Question 1

Compare two fractions via division 3/48 is what part of 1/12? Express the ratio as a single simplified fraction.

Accepted Answer

Introduction / Context: “What part of” questions translate to a division of fractions: (given fraction) ÷ (reference fraction). After forming the division, multiply by the reciprocal and simplify completely to obtain the final ratio in lowest terms. Given Data / Assumptions: Given fraction: 3/48. Reference fraction: 1/12. Compute (3/48) ÷ (1/12). Concept / Approach: Division of fractions a/b ÷ c/d equals (a/b) * (d/c). Here, multiply 3/48 by 12/1 and reduce by common factors. Aim for simplest terms to match options or determine if “None of these” applies. Step-by-Step Solution: Form the division: (3/48) ÷ (1/12) = (3/48) * (12/1).Simplify 3 * 12 / 48 = 36 / 48.Reduce: divide numerator and denominator by 12 ⇒ 36/48 = 3/4.Therefore, 3/48 is 3/4 of 1/12. Verification / Alternative check: Compute decimals: 3/48 = 0.0625 and 1/12 ≈ 0.08333. The ratio 0.0625 / 0.08333… = 0.75 = 3/4, confirming the fractional result. Why Other Options Are Wrong: 3/7, 1/12, and 4/3 do not equal 3/4; each represents a different magnitude relative to 1.Thus, “None of these” is correct because 3/4 is not listed among the options. Common Pitfalls: Forgetting to multiply by the reciprocal; failing to reduce 36/48; or switching the division order to (1/12) ÷ (3/48), which would invert the intended ratio. Final Answer: None of these

Question 2

Error-analysis in fraction operations:  A boy was instructed to multiply a given number by 8/17. Instead, he divided the number by 8/17 (i.e., multiplied by 17/8) and thus obtained a result that was 225 more than the correct product. What was the or…

Accepted Answer

Introduction / Context: This problem checks understanding of how dividing by a fraction compares to multiplying by that fraction, and how to set up an equation from a stated excess (the result is 225 more). It is a classic error-analysis question common in aptitude tests on fractions. Given Data / Assumptions: The intended operation is multiply by 8/17. The mistaken operation is divide by 8/17, which equals multiply by 17/8. The mistaken result exceeds the correct result by 225. Let the original number be x. Concept / Approach: Translate the narrative into algebraic expressions for both the correct and mistaken results, then equate their difference to 225. Use a ÷ (p/q) = × (q/p). Solve the linear equation cleanly by clearing denominators. Step-by-Step Solution: Correct result = x * (8/17).Mistaken result = x ÷ (8/17) = x * (17/8).Excess = (x * 17/8) − (x * 8/17) = 225.Compute the bracket: 17/8 − 8/17 = (289 − 64)/136 = 225/136.So x * (225/136) = 225 ⇒ x = 225 * (136/225) = 136. Verification / Alternative check: Plug x = 136: correct = 136 * 8/17 = 64; mistaken = 136 * 17/8 = 289; difference = 289 − 64 = 225, matching perfectly. Why Other Options Are Wrong: 8 and 17 are much too small; 64 is the correct result, not the original number; 272 doubles the true value and violates the 225 difference. Common Pitfalls: Forgetting that dividing by 8/17 multiplies by 17/8; subtracting in the wrong order; arithmetic slips when finding 17/8 − 8/17. Final Answer: 136

Question 3

Careful with fractions in word problems:  In an exam, a student was asked to compute 3/14 of a certain number N, but by mistake he computed 3/4 of N. His (wrong) answer was 150 greater than the correct answer. What is the value of N?

