Question 1

A number multiplied by five times itself equals 1445. Form the equation n * (5n) = 1445 and determine the positive value of n.

Accepted Answer

Introduction / Context:This algebraic simplification question checks your ability to translate an English statement into an equation and solve a simple quadratic for its positive root. Given Data / Assumptions: Statement: A number multiplied by five times itself equals 1445. Let the number be n (assume n is real). Equation: n * (5n) = 1445. Concept / Approach:Translate directly to an equation and solve for n. Since n * (5n) = 5n^2, isolate n^2 and take the positive square root (aptitude questions typically ask for the positive value unless specified otherwise). Step-by-Step Solution: From n * (5n) = 1445, we have 5n^2 = 1445.Divide both sides by 5: n^2 = 1445 / 5 = 289.Take square root: n = √289 = 17 (positive value). Verification / Alternative check:Compute 5n^2 with n = 17: 5 * 17^2 = 5 * 289 = 1445, which matches the given total exactly. Why Other Options Are Wrong:18, 15, 19, and 13 do not satisfy 5n^2 = 1445. Substituting any of these yields values other than 1445. Common Pitfalls:Interpreting “five times of itself” as n + 5n instead of n * (5n). The phrase clearly intends multiplication leading to 5n^2. Final Answer:17

Question 2

Equal partition of a board — unit conversion and division A wooden board is 80 inches long and is cut into 4 equal parts in a workshop. What is the exact length of each part, expressed in inches?

Accepted Answer

Introduction / Context:This problem assesses basic numerical simplification with real-world wording. A single board of known length is evenly divided. The task is to compute the length of one equal segment, reinforcing division of a whole by an integer and careful attention to units (inches). Given Data / Assumptions: Total length of the board = 80 inches. The board is divided into 4 equal parts. All parts are identical in length. Final answer should be reported in inches. Concept / Approach:When a quantity is split into n equal parts, each part has size total / n. Here, that is 80 / 4. Because the units are inches and there is no conversion needed, the computation is a straightforward division. Step-by-Step Solution:Let L_total = 80 inches.Number of equal parts, n = 4.Length per part = L_total / n = 80 / 4.Compute: 80 / 4 = 20.Therefore, each part measures 20 inches. Verification / Alternative check:Add the four equal parts to recover the total: 20 + 20 + 20 + 20 = 80 inches. This confirms the division is consistent with the original length. Why Other Options Are Wrong:25 inches, 15 inches, 18 inches, and 22 inches do not multiply by 4 to return 80 inches; each would yield an incorrect total when recombined. Common Pitfalls:Confusing “divide by 4” with “subtract 4,” or attempting unnecessary unit conversions. Another mistake is averaging unrelated options rather than performing the exact division. Final Answer:20 inches

Question 3

Custom operator evaluation — apply the rule a * b = a + b + a/b Using the defined operation, compute the numerical value of 12 * 4.

Accepted Answer

Introduction / Context:Algebraic operator questions define a nonstandard operation and require direct substitution. Here, a * b is not multiplication; it is defined as a + b + a/b. The goal is to accurately apply this rule to the pair (12, 4). Given Data / Assumptions: Custom definition: a * b = a + b + a/b. We must evaluate 12 * 4. Standard arithmetic conventions apply inside the right-hand side. Concept / Approach:Replace a by 12 and b by 4 in the given definition, then compute using normal precedence: perform the division a/b before the final addition to avoid mistakes. Step-by-Step Solution:Start with a * b = a + b + a/b.Substitute: 12 * 4 = 12 + 4 + 12/4.Compute the division: 12/4 = 3.Sum: 12 + 4 + 3 = 19. Verification / Alternative check:Break into parts: (12 + 4) = 16, then add (12/4) = 3 to get 19. Both routes confirm the same value. Why Other Options Are Wrong:48 and 20 come from mistaking * for ordinary multiplication or partial sums; 21 results from adding an extra 2; 16 ignores the a/b term entirely. Common Pitfalls:Assuming * means multiply; forgetting the division a/b; or doing 12/(4 + 12) instead of 12/4 as defined. Final Answer:19

Question 4

Identity-based simplification with cubes and squares Evaluate [137^2 + 137×133 + 133^2] / [137^3 − 133^3] using algebraic factorization.

