Question 1

Mixed operations with division inside the denominator Evaluate (48 − 12 × 3 + 9) / (12 − 9 ÷ 3).

Accepted Answer

Introduction / Context: This problem uses multiplication and division within larger sums and differences. It emphasizes performing operations in the correct order to avoid common mistakes. Given Data / Assumptions: Numerator: 48 − 12 × 3 + 9. Denominator: 12 − 9 ÷ 3. Concept / Approach: Apply multiplication/division before addition/subtraction. Reduce numerator and denominator separately to simplest forms, and then divide to get the final fraction. Step-by-Step Solution: Numerator: 12 × 3 = 36 → 48 − 36 + 9 = 12 + 9 = 21.Denominator: 9 ÷ 3 = 3 → 12 − 3 = 9.Value = 21 / 9 = 7 / 3 after simplifying by 3. Verification / Alternative check: Convert 7/3 to mixed number 2 and 1/3; quickly re-check numerator and denominator computations to ensure consistency. Why Other Options Are Wrong: 3 and 21 are interim values (denominator or numerator alone); 1/3 is the reciprocal of the simplified ratio; 9/7 reverses the fraction. Common Pitfalls: Adding 12 + 9 before multiplying; dividing 9 by 3 incorrectly; forgetting to simplify 21/9. Final Answer: 7 / 3

Question 2

Decimals with chained multiplication and division Evaluate 0.01 × 0.3 ÷ 0.4 × 0.5 (use left-to-right precedence for multiplication/division).

Accepted Answer

Introduction / Context: Chained decimal operations require attention to both decimal placement and operation order. Multiplication and division share the same precedence and are evaluated from left to right. Given Data / Assumptions: Expression: 0.01 × 0.3 ÷ 0.4 × 0.5. Left-to-right evaluation for × and ÷. Concept / Approach: Compute sequentially: first multiply 0.01 by 0.3, then divide the result by 0.4, then multiply by 0.5. Convert divisions by decimals into multiplications by reciprocals if helpful. Step-by-Step Solution: First: 0.01 × 0.3 = 0.003.Second: 0.003 ÷ 0.4 = 0.003 / 0.4 = 0.0075.Third: 0.0075 × 0.5 = 0.00375. Verification / Alternative check: As fractions: 0.01 = 1/100; 0.3 = 3/10; 0.4 = 2/5; 0.5 = 1/2. Product = (1/100)*(3/10)/(2/5)*(1/2) = (3/1000)*(5/2)*(1/2) = 15/4000 = 3/800 = 0.00375. Why Other Options Are Wrong: .015 and .0375 arise from skipping the division by 0.4 or mis-ordering steps; .1 is far too large; .0125 corresponds to partial computation errors. Common Pitfalls: Applying right-to-left instead of left-to-right for same-precedence operations; decimal point misplacement during division by 0.4. Final Answer: .00375

Question 3

Decimal operations with order of operations Evaluate the expression precisely: 0.05 * 5 − 0.005 * 5. Show the simplified result.

Accepted Answer

Introduction / Context: Decimal simplification questions test careful application of multiplication before subtraction and accurate place-value handling. Here we evaluate two decimal products and then subtract, which is a common pattern in bank PO and aptitude exams. Given Data / Assumptions: Expression: 0.05 * 5 − 0.005 * 5. Standard operator precedence: multiplication before subtraction. All numbers are exact decimals; no rounding is needed. Concept / Approach: Perform each multiplication independently, then subtract. Converting to fractions can also help: 0.05 = 5/100 and 0.005 = 5/1000. Both strategies lead to the same exact value. Step-by-Step Solution: Compute A = 0.05 * 5 = 0.25.Compute B = 0.005 * 5 = 0.025.Subtract: A − B = 0.25 − 0.025 = 0.225.Therefore, the simplified result is 0.225. Verification / Alternative check: Fraction method: 0.05 = 5/100, so A = (5/100)*5 = 25/100 = 0.25. Also, 0.005 = 5/1000 ⇒ B = (5/1000)*5 = 25/1000 = 0.025. Their difference is 0.225, confirming the decimal computation. Why Other Options Are Wrong: 2.25 and 2.025 arise from misplaced decimal points or ignoring place values.0.29875 does not follow any correct intermediate calculation for these factors. Common Pitfalls: Treating 0.05*5 as 0.5 (sliding the decimal incorrectly) or subtracting before multiplying. Always multiply first, then subtract, while carefully tracking decimal places. Final Answer: 0.225

Question 4

Apply BODMAS/PEMDAS to mixed decimals Evaluate: 4 − 3.6 ÷ 4 + 0.2 * 0.5 (division and multiplication before addition/subtraction).