Accepted Answer

Introduction / Context: This is a straightforward linear equation framed in words. It tests converting “more than” into a difference of fractional multiples and solving for the base number. Precision with common denominators is vital. Given Data / Assumptions: Correct value should be (3/14) * N. Mistake value computed was (3/4) * N. Difference (mistake − correct) equals 150. Concept / Approach: Model the statement as ((3/4) − (3/14)) * N = 150. Reduce the fractional difference, then isolate N. Avoid mixing decimals to keep exactness. Step-by-Step Solution: Compute the difference: 3/4 − 3/14 = (42 − 12)/56 = 30/56 = 15/28.Equation: (15/28) * N = 150.Solve for N: N = 150 * (28/15) = 10 * 28 = 280. Verification / Alternative check: Correct: (3/14) * 280 = 60. Mistake: (3/4) * 280 = 210. Difference: 210 − 60 = 150, consistent. Why Other Options Are Wrong: 180, 240, and 196 yield differences other than 150; 290 is not a nice multiple that satisfies the derived equation. Common Pitfalls: Reversing the order of subtraction; mishandling 3/14; prematurely rounding; skipping fraction reduction before solving. Final Answer: 280

Question 4

Fractional exponents disguised:  A fraction x is multiplied by itself and the product is then divided by its reciprocal. The resulting value is 18 26/27. Determine the original fraction x.

Accepted Answer

Introduction / Context: Although the description seems long, it simply asks for a cube root in fractional form. Multiplying a fraction by itself gives x^2, and dividing that by the reciprocal (1/x) gives x^3. The given mixed fraction must be converted to an improper fraction before taking the cube root. Given Data / Assumptions: Operation described: (x * x) / (1/x) = x^3. Result equals 18 26/27. We must find x as a positive rational in simplest terms. Concept / Approach: Convert 18 26/27 to an improper fraction, then equate x^3 to that value. Finally, take the cube root by recognizing perfect cubes in numerator and denominator. Step-by-Step Solution: Convert 18 26/27: 18 + 26/27 = (18*27 + 26)/27 = (486 + 26)/27 = 512/27.Thus x^3 = 512/27.Notice 512 = 8^3 and 27 = 3^3.Therefore x = (8/3). Verification / Alternative check: Compute x^3 with x = 8/3: (8/3)^3 = 512/27, which matches the given result exactly. Why Other Options Are Wrong: 8/27 is the reciprocal cube, not the cube root; 22/3 and 11/3 are much larger than 8/3; thus none of the provided options matches the correct 8/3. Common Pitfalls: Forgetting that dividing by a reciprocal multiplies by the original fraction; misreading the mixed number; not recognizing perfect cubes. Final Answer: None of these (the correct fraction is 8/3).

Question 5

Make the sum an integer:  Add 1 3/4, 2 1/2, 5 7/12, 3 1/3, and 2 1/4. What is the smallest fraction that must be subtracted from this sum to make the final result a whole number?

Accepted Answer

Introduction / Context: Questions of this type focus on fractional parts. To convert a total to the next lower whole number, we only need to subtract the fractional part of the sum. Efficient handling of common denominators speeds up the process. Given Data / Assumptions: Numbers: 1 3/4, 2 1/2, 5 7/12, 3 1/3, 2 1/4. We seek the smallest fraction to subtract so the total becomes an integer. Concept / Approach: Extract and add only the fractional parts; the whole-number parts do not affect the needed subtraction. The fractional sum modulo 1 is the minimum fraction to remove to land exactly on an integer. Step-by-Step Solution: Fractional parts: 3/4, 1/2, 7/12, 1/3, 1/4.Use denominator 12: 3/4 = 9/12; 1/2 = 6/12; 7/12 = 7/12; 1/3 = 4/12; 1/4 = 3/12.Sum = (9 + 6 + 7 + 4 + 3)/12 = 29/12 = 2 + 5/12.The fractional part is 5/12, so subtracting 5/12 makes the total an integer. Verification / Alternative check: Any full computation of the entire mixed-number sum will have a fractional part of 5/12. Removing exactly that amount yields a whole number. Why Other Options Are Wrong: 7/12 and 1/2 overshoot the fractional remainder; 7 is not a fraction and changes the integer part; 1/12 is too small and leaves a fractional residue. Common Pitfalls: Adding whole and fractional parts together and then trying to reduce; picking the complement-to-1 (7/12) instead of the actual fractional part. Final Answer: 5/12

Question 6

Operation mistake on a fraction:  Gopal had to find 7/9 of a fraction x. By mistake, he divided x by 7/9 (i.e., multiplied by 9/7). His answer exceeded the correct answer by 8/21. What is the correct answer (i.e., 7/9 of x)?