Accepted Answer

Introduction / Context:This expression is designed to test recognition of classic algebraic identities. The numerator matches a^2 + ab + b^2 and the denominator matches a^3 − b^3 for a = 137 and b = 133. Recognizing these patterns allows immediate reduction without heavy arithmetic. Given Data / Assumptions: a = 137, b = 133. Numerator: a^2 + ab + b^2. Denominator: a^3 − b^3. Concept / Approach:Use the identity a^3 − b^3 = (a − b)(a^2 + ab + b^2). When a^3 − b^3 is divided by a^2 + ab + b^2, the result is simply a − b. Therefore, (a^2 + ab + b^2) / (a^3 − b^3) = 1 / (a − b). Step-by-Step Solution:Numerator N = a^2 + ab + b^2.Denominator D = a^3 − b^3 = (a − b)(a^2 + ab + b^2) = (a − b)·N.Therefore, N / D = N / [(a − b)·N] = 1 / (a − b).Compute a − b = 137 − 133 = 4.Value = 1/4. Verification / Alternative check:Direct cancellation is valid because (a^2 + ab + b^2) is nonzero for distinct positive a and b. Hence, the simplified ratio is exact. Why Other Options Are Wrong:4 and 270 invert the correct ratio; 1/270 confuses the difference with a product; 1/8 is an arbitrary fraction not supported by the identity. Common Pitfalls:Attempting long numerical computations instead of using identities; misremembering a^3 − b^3 factorization; cancelling the wrong factor. Final Answer:1/4

Question 5

Apply the cube difference identity to simplify Compute (117^3 − 98^3) / (117^2 + 117×98 + 98^2) by using standard algebraic formulas.

Accepted Answer

Introduction / Context:This problem is a direct application of the identity a^3 − b^3 = (a − b)(a^2 + ab + b^2). The denominator matches the quadratic factor of the identity, allowing an immediate simplification to a − b. Given Data / Assumptions: a = 117, b = 98. Expression: (a^3 − b^3) / (a^2 + ab + b^2). Concept / Approach:From the identity, (a^3 − b^3) / (a^2 + ab + b^2) = (a − b). No further arithmetic is required beyond subtracting b from a. Step-by-Step Solution:Use identity: a^3 − b^3 = (a − b)(a^2 + ab + b^2).Divide by (a^2 + ab + b^2): result = a − b.Compute a − b = 117 − 98 = 19. Verification / Alternative check:If expanded numerically, the ratio will indeed equal 19. The algebraic route is more efficient and error-resistant. Why Other Options Are Wrong:215 and 311 are unrelated large values; 29 and 39 are plausible-looking integers but do not match a − b. Common Pitfalls:Forgetting the identity and trying to cube large numbers; arithmetic slips during lengthy calculations. Final Answer:19

Question 6

Recovering the intended identity: (a^3 + b^3)/(a^2 − ab + b^2) Assuming the denominator is 343^2 − 343×257 + 257^2, evaluate (343^3 + 257^3)/(343^2 − 343×257 + 257^2).

Accepted Answer

Introduction / Context:The printed denominator appears malformed in the raw text. By the Recovery-First Policy, we minimally repair it to the standard identity form a^2 − ab + b^2 so that the expression aligns with (a^3 + b^3)/(a^2 − ab + b^2). This is a classic result that simplifies to a + b. Given Data / Assumptions: Assume denominator is 343^2 − 343×257 + 257^2 (standard form). Numerator: 343^3 + 257^3. a = 343, b = 257. Concept / Approach:Use the identity a^3 + b^3 = (a + b)(a^2 − ab + b^2). Therefore, (a^3 + b^3)/(a^2 − ab + b^2) = a + b. This bypasses any need to compute large powers explicitly. Step-by-Step Solution:Recognize identity match: numerator a^3 + b^3 and denominator a^2 − ab + b^2.Apply formula: (a^3 + b^3)/(a^2 − ab + b^2) = a + b.Compute a + b = 343 + 257 = 600. Verification / Alternative check:Expanding a^3 + b^3 and dividing by the quadratic factor yields the linear factor (a + b). The identity guarantees exactness for all real a and b. Why Other Options Are Wrong:8600, 2600, and 800 are unrelated sums/products; only 600 equals a + b. The distractors are plausible numerically but inconsistent with the identity. Common Pitfalls:Not repairing the misprinted denominator; attempting massive exponent calculations instead of using the identity. Final Answer:600

Question 7

Bracketed fractional simplification — nested division and “of” as multiplication Evaluate 3 ÷ [(8 − 5) ÷ {(4 − 2) ÷ (2 + 8/13)}] carefully using left-to-right precedence for × and ÷.