Accepted Answer

Introduction / Context: This problem checks correct use of BODMAS/PEMDAS where division and multiplication precede addition and subtraction. With decimals, it also assesses your comfort handling tenths correctly. Given Data / Assumptions: Expression: 4 − 3.6 ÷ 4 + 0.2 * 0.5. Compute division and multiplication left to right before combining with addition/subtraction. No rounding is needed because all results are terminating decimals. Concept / Approach: Execute the operations in two passes: first compute 3.6 ÷ 4 and 0.2 * 0.5, then aggregate with 4 using addition/subtraction signs as given. Step-by-Step Solution: Compute 3.6 ÷ 4 = 0.9.Compute 0.2 * 0.5 = 0.1.Now evaluate: 4 − 0.9 + 0.1 = 3.2.Hence, the value is 3.2. Verification / Alternative check: Convert to fractions if desired: 3.6 = 36/10, 0.2 = 1/5, 0.5 = 1/2. Then 36/10 ÷ 4 = 36/40 = 9/10 = 0.9 and (1/5)*(1/2) = 1/10 = 0.1, leading to the same final sum 3.2. Why Other Options Are Wrong: 0.2 and 0.15 ignore part of the expression or misapply operator precedence.1.65 arises from subtracting 3.6 first then dividing, which is incorrect order. Common Pitfalls: Performing subtraction before division, or multiplying 0.2 and 0.5 incorrectly as 0.01. Remember: 2 tenths times 5 tenths is 10 hundredths = 0.1. Final Answer: 3.2

Question 5

Decimals and mixed operations Evaluate: 0.5 * 0.5 + 0.5 ÷ 5 (apply multiplication/division before addition).

Accepted Answer

Introduction / Context: This question reinforces the precedence of multiplication and division over addition with simple decimal values. It is a quick calculation if the order is respected and place values are handled correctly. Given Data / Assumptions: Expression: 0.5 * 0.5 + 0.5 ÷ 5. Perform product and quotient first, then sum the results. Decimals correspond to simple halves and tenths. Concept / Approach: Compute each operation separately: 0.5 * 0.5 and 0.5 ÷ 5. Then add the two results. Converting to fractions can also help: 0.5 = 1/2 and 5 = 5/1. Step-by-Step Solution: 0.5 * 0.5 = 0.25.0.5 ÷ 5 = 0.1.Sum: 0.25 + 0.1 = 0.35.Therefore, the expression equals 0.35. Verification / Alternative check: Fraction form: (1/2)*(1/2) = 1/4 = 0.25, and (1/2)/5 = 1/10 = 0.1. Their sum is 7/20 = 0.35, perfectly matching the decimal result. Why Other Options Are Wrong: 0.25 omits the division term; 0.45 adds incorrectly; 0.15 results from multiplying 0.5 and 0.5 incorrectly or dividing 0.5 by 5 wrongly. Common Pitfalls: Treating 0.5 ÷ 5 as 0.5/0.5; misplacing decimal points; or adding before finishing both operations of higher precedence. Final Answer: 0.35

Question 6

Implied multiplication with parentheses; follow left-to-right for * and ÷ Evaluate: 8 ÷ 4 (3 − 2) * 4 + 3 − 7.

Accepted Answer

Introduction / Context: Expressions with implied multiplication (e.g., 4(3−2)) often cause mistakes if parentheses and left-to-right rules are ignored. This item checks that you reduce the bracket first, then process division and multiplication strictly left to right before addition/subtraction. Given Data / Assumptions: Expression: 8 ÷ 4 (3 − 2) * 4 + 3 − 7. (3 − 2) is evaluated first. Multiplication and division share precedence; evaluate them left to right. Concept / Approach: After simplifying the bracket, rewrite the chain of * and ÷ so you can proceed sequentially from left to right. This avoids the common error of multiplying by 4 first or reordering operations incorrectly. Step-by-Step Solution: Compute bracket: (3 − 2) = 1.Expression becomes: 8 ÷ 4 * 1 * 4 + 3 − 7.Left to right: 8 ÷ 4 = 2; 2 * 1 = 2; 2 * 4 = 8.Then add/subtract: 8 + 3 − 7 = 4. Verification / Alternative check: Replace implied multiplication by explicit symbol to verify: 8 ÷ 4 * (3 − 2) * 4 + 3 − 7, which yields the same sequence and result 4. Why Other Options Are Wrong: −3 or −4 come from mishandling signs or subtracting 7 too early.5 appears if one multiplies by 4 prematurely or misorders operations after removing brackets. Common Pitfalls: Treating implied multiplication as higher precedence than division; skipping the left-to-right rule; or forgetting to apply the +3 − 7 at the end. Final Answer: 4