Accepted Answer

Introduction / Context: This problem is a direct application of turning a word statement into an equation comparing a mistaken operation with the correct one. It highlights the reversal of division by a fraction, and finding the exact amount by which the wrong answer exceeds the right answer. Given Data / Assumptions: Correct result C = (7/9) * x. Mistaken result M = x ÷ (7/9) = x * (9/7). Excess M − C = 8/21. Concept / Approach: Express the excess as a multiple of x, solve for x, and finally compute C = (7/9) * x. Keep all steps exact with fractions to avoid rounding. Step-by-Step Solution: M − C = x*(9/7) − x*(7/9) = x * ( (81 − 49)/63 ) = x * (32/63).So x * (32/63) = 8/21.Solve for x: x = (8/21) * (63/32) = 24/32 = 3/4.Correct answer C = (7/9) * (3/4) = 21/36 = 7/12. Verification / Alternative check: Compute M: (3/4) * (9/7) = 27/28. Then M − C = 27/28 − 7/12 = (81 − 49)/84 = 32/84 = 8/21, confirming consistency. Why Other Options Are Wrong: 3/7, 2/21, and 1/3 do not match the derived value; 5/12 is a common but incorrect simplification slipped from intermediate steps. Common Pitfalls: Forgetting to invert while dividing by 7/9; arithmetic error in (81 − 49); simplifying 21/36 incorrectly. Final Answer: 7/12

Question 7

Proportional scoring in an innings:  The highest score equals 3/11 of the team total T. The second-highest equals 3/11 of the remainder after removing the highest. If the difference between these two scores is 9 runs, what is the total score T?

Accepted Answer

Introduction / Context: This question involves expressing parts of a total in fractions and then comparing those parts. It is a clean application of linear relationships and fraction arithmetic, typical in quantitative reasoning sections. Given Data / Assumptions: Highest score H = (3/11) * T. Remainder after removing H: T − H = T − 3T/11 = 8T/11. Second-highest N = (3/11) * (8T/11) = 24T/121. Difference H − N = 9. Concept / Approach: Form expressions for both scores, subtract them, and solve for T. The denominators are manageable (11 and 121), making exact manipulation easy and reliable. Step-by-Step Solution: H − N = (3T/11) − (24T/121).Common denominator 121: H − N = (33T − 24T)/121 = 9T/121.Set 9T/121 = 9 ⇒ T = 121. Verification / Alternative check: Compute H = 3*121/11 = 33; N = 24*121/121 = 24; difference = 9, as required. Why Other Options Are Wrong: 99, 110, 132, and 143 do not yield a 9-run difference with the specified fractional structure. Common Pitfalls: Taking the second-highest as 3/11 of T instead of the remainder; arithmetic slips in converting 3/11 of 8/11 T. Final Answer: 121

Question 8

Sharing a cake unevenly:  In a family, the father took 1/4 of a cake and this amount was exactly three times as much as each of the other family members received (all others got equal shares). How many family members are there in total?

Accepted Answer

Introduction / Context: This is a classic partition problem. One person takes a fixed fraction, and that share is a simple multiple of everyone else’s equal share. We build an equation for the total and solve for the number of people. Given Data / Assumptions: Father’s share = 1/4 of the cake. Each other member’s share = x (equal for all others). Father’s share is 3 times any other member’s share: 1/4 = 3x. Let total members be n (1 father + n − 1 others). Concept / Approach: From 1/4 = 3x, find x. Then write the total cake as father’s share plus the sum of the others’ shares. Solve for n using a simple linear equation. Step-by-Step Solution: From 1/4 = 3x ⇒ x = 1/12.Total cake = father + others = 1/4 + (n − 1)*(1/12).Set sum to 1: 1/4 + (n − 1)/12 = 1.Multiply by 12: 3 + (n − 1) = 12 ⇒ n − 1 = 9 ⇒ n = 10. Verification / Alternative check: With 10 members: father has 1/4; each of the 9 others has 1/12; total = 3/12 + 9/12 = 12/12 = 1, consistent. Why Other Options Are Wrong: 3, 7, 9, and 12 do not satisfy both the equality of the others’ shares and the 3-times relationship with a 1/4 father’s share. Common Pitfalls: Assuming the father’s share is three times the sum of the others (it is three times each other’s share); miscounting the number of “other” members. Final Answer: 10

Question 9

Weighted month in annual income:  Ravi earns exactly twice as much in January as he earns in each of the other months of the year. What fraction of his total annual earnings does he earn in January?