Accepted Answer

Introduction / Context:Complex fractional expressions with nested parentheses test precision with order of operations. Here, “of” means multiplication, and division and multiplication share the same precedence, evaluated left to right. The main skill is systematic simplification from the innermost bracket outward. Given Data / Assumptions: Expression: 3 ÷ [(8 − 5) ÷ {(4 − 2) ÷ (2 + 8/13)}]. Interpret “of” (if present) as multiplication. Perform operations in the structured bracket order. Concept / Approach:Compute from the deepest inner group: first (2 + 8/13), then (4 − 2), then form the division (4 − 2)/(2 + 8/13). Next, evaluate (8 − 5) ÷ {previous result}. Finally, divide 3 by that outcome. Step-by-Step Solution:Compute inner sum: 2 + 8/13 = 34/13.Compute 4 − 2 = 2.Form ratio: (4 − 2)/(2 + 8/13) = 2 / (34/13) = 2 * 13/34 = 26/34 = 13/17.Compute 8 − 5 = 3.Now 3 ÷ (13/17) = 3 * 17/13 = 51/13.Finally, 3 ÷ [51/13] = 3 * 13/51 = 39/51 = 13/17. Verification / Alternative check:Convert intermediate numbers to decimals to sanity-check: 13/17 ≈ 0.7647. The inner ratio 2/(2.615...) ≈ 0.7647; the outer sequence returns to the same exact fraction. Why Other Options Are Wrong:68/13 and 17/13 invert or misplace a division; 13/68 is an over-inversion; 51/13 is an intermediate value, not the final result. Common Pitfalls:Reversing the order of division; dropping the final division by 3; mishandling the conversion of mixed fractions to improper fractions. Final Answer:13/17

Question 8

Mixed numbers and fractional operations — follow left-to-right for × and ÷ Compute the value of {7 1/2 + 1/2 ÷ 1/2 of 1/4 − 2/5 × 2 1/3 ÷ 1 7/8 of (1 2/5 − 1 1/3)}.

Accepted Answer

Introduction / Context:This expression involves mixed numbers, “of” (multiplication), division, and subtraction within parentheses. The key is to convert mixed numbers to improper fractions and then evaluate products and divisions left to right for operators of the same precedence. Careful bookkeeping prevents sign and order errors. Given Data / Assumptions: Interpret “of” as multiplication. Work left to right for × and ÷ of equal precedence. 7 1/2 = 15/2, 2 1/3 = 7/3, 1 7/8 = 15/8, (1 2/5 − 1 1/3) = (7/5 − 4/3) = 1/15. Concept / Approach:Simplify term by term: compute the small parenthesis first, then evaluate each multiplication/division sequence. Sum and subtract the resulting fractions at the end using a common denominator. Step-by-Step Solution:Term A: 7 1/2 = 15/2.Term B: 1/2 ÷ 1/2 of 1/4 = (1/2 ÷ 1/2) × 1/4 = 1 × 1/4 = 1/4.Term C core: 2/5 × 2 1/3 ÷ 1 7/8 of (1 2/5 − 1 1/3).Convert: 2/5 × 7/3 ÷ 15/8 × 1/15.Compute: 2/5 × 7/3 = 14/15; then ÷ 15/8 → 14/15 × 8/15 = 112/225; then × 1/15 → 112/3375.Assemble: 15/2 + 1/4 − 112/3375.Combine 15/2 and 1/4 → 31/4; then subtract 112/3375.Common denominator 13500 → result = (31×3375 − 112×4)/13500 = (104625 − 448)/13500 = 104177/13500. Verification / Alternative check:Approximate: 31/4 = 7.75; 112/3375 ≈ 0.033185; net ≈ 7.7168. None of the listed fractions (31/5 = 6.2, 21/24 ≈ 0.875, 41/30 ≈ 1.3667) match. Why Other Options Are Wrong:31/5, 21/24, and 41/30 are far from the computed value; therefore, the only consistent choice is “None of these.” Common Pitfalls:Misinterpreting “of,” reversing division order, or forgetting to convert mixed numbers before multiplying/dividing. Final Answer:None of these

Question 9

Reconstructing a bracketed expression with “of” and ratios Evaluate 3/4 ÷ (2 1/4 of 2/3) − [ (1/2 − 1/3) / (1/2 + 1/3) × 3 1/3 ] + 5/6.