Question 7

Evaluate a sum of three rational subexpressions Compute: (20 ÷ 5) ÷ 2 + (16 ÷ 8) * 2 + (10 ÷ 5) * (3 ÷ 2).

Accepted Answer

Introduction / Context: This computation combines nested divisions and multiplications. Breaking each parenthesized term into a simple number first keeps the process clean and avoids left-to-right precedence mistakes inside the groups. Given Data / Assumptions: Expression: (20 ÷ 5) ÷ 2 + (16 ÷ 8) * 2 + (10 ÷ 5) * (3 ÷ 2). Parentheses isolate each mini-calculation. No rounding; all results are exact. Concept / Approach: Evaluate the three grouped parts individually, then add. Each part simplifies to an integer or a simple fraction: this reduces error and makes mental arithmetic easy. Step-by-Step Solution: Part 1: (20 ÷ 5) ÷ 2 = 4 ÷ 2 = 2.Part 2: (16 ÷ 8) * 2 = 2 * 2 = 4.Part 3: (10 ÷ 5) * (3 ÷ 2) = 2 * 3/2 = 3.Sum: 2 + 4 + 3 = 9. Verification / Alternative check: Fraction approach: (20/5)/2 = 4/2 = 2; (16/8)*2 = 2*2 = 4; (10/5)*(3/2) = 2*(3/2) = 3. Totals to 9 again, confirming consistency. Why Other Options Are Wrong: 12, 15, 18 result from incorrectly doubling one term or miscomputing 3/2 as 1 or 2. Common Pitfalls: Dropping parentheses and applying left-to-right globally; misreading 3 ÷ 2 as 2 ÷ 3; or assuming every part must be an integer before summing. Final Answer: 9

Question 8

Mixed operations with integer division and multiplication Evaluate: 3120 ÷ 26 + 13 * 30, applying division and multiplication before addition.

Accepted Answer

Introduction / Context: This problem combines a clean division with a straightforward multiplication. After computing both parts under the standard precedence rules, we add to obtain the final result and then compare against options (including the common “None of these”). Given Data / Assumptions: Expression: 3120 ÷ 26 + 13 * 30. Operators: division and multiplication precede addition. Integers chosen to give an exact quotient for the first term. Concept / Approach: Evaluate each high-precedence operation first. Notice that 3120 ÷ 26 simplifies exactly, avoiding any decimal. Then perform the multiplication 13 * 30 and add the two results. Step-by-Step Solution: Compute 3120 ÷ 26. Since 26 * 120 = 3120, this quotient is 120 exactly.Compute 13 * 30 = 390.Add: 120 + 390 = 510.Therefore, the exact value is 510. Verification / Alternative check: You can reduce 3120/26 by dividing numerator and denominator by 13: (3120/13)/(26/13) = 240/2 = 120, confirming the first term. The sum to 510 remains consistent. Why Other Options Are Wrong: 2400 and 3900 ignore the addition structure or multiply incorrectly.536 is close but incorrect; it may come from mistakenly adding 26 to 510 or using 3120/26 ≈ 120 then adding 416. Common Pitfalls: Treating 13 * 30 as 3900 by appending an extra zero; miscomputing 3120 ÷ 26 as a non-integer; or forgetting that addition occurs after both operations are computed. Final Answer: None of these

Question 9

Respect left-to-right for multiplication/division Evaluate: 7 + 7 ÷ 7 * 7 (do division, then multiplication, then addition).