Accepted Answer

Introduction / Context: This question tests the ability to convert a proportional monthly statement into a fraction of an annual total. It uses uniform earnings across months except for one weighted month (January). Given Data / Assumptions: Let earnings in each non-January month be k. January earnings are 2k. There are 12 months in total. Concept / Approach: Compute total annual income as the sum of 11 months at k plus January at 2k. Then express January’s income as a fraction of the total and reduce to lowest terms. Step-by-Step Solution: Total annual = 11*k + 2k = 13k.January share = 2k.Required fraction = January / Total = 2k / 13k = 2/13. Verification / Alternative check: Pick any k (e.g., k = 1). Then January = 2, yearly total = 13, so January is 2/13 of the annual income, independent of k. Why Other Options Are Wrong: 1/10, 1/5, 1/6 do not reflect the 13k total; 5/7 is unrelated to the 12-month structure with a single doubled month. Common Pitfalls: Forgetting there are 11 non-January months; adding an extra month; reducing 2/13 incorrectly. Final Answer: 2/13

Question 10

Use identities to evaluate a ratio:  Given a^2 + b^2 = 234 and ab = 108, compute the value of (a + b)/(a − b). Assume real numbers where the expression is defined.

Accepted Answer

Introduction / Context: This item examines the use of algebraic identities to transform sums and differences into expressions of a^2 + b^2 and ab. It is a staple technique for speed-solving symmetric expressions without solving for a and b individually. Given Data / Assumptions: a^2 + b^2 = 234. ab = 108. We need (a + b)/(a − b), assuming a ≠ b so the denominator is nonzero. Concept / Approach: Use (a + b)^2 = a^2 + b^2 + 2ab and (a − b)^2 = a^2 + b^2 − 2ab. Then the square of the desired ratio equals (a + b)^2 / (a − b)^2, letting us avoid computing a and b explicitly. Take the positive root for the standard magnitude unless a sign is specified. Step-by-Step Solution: Compute (a + b)^2 = 234 + 2*108 = 234 + 216 = 450.Compute (a − b)^2 = 234 − 2*108 = 234 − 216 = 18.Thus ((a + b)/(a − b))^2 = 450/18 = 25.Therefore (a + b)/(a − b) = 5 (taking the principal value). Verification / Alternative check: If needed, one can construct values of a and b satisfying the conditions (e.g., solving a quadratic with sum s and product p) and confirm the ratio equals 5; however, the identity-based route is faster and exact. Why Other Options Are Wrong: 10 and 8 do not respect the derived ratio; 4 gives a squared ratio of 16, not 25; 6 gives 36, inconsistent with 450/18. Common Pitfalls: Forgetting the 2ab term in the identities; attempting to solve for a and b directly; sign confusion about taking the square root. Final Answer: 5

Question 11

If a is a real number satisfying a^2 + 1 = a, compute the exact value of a^12 + a^6 + 1. Show the reasoning clearly and simplify powers using any useful identities.

Accepted Answer

Introduction / Context: This algebraic simplification question assesses your ability to use polynomial identities and properties of special roots. The equation a^2 + 1 = a can be rearranged into a quadratic with complex roots. From that relation, we can deduce periodic behavior of powers of a and evaluate expressions such as a^12 + a^6 + 1 without expanding naively. Given Data / Assumptions: a is a real or complex number satisfying a^2 + 1 = a. We must find the exact value of a^12 + a^6 + 1. Standard algebraic identities and power-cycling techniques may be used. Concept / Approach: Start by forming the quadratic: a^2 − a + 1 = 0. The roots of this quadratic are complex and lie on the unit circle. Numbers on the unit circle often have powers that repeat in cycles. In particular, the roots are primitive 6th roots of unity excluding 1, which implies a^6 = 1. Once a^6 is known, higher powers like a^12 follow immediately, enabling a quick evaluation of the target sum. Step-by-Step Solution: From a^2 + 1 = a, rearrange to a^2 − a + 1 = 0.This quadratic has roots a = (1 ± i√3)/2, which satisfy a^6 = 1.Therefore a^6 = 1 and a^12 = (a^6)^2 = 1.Compute the sum: a^12 + a^6 + 1 = 1 + 1 + 1 = 3. Verification / Alternative check: Another route is to note a^2 = a − 1, then power-reduce systematically: a^3 = a·a^2 = a(a − 1) = a^2 − a = (a − 1) − a = −1. Hence a^6 = (a^3)^2 = (−1)^2 = 1. The same conclusion leads to a^12 = 1 and the sum equals 3. Why Other Options Are Wrong: 2 and 1: These ignore that both a^6 and a^12 equal 1, causing undercount. −3 and −1: Sign errors from misusing the relation a^2 = a − 1 or mis-evaluating a^3. Common Pitfalls: Expanding powers directly without using the quadratic relation; forgetting that the roots are 6th roots of unity; arithmetic slips in deriving a^3. Final Answer: 3