Accepted Answer

Introduction / Context:Complex fractional expressions often compress multiple ideas: “of” as multiplication, ratios of sums/differences, and mixed numbers. Interpreting the brackets sensibly and proceeding stepwise yields a clean exact fraction at the end. Given Data / Assumptions: Interpret “of” as multiplication and use left-to-right evaluation for × and ÷. Assumed grouping: 3/4 ÷ (2 1/4 of 2/3) − [ (1/2 − 1/3) / (1/2 + 1/3) × 3 1/3 ] + 5/6. 2 1/4 = 9/4, 3 1/3 = 10/3. Concept / Approach:Compute the “of” products first inside their brackets, then resolve divisions and multiplications from left to right. Reduce intermediate fractions whenever possible to keep numbers small and accurate. Step-by-Step Solution:Term A: 2 1/4 of 2/3 = (9/4) × (2/3) = 18/12 = 3/2.Then 3/4 ÷ (3/2) = 3/4 × 2/3 = 1/2.Term B ratio: (1/2 − 1/3) / (1/2 + 1/3) = (1/6) / (5/6) = 1/5.Multiply by 3 1/3: (1/5) × (10/3) = 2/3.Assemble expression: 1/2 − 2/3 + 5/6.Convert to sixths: 1/2 = 3/6; −2/3 = −4/6; +5/6 remains 5/6.Sum: (3 − 4 + 5)/6 = 4/6 = 2/3. Verification / Alternative check:Compute decimals: 1/2 = 0.5; 2/3 ≈ 0.6667; 5/6 ≈ 0.8333 → 0.5 − 0.6667 + 0.8333 ≈ 0.6666…, confirming 2/3. Why Other Options Are Wrong:7/18, 49/54, 1/6, and 5/9 arise from mis-grouping or mishandling division by a fraction. Common Pitfalls:Dropping parentheses; interpreting “of” incorrectly; forgetting that dividing by a fraction multiplies by its reciprocal. Final Answer:2/3

Question 10

Solve for the unknown in a mixed-fraction equation Find the value of ? if  (5/6 ÷ 6/7) × ? − (8/9 ÷ 1 3/5) + (3/4 × 3 1/3) = 2 7/9.

Accepted Answer

Introduction / Context:This equation mixes division by fractions, multiplication by mixed numbers, and solving for an unknown factor. The strategy is to simplify each known term, isolate the variable term, and then solve for the unknown by dividing both sides by its coefficient. Given Data / Assumptions: (5/6 ÷ 6/7) × ? − (8/9 ÷ 1 3/5) + (3/4 × 3 1/3) = 2 7/9. 1 3/5 = 8/5, 3 1/3 = 10/3, 2 7/9 = 25/9. Concept / Approach:Compute each operation stepwise. Division by a fraction equals multiplication by its reciprocal. Convert all mixed numbers to improper fractions early to avoid mistakes. Collect constants on one side, then solve for the unknown. Step-by-Step Solution:Coefficient of ?: 5/6 ÷ 6/7 = 5/6 × 7/6 = 35/36.Second term: 8/9 ÷ 8/5 = 8/9 × 5/8 = 5/9 (to be subtracted).Third term: 3/4 × 10/3 = 10/4 = 5/2.Right side: 2 7/9 = 25/9.Form equation: (35/36)·x − 5/9 + 5/2 = 25/9.Combine constants: −5/9 + 5/2 = −10/18 + 45/18 = 35/18.So (35/36)·x + 35/18 = 25/9 = 50/18.Subtract: (35/36)·x = 50/18 − 35/18 = 15/18 = 5/6.Solve: x = (5/6) ÷ (35/36) = (5/6) × (36/35) = 6/7. Verification / Alternative check:Substitute x = 6/7 back into the original to confirm both sides equal 25/9. Why Other Options Are Wrong:7/6, 1, 5/6 lead to a left-hand side different from 25/9; “None of these” is invalid since 6/7 works exactly. Common Pitfalls:Failing to convert mixed numbers; forgetting to apply reciprocals during division by fractions; arithmetic mistakes when combining constants. Final Answer:6/7

Question 11

Balance an equation with products of mixed numbers Compute the missing term ? such that 15 2/3 × 3 1/6 + 6 1/3 = 11 7/18 + ?.