Accepted Answer

Introduction / Context: This classic trap checks whether you apply the left-to-right rule for multiplication and division of equal precedence before doing any addition. Doing it correctly yields an integer result without ambiguity. Given Data / Assumptions: Expression: 7 + 7 ÷ 7 * 7. Division and multiplication have equal precedence; evaluate left to right. Addition is done last. Concept / Approach: First compute 7 ÷ 7, then multiply the result by 7, and finally add the leading 7. This prevents the common mistake of adding first or multiplying 7*7 first. Step-by-Step Solution: 7 ÷ 7 = 1.1 * 7 = 7.Now add: 7 + 7 = 14.So the value is 14. Verification / Alternative check: Insert parentheses to mirror precedence: 7 + ((7 ÷ 7) * 7) = 7 + (1 * 7) = 14, matching the left-to-right rule for equal-precedence ops. Why Other Options Are Wrong: 42 comes from multiplying 7*7 first to get 49, then adding 7 and dividing by 7 unsystematically.71/7 or 2/7 reflect mishandled grouping or treating the entire expression as a single fraction. Common Pitfalls: Violating left-to-right for * and ÷; distributing the 7 incorrectly; or treating 7 + 7 ÷ 7 as (7 + 7)/7 by mistake. Final Answer: 14

Question 10

Subtraction after division with decimals Evaluate: 5005 − 5000 ÷ 10.00 (remember division before subtraction).

Accepted Answer

Introduction / Context: Many slip-ups occur when subtraction appears before a division term. This item checks that you execute the division first—even with a decimal divisor—before performing the subtraction. Given Data / Assumptions: Expression: 5005 − 5000 ÷ 10.00. Operator precedence: division before subtraction. 10.00 equals 10 exactly; the trailing zeros are cosmetic. Concept / Approach: Compute 5000 ÷ 10.00 first to get an exact integer. Then subtract that result from 5005. Converting 10.00 to 10 makes no difference to the value. Step-by-Step Solution: Compute division: 5000 ÷ 10.00 = 500.Subtract: 5005 − 500 = 4505.Therefore, the expression evaluates to 4505. Verification / Alternative check: View as 5005 − (5000/10) = 5005 − 500 = 4505, which is unchanged by representing 10 as 10.00. The decimal formatting does not alter the arithmetic. Why Other Options Are Wrong: 50 confuses subtraction with division of the whole number; 5000 ignores the division completely; 0.5 arises from dividing 5 by 10 erroneously. Common Pitfalls: Subtracting first (5005 − 5000) and then dividing; misreading 10.00 as 100; or believing decimal points change the numeric value of 10. Final Answer: 4505

Question 11

Chaining division and multiplication (left-to-right) Evaluate: 171 ÷ 19 * 9 exactly.

Accepted Answer

Introduction / Context: Problems like this reward noticing exact cancellations. If the first division yields an integer, the remaining multiplication is quick and prevents rounding errors. Given Data / Assumptions: Expression: 171 ÷ 19 * 9. Division and multiplication evaluated left to right. All values are integers; no rounding needed. Concept / Approach: Check whether 171 is a multiple of 19. If so, perform the exact division to simplify the expression, then multiply by 9. This is faster than turning to decimals too early. Step-by-Step Solution: Observe 19 * 9 = 171, hence 171 ÷ 19 = 9.Then compute 9 * 9 = 81.Thus, the final value is 81. Verification / Alternative check: Reverse grouping: (171/19)*9 = 9*9 = 81. Direct computation without simplification would also yield 81 but with more steps, so the factor observation is preferred. Why Other Options Are Wrong: 0 or 1 ignore the multiplication; 18 comes from stopping after the first division and forgetting the final multiplication. Common Pitfalls: Converting 171/19 to a decimal approximation and introducing rounding; losing the left-to-right flow by multiplying 19 * 9 first in the denominator mentally. Final Answer: 81

Question 12

Recovery fix for ambiguous “of” usage; clarify intended operations Evaluate the clarified expression: 108 ÷ (36 of 1/3) + (2/5 of 3 3/4). (Here “of” means multiplication, and the grouping removes ambiguity.)