Question 12

If a, b, and c are non-zero and satisfy a + 1/b = 1 and b + 1/c = 1, determine the exact value of the product abc.

Accepted Answer

Introduction / Context: This problem tests manipulation of linked fractional equations and recognition of patterns that eliminate variables. Even though there are two equations with three unknowns, the target expression abc may still be uniquely determined if the relationships are structured appropriately. Given Data / Assumptions: a + 1/b = 1 b + 1/c = 1 a, b, c are all non-zero. We must compute abc. Concept / Approach: Express a and b in terms of c, then multiply. The key is to transform each equation into a rational expression in a single variable. From b + 1/c = 1, we can write b in terms of c; then substitute into a + 1/b = 1 to express a in terms of c. Multiplying a·b·c often causes factors to cancel, leading to a constant independent of c. Step-by-Step Solution: From b + 1/c = 1 ⇒ b = 1 − 1/c = (c − 1)/c.Then 1/b = c/(c − 1).From a + 1/b = 1 ⇒ a = 1 − 1/b = 1 − c/(c − 1) = −1/(c − 1).Now compute abc = [−1/(c − 1)] * [(c − 1)/c] * c = −1. Verification / Alternative check: Pick a convenient c (e.g., c = 2). Then b = (2 − 1)/2 = 1/2; so a = 1 − 1/b = 1 − 2 = −1. Hence abc = (−1) * (1/2) * 2 = −1, confirming the general result. Why Other Options Are Wrong: 1 and 3 / −3 / −2: These arise from sign mistakes when forming 1/b or from substituting b incorrectly into a. Common Pitfalls: Assuming there is not enough information because there are more unknowns than equations; dropping the minus sign when simplifying a = −1/(c − 1). Final Answer: -1

Question 13

Given a + b + c = 0, evaluate the product: [ (a + b)/c + (b + c)/a + (c + a)/b ] * [ a/(b + c) + b/(c + a) + c/(a + b) ].

Accepted Answer

Introduction / Context: Algebraic manipulations often simplify dramatically when a + b + c = 0. This question checks whether you can exploit that condition to reduce complicated rational expressions. Recognizing that sums like a + b equal −c is the main lever to collapse each fraction to simple constants. Given Data / Assumptions: a + b + c = 0 with non-zero a, b, c to keep denominators valid. Compute the product of two bracketed sums. Concept / Approach: Use the identity implied by the condition: a + b = −c, b + c = −a, c + a = −b. Substituting these into all numerators and denominators instantly converts each fraction into −1, which makes each bracket sum straightforward. Finally, multiply the two summed values. Step-by-Step Solution: Since a + b = −c, (a + b)/c = (−c)/c = −1.Similarly, (b + c)/a = (−a)/a = −1, and (c + a)/b = (−b)/b = −1.Therefore the first bracket equals (−1) + (−1) + (−1) = −3.In the second bracket: a/(b + c) = a/(−a) = −1; b/(c + a) = b/(−b) = −1; c/(a + b) = c/(−c) = −1.So the second bracket also equals −3.Product = (−3) * (−3) = 9. Verification / Alternative check: Pick a concrete triple with sum zero, e.g., a = 1, b = 2, c = −3. Substitute to verify each fraction is −1. The product again equals 9. Why Other Options Are Wrong: 0, 8, −3, 3: These values come from arithmetic slips or forgetting that both brackets sum to −3. Common Pitfalls: Attempting to find a common denominator instead of exploiting a + b + c = 0; sign errors when converting a + b to −c. Final Answer: 9

Question 14

If a + b + c = 14 and a^2 + b^2 + c^2 = 96, find the value of ab + bc + ca using the standard identity that links sums and squares.