Accepted Answer

Introduction / Context:This question requires accurate conversion of mixed numbers to improper fractions, multiplication of fractions, and isolating a missing term in an equality. Precision with numerators and denominators is crucial when dealing with non-integer products. Given Data / Assumptions: Equation: 15 2/3 × 3 1/6 + 6 1/3 = 11 7/18 + ? 15 2/3 = 47/3, 3 1/6 = 19/6, 6 1/3 = 19/3, 11 7/18 = 205/18. Concept / Approach:Convert all mixed numbers to improper fractions. Multiply fractions by multiplying numerators and denominators. Then add the known terms on the left, and subtract the right-hand known term to solve for the unknown ?. Reduce the final fraction if possible. Step-by-Step Solution:Left product: (47/3) × (19/6) = 893/18.Add 6 1/3 = 19/3 = 114/18.Left sum: 893/18 + 114/18 = 1007/18.Right known term: 11 7/18 = 205/18.Solve for ?: 1007/18 = 205/18 + ? → ? = (1007 − 205)/18 = 802/18 = 401/9. Verification / Alternative check:Convert 401/9 back to a mixed number: 44 5/9. Substituting creates equality on both sides, confirming correctness. Why Other Options Are Wrong:395/9, 1374/9, and 297/9 do not balance the equation; therefore, “None of these” is the correct choice since 401/9 is not listed among the given options. Common Pitfalls:Misconverting mixed numbers; forgetting to bring terms to a common denominator; arithmetic slips when subtracting numerators. Final Answer:None of these

Question 12

Carefully simplify the nested fractional expression Compute the exact value of: 7 1/2 − [ 2 1/4 ÷ { 1 1/4 − 1/2(1 1/2 − 1/3 − 1/6) } ].

Accepted Answer

Introduction / Context: This problem assesses careful simplification of a layered fractional expression containing mixed numbers, brackets, and the factor “1/2” multiplying a parenthetical group. It is a classic test of order of operations, conversion between mixed and improper fractions, and accurate arithmetic with rational numbers. Given Data / Assumptions: Target expression: 7 1/2 − [ 2 1/4 ÷ { 1 1/4 − 1/2(1 1/2 − 1/3 − 1/6) } ]. Mixed numbers should be converted to improper fractions before operations. Multiplication inside braces must be carried out before the outer subtraction and division. Concept / Approach: Work from the innermost parentheses outward. First simplify the small difference (1 1/2 − 1/3 − 1/6). Then apply the factor 1/2 to that result. Use the result to evaluate the expression inside braces. Next carry out the division in the bracket. Finally, subtract this bracket from 7 1/2. Keep all steps in exact fractional form to avoid rounding errors. Step-by-Step Solution:Convert: 7 1/2 = 15/2, 2 1/4 = 9/4, 1 1/4 = 5/4, 1 1/2 = 3/2.Innermost: 3/2 − 1/3 − 1/6. Since 1/3 + 1/6 = 1/2, this becomes 3/2 − 1/2 = 1.Apply 1/2: 1/2 * 1 = 1/2.Braces: 5/4 − 1/2 = 5/4 − 2/4 = 3/4.Bracket: 9/4 ÷ 3/4 = (9/4) * (4/3) = 9/3 = 3.Total: 15/2 − 3 = 15/2 − 6/2 = 9/2. Verification / Alternative check: Each stage can be double-checked by converting the mixed numbers first and verifying cancellations (for example, dividing by 3/4 equals multiplying by 4/3). The arithmetic aligns perfectly to give 9/2 = 4.5. Why Other Options Are Wrong:1: Ignores most of the interior arithmetic and the division by 3/4.4: Likely from subtracting 3 from 7 without converting to halves correctly.177/288: An unrelated small fraction produced by mishandling multiple denominators at once. Common Pitfalls: Forgetting to distribute the 1/2 across the inner parentheses; mixing up order of operations; or converting mixed numbers inconsistently, which can lead to sign or denominator errors. Always reduce the innermost group first and proceed outward in a structured manner. Final Answer: 9/2

Question 13

Proportion with a missing mean: solve for the number x in 50 / x = x / 12 1/2 (treat 12 1/2 as 12.5 and find x exactly).

