wrong: 79.2 and 79.02: Fail to apply the 0.1 scaling factor correctly. 7.092: Incorrect adjustment of decimal places leads to an off-by-scale result. Common Pitfalls: Mismatching the scaling of numerator and denominator. Dropping or adding extra zeros during decimal shifts. Final Answer: 7.92

Question 20

Apply the difference of squares identity with decimals: Evaluate (.538^2 − .462^2) / (1 − 0.924) by recognizing algebraic factorization patterns and simplifying cleanly.

Accepted Answer

Introduction / Context: This question highlights the algebraic identity a^2 − b^2 = (a − b)(a + b). Recognizing this pattern avoids tedious decimal squaring. Pairing that with a straightforward subtraction in the denominator gives a clean simplification to an integer answer. Given Data / Assumptions: Numerator: 0.538^2 − 0.462^2. Denominator: 1 − 0.924. Concept / Approach: Use the identity a^2 − b^2 = (a − b)(a + b). Compute (a − b) and (a + b) once; then multiply. For the denominator, simple subtraction suffices. Compare numerator and denominator to see if cancellation occurs. Step-by-Step Solution: Set a = 0.538 and b = 0.462.a − b = 0.538 − 0.462 = 0.076.a + b = 0.538 + 0.462 = 1.000.So numerator = (a − b)(a + b) = 0.076 × 1.000 = 0.076.Denominator: 1 − 0.924 = 0.076.Quotient = 0.076 / 0.076 = 1. Verification / Alternative check: Because a + b = 1 exactly, the numerator equals a − b. Direct subtraction confirms both numerator and denominator equal 0.076. Why Other Options Are Wrong: .076: That is the numerator alone, not the final ratio. 1.042 or 2: These result from arithmetic or identity misuse. Common Pitfalls: Squaring decimals individually, increasing error risk. Forgetting the identity and missing the perfect cancellation. Final Answer: 1

Decimal Fraction Questions