Accepted Answer

Introduction / Context: This question targets a staple algebra identity connecting the square of a sum to the sum of squares and pairwise products. It is frequently used to compute ab + bc + ca when you know a + b + c and a^2 + b^2 + c^2. Mastery of this identity saves time in many contest and exam problems. Given Data / Assumptions: a + b + c = 14 a^2 + b^2 + c^2 = 96 Real numbers (no special constraints beyond this are required). Concept / Approach: Use the identity (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca). Rearranging immediately yields ab + bc + ca in one step. This avoids solving for individual variables, which is unnecessary and inefficient here. Step-by-Step Solution: Compute (a + b + c)^2 = 14^2 = 196.Apply identity: 196 = 96 + 2(ab + bc + ca).Thus 2(ab + bc + ca) = 196 − 96 = 100.Therefore ab + bc + ca = 100 / 2 = 50. Verification / Alternative check: Sanity check: ab + bc + ca = 50 is consistent with many possible triples; for example, symmetric choices near the mean 14/3 will produce reasonable non-negative pairwise sums. Why Other Options Are Wrong: 51 / 55 / 60 / 65: These are common arithmetic slips (e.g., squaring 14 incorrectly or subtracting 96 from 196 wrongly). Common Pitfalls: Forgetting the factor 2 in the identity; attempting to solve for a, b, and c individually; mixing up a^2 + b^2 + c^2 with (a + b + c)^2. Final Answer: 50

Question 15

Evaluate the nested fractional expression exactly: [ 5 − ( 3/4 + { 2 1/2 − ( 1/2 + 1/6 − 1/7 ) } ) ] / 2. Convert mixed numbers to improper fractions and simplify.

Accepted Answer

Introduction / Context: Nestings with mixed numbers and multiple signs test attention to order of operations and accurate handling of fractions. This problem requires converting a mixed number to an improper fraction, simplifying the inner parentheses first, and then carefully executing additions and subtractions before the final division by 2. Given Data / Assumptions: Expression: [ 5 − ( 3/4 + { 2 1/2 − ( 1/2 + 1/6 − 1/7 ) } ) ] / 2. “2 1/2” denotes the mixed number 5/2. Perform parentheses and brackets from inside out; apply division by 2 to the whole bracket. Concept / Approach: Work from the innermost parentheses outward. Combine the small fractions inside, subtract from the mixed number, add 3/4, subtract from 5, and finally divide by 2. Keeping everything as exact fractions avoids rounding errors and ensures a simplified result. Step-by-Step Solution: Innermost parentheses: 1/2 + 1/6 − 1/7 → common denominator 42 gives (21 + 7 − 6)/42 = 22/42 = 11/21.Compute the curly bracket: 2 1/2 − (11/21) = (5/2) − (11/21) = (105 − 22)/42 = 83/42.Add 3/4: 3/4 + 83/42 = (63 + 166)/84 = 229/84.Subtract from 5: 5 − 229/84 = 420/84 − 229/84 = 191/84.Divide by 2: (191/84) / 2 = 191/168. Verification / Alternative check: Convert to decimals as a check: 1/2 + 1/6 − 1/7 ≈ 0.523809; 2.5 − 0.523809 ≈ 1.976190; + 0.75 ≈ 2.726190; 5 − 2.726190 ≈ 2.273810; ÷ 2 ≈ 1.136905. Now 191/168 ≈ 1.136905 — exact match. Why Other Options Are Wrong: 223/168, 183/168, 205/168, 171/168: Each is close but results from a minor sign error or denominator mismatch in one of the inner steps. Common Pitfalls: Misreading the scope of the final “/ 2” (it applies to the entire bracket); forgetting to convert 2 1/2 to 5/2; adding fractions without a common denominator. Final Answer: 191/168

Question 16

Evaluate the ratio: [0.5*0.5*0.5 + 0.2*0.2*0.2 + 0.3*0.3*0.3 − 3*0.5*0.3*0.2] / [0.5*0.5 + 0.2*0.2 + 0.3*0.3 − 0.5*0.2 − 0.2*0.3 − 0.5*0.3]. Use the identity for x^3 + y^3 + z^3 − 3xyz.