Accepted Answer

Introduction / Context: Proportion problems of the form a/x = x/b are common in aptitude tests. They test recognition that such a relation implies x^2 = a*b, so x is the positive square root of the product (for positive a and b). Here, we handle a mixed number and keep arithmetic clean and exact. Given Data / Assumptions: Equation: 50 / x = x / 12 1/2. Interpret 12 1/2 as 12.5. All quantities are positive; take the positive root for x. Concept / Approach: Cross-multiplying a proportion gives 50 * 12.5 = x * x. This is a direct application of the mean-proportional idea: if a/x = x/b, then x^2 = a*b. After computing the product, take the square root to find x exactly. Step-by-Step Solution:Write 12 1/2 as 12.5.Set x^2 = 50 * 12.5.Compute product: 50 * 12.5 = 625.Therefore x = sqrt(625) = 25 (positive value). Verification / Alternative check: Substitute x = 25 back: 50/25 = 2 and 25/12.5 = 2. Both sides match, confirming x = 25. Why Other Options Are Wrong:25/2 = 12.5 gives 50/12.5 = 4 while 12.5/12.5 = 1, not equal.4 leads to 50/4 = 12.5 but 4/12.5 = 0.32, not equal.50 makes 50/50 = 1 while 50/12.5 = 4, not equal. Common Pitfalls: Misreading 12 1/2 as 12/5; forgetting to square-root after forming x^2; or introducing sign errors. Remember that for positive magnitudes, the positive square root is the meaningful solution. Final Answer: 25

Question 14

Algebraic substitution with a given ratio If x / y = 4 / 5, evaluate 4/7 + (2y − x) / (2y + x).

Accepted Answer

Introduction / Context: This question tests the standard technique of substituting proportional values from a given ratio to simplify an algebraic expression. By expressing x and y in terms of a single parameter, you can reduce the compound fraction to a simple rational number and then add the constant 4/7. Given Data / Assumptions: x / y = 4 / 5. Expression to evaluate: 4/7 + (2y − x) / (2y + x). Assume y ≠ 0 and denominators remain nonzero. Concept / Approach: Let x = 4k and y = 5k for some nonzero k. This preserves the ratio x/y = 4/5. Substitute into the expression and simplify the resulting rational terms. Parameter k cancels out, leaving a pure number that can be added to 4/7. Step-by-Step Solution:Set x = 4k and y = 5k.Compute numerator: 2y − x = 10k − 4k = 6k.Compute denominator: 2y + x = 10k + 4k = 14k.Thus (2y − x)/(2y + x) = 6k/14k = 3/7.Add: 4/7 + 3/7 = 1. Verification / Alternative check: Pick a concrete pair (x, y) with ratio 4:5, e.g., x = 4 and y = 5. Then (2*5 − 4)/(2*5 + 4) = 6/14 = 3/7, and 4/7 + 3/7 = 1, confirming the result. Why Other Options Are Wrong:3/7 is the value of just the fractional part, not the whole expression.11/7 and 2 come from arithmetic slips while adding or simplifying the ratio. Common Pitfalls: Forgetting to scale both x and y consistently; not canceling k; or adding 4/7 incorrectly. Keep the ratio substitution clean and perform fraction addition with like denominators. Final Answer: 1

Question 15

Solve using a given proportional relation If x / (2y) = 3 / 2, find the exact value of (2x + y) / (x − 2y).

Accepted Answer

Introduction / Context: Converting a proportion into an explicit relationship between x and y simplifies complex rational expressions. This is a common step in algebraic simplification and aptitude problems. Given Data / Assumptions: x / (2y) = 3 / 2. Compute (2x + y) / (x − 2y). Assume y ≠ 0 and denominators stay nonzero. Concept / Approach: From x / (2y) = 3 / 2, deduce x = 3y. Substituting this into the target expression collapses both numerator and denominator to simple multiples of y. The variable cancels out, leaving a pure number. Step-by-Step Solution:Given x/(2y) = 3/2 ⇒ x = 3y.Numerator: 2x + y = 2*(3y) + y = 6y + y = 7y.Denominator: x − 2y = 3y − 2y = y.Thus (2x + y)/(x − 2y) = 7y / y = 7. Verification / Alternative check: Choose y = 1 ⇒ x = 3. Then (2*3 + 1)/(3 − 2) = 7/1 = 7, agreeing with the symbolic solution. Why Other Options Are Wrong:1/7 is the reciprocal of the correct result, often from flipping numerator and denominator.7.1 is a rounding-type distractor; all arithmetic here is exact.None of these is incorrect because 7 is attainable exactly. Common Pitfalls: Misreading x/(2y) as (x/2)/y; substituting x = (3/2)y incorrectly; or forgetting to subtract 2y in the denominator. Maintain the relation x = 3y exactly. Final Answer: 7

Question 16

Recognize and apply algebraic identities Evaluate (885^3 + 115^3) / (885^2 + 115^2 − 885*115) exactly.