Accepted Answer

Introduction / Context: The original stem “108 ÷ 36 of 1/3 + 2/5 + 3 3/4” is ambiguous because “of” (multiplication) appears without clear grouping, and many learners are unsure whether “of” has higher precedence than division. Applying the Recovery-First Policy, we minimally repair the item to the most standard, testable reading: compute 108 ÷ (36 of 1/3) and then add (2/5 of 3 3/4). This clarification matches well-known conventions in school arithmetic where “of” means multiplication and parentheses remove ambiguity. Given Data / Assumptions: Treat “of” as multiplication. First grouped term: 108 ÷ (36 * 1/3). Second grouped term: (2/5) * (3 3/4). Mixed number 3 3/4 is 15/4. Concept / Approach: Evaluate each group separately. Using fractions keeps results exact and avoids decimal rounding. After simplifying each, add the results and convert to an improper fraction if needed to match the answer style. Step-by-Step Solution: Compute inner product: 36 of 1/3 = 36 * 1/3 = 12.Now 108 ÷ 12 = 9.Convert the mixed number: 3 3/4 = 15/4.Compute (2/5) of (15/4) = (2/5) * (15/4) = 30/20 = 3/2 = 1.5.Add both parts: 9 + 3/2 = 9 + 1.5 = 10.5 = 21/2. Verification / Alternative check: Reverse order: find (2/5)*(15/4) = 3/2 first, then evaluate 108 ÷ (36*1/3) = 108 ÷ 12 = 9, sum to 21/2. Both routes agree. Why Other Options Are Wrong: 83/4 = 20.75 and 61/4 = 15.25 are larger than the clarified sum; they result from misplacing parentheses or adding 3 3/4 directly without the “of.”101/2 = 50.5 is far too large and reflects ignoring division completely. Common Pitfalls: Misinterpreting “of” as having different precedence than multiplication; treating 3 3/4 as 33/4; or forgetting to convert the mixed number before multiplying. Clear grouping avoids these mistakes. Final Answer: 21/2

Question 13

Compute the fractional expression with correct order of operations:  Evaluate 1 3/5 − (2/3 ÷ 12/13) + (7/5 × 1/3). Express the final result as a single simplified fraction.

Accepted Answer

Introduction / Context: This simplification tests mastery of mixed numbers, fraction division and multiplication, and adding/subtracting with unlike denominators. The key is to convert the mixed number to an improper fraction, respect operation order, and reduce at the end for a neat, exact answer. Given Data / Assumptions: Treat “×” as multiplication and “÷” as division. Expression: 1 3/5 − (2/3 ÷ 12/13) + (7/5 × 1/3). Final output should be a single simplified fraction. Concept / Approach: Use the rule a ÷ b = a * (1/b) for fractions. Do products and divisions before the outer additions/subtractions when grouped. Convert the mixed number 1 3/5 into an improper fraction to combine results over a common denominator. Step-by-Step Solution: Convert 1 3/5 = (5*1 + 3)/5 = 8/5.Evaluate division: 2/3 ÷ 12/13 = (2/3) * (13/12) = 26/36 = 13/18.Evaluate product: 7/5 × 1/3 = 7/15.Assemble: 8/5 − 13/18 + 7/15.Common denominator 90: 8/5 = 144/90, 13/18 = 65/90, 7/15 = 42/90.Compute: 144/90 − 65/90 + 42/90 = (144 − 65 + 42)/90 = 121/90. Verification / Alternative check: Reduce along the way if possible. None of 8/5, 13/18, or 7/15 further reduce together to a simpler path than using 90, and 121/90 is already in lowest terms (121 = 11^2, denominator 90 = 2 * 3^2 * 5, no common factor). Why Other Options Are Wrong: 131/90 results from adding an extra 10/90; 19/30 and 11/30 come from incorrect common denominators; 30 ignores fraction arithmetic and order of operations. Common Pitfalls: Forgetting to invert the second fraction during division; treating 1 3/5 as 1 + 3/5 but then mixing denominators incorrectly; skipping the common denominator step. Final Answer: 121/90

Question 14

Fill the missing fraction to complete the sum:  3 1/4 + 4 1/6 + ? + 1/4 = 10. Determine the exact value of the missing term as a simplified fraction.