Accepted Answer

Introduction / Context: This expression is a classic application of the identity x^3 + y^3 + z^3 − 3xyz = (x + y + z)(x^2 + y^2 + z^2 − xy − yz − zx). When the numerator and denominator match the two factors in this identity, the ratio collapses to x + y + z, dramatically simplifying the computation. Given Data / Assumptions: x = 0.5, y = 0.2, z = 0.3. Numerator: x^3 + y^3 + z^3 − 3xyz. Denominator: x^2 + y^2 + z^2 − xy − yz − zx. Concept / Approach: Invoke the factorization identity directly. Since the numerator equals (x + y + z)(denominator), the entire fraction equals x + y + z (provided x + y + z ≠ 0). This avoids computing each cube and product explicitly, though you can still verify numerically if you wish. Step-by-Step Solution: Recognize identity: x^3 + y^3 + z^3 − 3xyz = (x + y + z)(x^2 + y^2 + z^2 − xy − yz − zx).Given denominator equals the second factor, the ratio = x + y + z.Compute x + y + z = 0.5 + 0.2 + 0.3 = 1.0. Verification / Alternative check: Explicit calculation confirms the same: Numerator = 0.125 + 0.008 + 0.027 − 0.09 = 0.070; Denominator = 0.25 + 0.04 + 0.09 − 0.10 − 0.06 − 0.15 = 0.07; Ratio = 0.07 / 0.07 = 1. Why Other Options Are Wrong: 0.6 / 0.4 / 0.03 / 0.9: These result from misapplying the identity or arithmetic mistakes in the numerator/denominator. Common Pitfalls: Forgetting the exact structure of the identity; computing many decimals and rounding midstream; sign errors in −3xyz and the pairwise products in the denominator. Final Answer: 1

Question 17

Simplify the expression using the difference of squares: [ (999 + 588)^2 − (999 − 588)^2 ] / (999 * 588 ).

Accepted Answer

Introduction / Context: Expressions of the form A^2 − B^2 invite the identity A^2 − B^2 = (A − B)(A + B). Recognizing this turns intimidating large numbers into small, easily cancelable factors—perfect for mental math under time pressure. Given Data / Assumptions: A = 999 + 588, B = 999 − 588. Compute [A^2 − B^2] / (999 * 588). Concept / Approach: Use the difference-of-squares identity to factor the numerator. Then simplify the fraction by canceling common factors with the denominator. Because A + B and A − B become simple combinations of 999 and 588, the cancellation is straightforward. Step-by-Step Solution: A − B = (999 + 588) − (999 − 588) = 1176.A + B = (999 + 588) + (999 − 588) = 1998 = 2 * 999.Numerator = (A − B)(A + B) = 1176 * 1998 = 1176 * (2 * 999).Divide by denominator 999 * 588: cancellation of 999 gives 1176 * 2 / 588 = 2352 / 588 = 4. Verification / Alternative check: Compute 2352/588 by noticing 588 * 4 = 2352 — exact equality confirms the result. Why Other Options Are Wrong: 8, 3, 2, 6: These stem from partial cancellations or miscomputing A + B. Common Pitfalls: Forgetting that A + B = 2*999; arithmetic errors in 1176 or in the final division. Final Answer: 4

Question 18

Use the identity (a + b)^2 + (a − b)^2 = 2(a^2 + b^2) to evaluate: [ (238 + 131)^2 + (238 − 131)^2 ] / ( 238*238 + 131*131 ).