Accepted Answer

Introduction / Context: This item checks recognition of the sum-of-cubes identity and the common factor it shares with a^2 + b^2 − ab. Spotting these identities drastically reduces computation for large numbers like 885 and 115. Given Data / Assumptions: Expression: (a^3 + b^3) / (a^2 + b^2 − ab) with a = 885 and b = 115. Use standard algebraic identities; no numeric expansion is required. Concept / Approach: Identity: a^3 + b^3 = (a + b)(a^2 − ab + b^2). Observe that the denominator is exactly a^2 + b^2 − ab, which equals a^2 − ab + b^2. Therefore the fraction simplifies to (a + b)(a^2 − ab + b^2)/(a^2 − ab + b^2) = a + b. Step-by-Step Solution:Rewrite numerator: a^3 + b^3 = (a + b)(a^2 − ab + b^2).Denominator: a^2 + b^2 − ab = a^2 − ab + b^2.Cancel common factor: result = a + b.Compute a + b = 885 + 115 = 1000. Verification / Alternative check: You can verify numerically with smaller placeholders (e.g., a = 8, b = 1) to confirm the identity-based simplification holds. The structure is general, not specific to 885 and 115. Why Other Options Are Wrong:115 and 885 are just the individual terms, not the sum.770 is an attractive distractor (e.g., 885 − 115), but the correct operation is addition after cancellation. Common Pitfalls: Misremembering a^3 + b^3 as (a + b)^3; attempting long multiplication; or overlooking that a^2 + b^2 − ab equals a^2 − ab + b^2 exactly, enabling cancellation. Final Answer: 1000

Question 17

Mixed numbers to improper fractions; chained division and multiplication Evaluate exactly: 4 2/17 ÷ 1 2/5 * 2 2/33.

Accepted Answer

Introduction / Context: Converting mixed numbers to improper fractions is essential before performing sequential division and multiplication. This exercise ensures you can manage multiple conversions and simplifications without losing accuracy. Given Data / Assumptions: Expression: 4 2/17 ÷ 1 2/5 * 2 2/33. All operations are exact with rational numbers. No rounding should be used; simplify by cancellation where possible. Concept / Approach: Convert each mixed number to an improper fraction. Replace division by multiplication with the reciprocal. Then perform cancellations before multiplying to keep numbers small and avoid arithmetic errors. The final result should be reduced to lowest terms if any common factors remain. Step-by-Step Solution:Convert: 4 2/17 = 70/17 (since 4*17 + 2 = 70).Convert: 1 2/5 = 7/5; 2 2/33 = 68/33.Rewrite: (70/17) ÷ (7/5) * (68/33) = (70/17) * (5/7) * (68/33).Cancel 70/7 = 10: gives (10*5/17) * (68/33) = (50/17) * (68/33).Cancel 68/17 = 4: yields (50*4)/33 = 200/33. Verification / Alternative check: Decimal approximation: 200/33 ≈ 6.0606. Computing the original expression numerically (with adequate precision) produces the same value, confirming the rational arithmetic. Why Other Options Are Wrong:61/11 (~5.545) and 81/11 (~7.364) reflect partial cancellations or misplaced reciprocals.42/13 (~3.231) comes from incorrect conversions of mixed numbers. Common Pitfalls: Forgetting to invert the divisor; skipping the simplification 68/17 = 4; or mishandling the mixed-to-improper conversion, which often leads to off-by-one numerators. Final Answer: 200/33

Question 18

Evaluate a rational expression from a given ratio If a / b = 1 / 3, compute (3a + 2b) / (3a − 2b).