Accepted Answer

Introduction / Context: This completion problem checks comfort with mixed numbers, conversion to improper fractions, and solving a simple linear equation in one unknown fraction. Accuracy with common denominators ensures the missing term is identified exactly. Given Data / Assumptions: Total required: 10. Known terms: 3 1/4, 4 1/6, and 1/4. Unknown term represented by ?. Concept / Approach: Convert mixed numbers to improper fractions, combine known fractions, and subtract from 10 to isolate the missing fraction. Keep exact fractional arithmetic (avoid decimals) to reach a simplified result. Step-by-Step Solution: Convert 3 1/4 = 13/4.Convert 4 1/6 = 25/6.Add known parts: 13/4 + 1/4 = 14/4 = 7/2.Now 7/2 + 25/6 = (21/6 + 25/6) = 46/6 = 23/3.We need total 10 = 30/3, so ? = 30/3 − 23/3 = 7/3. Verification / Alternative check: Convert 7/3 to 2 1/3; adding to 3 1/4 + 4 1/6 + 1/4 indeed reaches 10 when recomputed carefully with common denominators. Why Other Options Are Wrong: 21/6 = 7/2 is too large; 41/3 and 11/3 overshoot or undershoot the needed difference; 21/3 = 7 is far from the fractional gap remaining. Common Pitfalls: Mistreating mixed numbers; adding 1/4 incorrectly; switching to decimal approximations and rounding off, which can mask the exact value. Final Answer: 7/3

Question 15

Apply multiplication and division of fractions carefully:  Evaluate 48 ÷ 12 × ( (9/8 of 4/3) ÷ (3/4) of 2/3 ). Give the result as a simplified fraction.

Accepted Answer

Introduction / Context: Expressions with “of” require treating “of” as multiplication. Multiplication and division among fractions should be evaluated left to right inside grouping symbols. This exercise emphasizes disciplined order of operations and simplification. Given Data / Assumptions: Compute 48 ÷ 12 first (outside the parentheses). Inside parentheses: (9/8 of 4/3) ÷ (3/4) of 2/3 = (9/8 * 4/3) ÷ (3/4) * 2/3, left to right. Report a simplified fractional answer. Concept / Approach: Use a ÷ b = a * (1/b) and simplify at each step. Cancel common factors early to avoid large numerators and denominators, then combine with the outside factor to finish. Step-by-Step Solution: Outside factor: 48 ÷ 12 = 4.Compute 9/8 * 4/3 = 36/24 = 3/2.Divide by 3/4: (3/2) ÷ (3/4) = (3/2) * (4/3) = 2.Multiply by 2/3: 2 * 2/3 = 4/3.Combine with outside factor: 4 * 4/3 = 16/3. Verification / Alternative check: Compute everything in one consolidated fraction to confirm: (48/12) * ((9/8 * 4/3) * (4/3) * (2/3)) after converting the division; this also yields 16/3 after cancellations. Why Other Options Are Wrong: 11/3 and 3 arise from premature addition or missed steps; 12 ignores the internal scaling by fractions; 51/3 = 17 is far from the balanced fraction result. Common Pitfalls: Not treating “of” as multiplication; reversing the divisor; skipping the left-to-right rule for multiply/divide. Final Answer: 16/3

Question 16

Evaluate the nested brackets accurately:  Compute 10 − [9 − {8 − (7 − 6)}] − 5 and choose the correct integer result.

Accepted Answer

Introduction / Context: Nesting of parentheses, braces, and brackets demands inside-out evaluation. This problem reinforces the disciplined approach to evaluate the innermost operation first and move outward step by step to avoid sign mistakes. Given Data / Assumptions: Expression: 10 − [9 − {8 − (7 − 6)}] − 5. Standard order: parentheses, then braces, then brackets, then remaining operations left to right. Concept / Approach: Work from the innermost parentheses outward. Keep track of minus signs carefully: subtracting a bracketed value changes signs when distributing over the terms inside. Step-by-Step Solution: Innermost: (7 − 6) = 1.Next: {8 − 1} = 7.Next: [9 − 7] = 2.Final: 10 − 2 − 5 = 8 − 5 = 3. Verification / Alternative check: Rewrite as 10 − 2 − 5 after evaluating the bracketed value to ensure no sign inversion errors. This still equals 3. Why Other Options Are Wrong: −5 results from sign mishandling; 1 and 9 arise from partially completed or incorrectly ordered steps; 5 ignores the second subtraction. Common Pitfalls: Dropping brackets early; forgetting that subtracting a bracket is subtracting the entire evaluated result; arithmetic slips in the final subtraction. Final Answer: 3

Question 17

Evaluate the layered fraction expression step by step:  Compute 1 + 1 ÷ {1 + 1 ÷ (1 + 1/3)} and express the result as a simplified fraction.