Accepted Answer

Introduction / Context: This problem is a direct application of the identity (a + b)^2 + (a − b)^2 = 2(a^2 + b^2). Recognizing this pattern quickly reduces the expression to a constant, independent of the specific values of a and b, as long as the structure matches the identity. Given Data / Assumptions: a = 238, b = 131. Expression: [ (a + b)^2 + (a − b)^2 ] / ( a^2 + b^2 ). Concept / Approach: Apply the identity to the numerator. Since the denominator equals a^2 + b^2, the ratio simplifies immediately. No numeric expansion is required, though you may compute to verify. Step-by-Step Solution: (a + b)^2 + (a − b)^2 = 2(a^2 + b^2).Therefore the whole ratio = [2(a^2 + b^2)] / (a^2 + b^2) = 2. Verification / Alternative check: Compute a^2 + b^2 numerically if desired and observe exact cancellation when doubling and dividing. Why Other Options Are Wrong: 4, 8, 9, 1: These values appear if you square and add incorrectly or forget to divide by a^2 + b^2. Common Pitfalls: Over-expanding and making arithmetic mistakes; ignoring the elegant identity that saves time. Final Answer: 2

Question 19

Given √2 ≈ 1.4142, compute the decimal value of 7 / (4 + √2) to four decimal places by rationalizing the denominator.

Accepted Answer

Introduction / Context: Fractions with radicals in the denominator are best handled by rationalizing the denominator. This simplifies numerical evaluation and reduces rounding errors. Here, a short rationalization step converts the expression to a simple combination of integers and √2. Given Data / Assumptions: Expression: 7 / (4 + √2), with √2 ≈ 1.4142. Accuracy requested: four decimal places (or as presented in the options). Concept / Approach: Multiply numerator and denominator by the conjugate (4 − √2). The product in the denominator becomes a difference of squares, eliminating the radical. Then simplify and evaluate using the given approximation for √2. Step-by-Step Solution: 7 / (4 + √2) = 7(4 − √2) / [(4 + √2)(4 − √2)].Denominator: 16 − (√2)^2 = 16 − 2 = 14.Thus the value = 7(4 − √2) / 14 = (4 − √2) / 2.Compute numerically: (4 − 1.4142)/2 = 2.5858/2 = 1.2929. Verification / Alternative check: Direct calculator check of 7/(4 + 1.4142) yields ≈ 1.2929, matching the rationalized computation. Why Other Options Are Wrong: 3.5858 / 4.4142 / 1.5858 / 1.4142: These correspond to 4 − √2, 4 + √2, or incomplete division by 2, not the final simplified value. Common Pitfalls: Forgetting to divide by 2 after rationalizing; using √2 ≈ 1.41 too early which can shift the thousandths place. Final Answer: 1.2929

Question 20

Evaluate the expression (0.96)^3 − (0.1)^3 divided by [ (0.96)^2 + (0.96*0.1) + (0.1)^2 ]. Use the a^3 − b^3 factorization to simplify.

Accepted Answer

Introduction / Context: This expression is designed for the identity a^3 − b^3 = (a − b)(a^2 + ab + b^2). When the denominator matches a^2 + ab + b^2, the overall fraction collapses to a − b. Recognizing such structures converts a potentially tedious decimal computation into a one-step result. Given Data / Assumptions: a = 0.96, b = 0.10. Expression: [a^3 − b^3] / [a^2 + ab + b^2]. Note that ab = 0.96 * 0.10 = 0.096 (as written). Concept / Approach: Apply the cube difference factorization. Since the denominator equals a^2 + ab + b^2 exactly, the expression simplifies to a − b immediately. Then compute 0.96 − 0.10 to reach the final answer without rounding issues. Step-by-Step Solution: Use identity: a^3 − b^3 = (a − b)(a^2 + ab + b^2).Given denominator = a^2 + ab + b^2, the ratio = a − b.Compute a − b = 0.96 − 0.10 = 0.86. Verification / Alternative check: Approximate directly: (0.884736 − 0.001) / (0.9216 + 0.096 + 0.01) ≈ 0.883736 / 1.0276 ≈ 0.86 (rounded), confirming the identity-based result. Why Other Options Are Wrong: 1.06 / 0.95 / 0.97 / 0.80: These values arise from arithmetic slips or from misreading the denominator as a^2 + a + b^2 or using 0.96 instead of 0.096 for ab. Common Pitfalls: Misreading 0.096 as 0.96; attempting to compute the cubes fully and rounding at intermediate steps; forgetting the identity and overcomplicating the work. Final Answer: 0.86

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