Accepted Answer

Introduction / Context: This item uses a basic ratio to enable substitution and simplification of a rational expression. It checks whether you can correctly manipulate signs in the denominator to obtain the final value. Given Data / Assumptions: a / b = 1 / 3 ⇒ a = b/3. Evaluate (3a + 2b) / (3a − 2b). Assume b ≠ 0 and denominator ≠ 0. Concept / Approach: Substitute a = b/3 into numerator and denominator. Factor out b to see immediate cancellation, leaving a ratio of small integers. Pay attention to the negative sign produced in the denominator. Step-by-Step Solution:Set a = b/3.Numerator: 3a + 2b = 3*(b/3) + 2b = b + 2b = 3b.Denominator: 3a − 2b = 3*(b/3) − 2b = b − 2b = −b.Therefore (3a + 2b)/(3a − 2b) = 3b/(−b) = −3. Verification / Alternative check: Choose b = 3 ⇒ a = 1. Then (3*1 + 2*3)/(3*1 − 2*3) = (3 + 6)/(3 − 6) = 9/(−3) = −3, confirming the result. Why Other Options Are Wrong:3 is the sign-ignored result; a common error is missing the negative in the denominator.−5 and −1 arise from arithmetic slips when combining like terms. Common Pitfalls: Dropping the negative sign in the denominator or substituting the ratio incorrectly (e.g., setting a = 3b instead of a = b/3). Keep track of signs carefully. Final Answer: −3

Question 19

Telescoping product over odd terms Compute (2 − 1/3)(2 − 3/5)(2 − 5/7)…(2 − 999/1001) exactly.

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Introduction / Context: Telescoping products are powerful when each factor can be expressed in a form that cancels with neighboring terms. Recognizing the general term and its simplification unlocks a dramatic reduction in effort and avoids massive multiplications. Given Data / Assumptions: Factors: (2 − 1/3), (2 − 3/5), (2 − 5/7), …, (2 − 999/1001). The pattern uses odd numerators k and denominators k + 2, with k = 1, 3, 5, …, 999. Concept / Approach: For a general odd k, 2 − k/(k + 2) = (2(k + 2) − k)/(k + 2) = (2k + 4 − k)/(k + 2) = (k + 4)/(k + 2). The product over successive odd k thus becomes a chain of fractions whose numerators cancel with later denominators, forming a telescoping product. Step-by-Step Solution:Write each factor as (k + 4)/(k + 2).Sequence samples: for k = 1: 5/3; k = 3: 7/5; k = 5: 9/7; … ; k = 999: 1003/1001.Multiplying cancels intermediate 5, 7, 9, …, 1001 between numerators and denominators.Product reduces to final numerator over initial denominator: 1003/3. Verification / Alternative check: Compute the first few and last few factors to observe the cancellation pattern explicitly. The only surviving terms are the very first denominator (3) and the very last numerator (1003). Why Other Options Are Wrong:991/1001 and 1001/13 do not follow the correct telescoping endpoints.“None of these” is incorrect because 1003/3 is obtained exactly. Common Pitfalls: Misidentifying the general factor; starting k at 0 or using even k; or forgetting that only the first denominator and last numerator remain after cancellation. Final Answer: 1003/3

Question 20

Unit-fraction counting in a mixed number How many parts of size 1/8 are contained in 37 1/2?

Accepted Answer

Introduction / Context: Problems that ask “how many 1/n are in a quantity” check a core ratio idea: the count equals total quantity divided by the unit size. When the quantity is a mixed number, convert to an improper fraction or a decimal to proceed cleanly. Given Data / Assumptions: Total amount: 37 1/2. Unit size: 1/8. We want the exact number of such unit parts contained in the total. Concept / Approach: The count of unit fractions equals (Total) / (Unit). Division by a unit fraction 1/8 is equivalent to multiplying by 8. Converting 37 1/2 to 37.5 or to an improper fraction makes the multiplication straightforward. Step-by-Step Solution:Convert 37 1/2 to 37.5 (or 75/2).Compute count: 37.5 ÷ (1/8) = 37.5 * 8.Multiply: 37.5 * 8 = 300 (since 8 * 30 = 240 and 8 * 7.5 = 60; 240 + 60 = 300).Therefore, there are exactly 300 parts of size 1/8. Verification / Alternative check: Fraction route: (75/2) ÷ (1/8) = (75/2) * 8 = 75 * 4 = 300. Both methods agree precisely. Why Other Options Are Wrong:400 and 500 result from multiplying by 10 or 16 incorrectly.“Cannot be determined” is not applicable; the values are exact and sufficient. Common Pitfalls: Dividing by 8 instead of multiplying by 8; misreading 37 1/2 as 37/2; or mixing decimal and fraction methods mid-stream. Keep a single, consistent method. Final Answer: 300

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