Accepted Answer

Introduction / Context: Nesting of unit fractions is common in aptitude tests. Resolving the inner-most parentheses first ensures clarity and prevents inversion errors when moving outward through the expression. Given Data / Assumptions: Expression: 1 + 1 ÷ {1 + 1 ÷ (1 + 1/3)}. Division by a fraction means multiplying by its reciprocal. All steps should remain fractional until the end. Concept / Approach: Start inside: compute (1 + 1/3), then take its reciprocal for the inner division, add to 1, invert again for the outer division, and finally add 1. Keeping every step exact avoids rounding issues. Step-by-Step Solution: Compute inner sum: 1 + 1/3 = 4/3.Inner division: 1 ÷ (4/3) = 3/4.Middle braces: 1 + 3/4 = 7/4.Outer division: 1 ÷ (7/4) = 4/7.Final addition: 1 + 4/7 = 11/7. Verification / Alternative check: Represent the whole expression as a continued fraction 1 + 1/(1 + 1/(1 + 1/3)) and evaluate bottom-up; the result is again 11/7. Why Other Options Are Wrong: 11/3 and 12/3 are too large; 14/7 equals 2 and ignores nesting; 11/8 comes from inverting the wrong term. Common Pitfalls: Inverting 1 + 1/3 instead of dividing by it; adding 1 at the wrong stage; switching to decimals mid-calculation. Final Answer: 11/7

Question 18

Handle “of”, division, and addition correctly:  Evaluate 2 of 3/4 ÷ 3/4 + 1/4, treating “of” as multiplication and following left-to-right rules for multiply/divide.

Accepted Answer

Introduction / Context: Many test-takers stumble on expressions with “of” and division. Remember that “of” indicates multiplication. Also, when only multiplication and division appear at the same precedence, evaluate strictly from left to right before doing the addition at the end. Given Data / Assumptions: Expression: 2 of 3/4 ÷ 3/4 + 1/4. “of” means multiply; multiply/divide left to right. Return a simplified fraction. Concept / Approach: Compute the left product, then divide by 3/4, and finally add 1/4. Convert division by a fraction to multiplication by its reciprocal to simplify cancellations. Step-by-Step Solution: Compute product: 2 of 3/4 = 2 * 3/4 = 3/2.Divide by 3/4: (3/2) ÷ (3/4) = (3/2) * (4/3) = 2.Add 1/4: 2 + 1/4 = 9/4. Verification / Alternative check: Convert 2 to 8/4; 8/4 + 1/4 = 9/4, confirming consistency. Why Other Options Are Wrong: 3/2 and 7/4 arise from stopping too early or misdividing; 5/2 and 8/3 result from mishandling the reciprocal during division. Common Pitfalls: Performing addition before completing multiply/divide; failing to invert 3/4 correctly during division; reducing incorrectly. Final Answer: 9/4

Question 19

Apply division before addition as required:  Calculate 1/2 + 1/2 ÷ 1/2 and provide the simplified fractional result.

Accepted Answer

Introduction / Context: When addition and division appear together, division must be completed first. This item checks whether you respect the precedence rules with basic fractions and then simplify to lowest terms. Given Data / Assumptions: Expression: 1/2 + 1/2 ÷ 1/2. Division precedes addition. All results should be simplified fractions. Concept / Approach: Compute 1/2 ÷ 1/2 by multiplying by the reciprocal, then add that result to 1/2. Keep fractions exact for clarity and simplicity. Step-by-Step Solution: Evaluate division: 1/2 ÷ 1/2 = (1/2) * (2/1) = 1.Add remaining term: 1/2 + 1 = 1 1/2 = 3/2. Verification / Alternative check: As decimals: 0.5 + (0.5 ÷ 0.5) = 0.5 + 1 = 1.5 = 3/2, confirming the fraction form. Why Other Options Are Wrong: 5/2, 8/3, and 9/4 result from adding before dividing or misinterpreting 1/2 ÷ 1/2; 2 equals 4/2 and is too large. Common Pitfalls: Doing addition first; thinking 1/2 ÷ 1/2 is 1/4; forgetting to simplify mixed numbers to improper fractions when needed. Final Answer: 3/2

Question 20

Solve for the missing divisor using decimal arithmetic:  (17.28 ÷ x) / (3.6 × 0.2) = 2. Find x.

Accepted Answer

Introduction / Context: This problem tests manipulation of decimal products and ratios. By simplifying the denominator and then isolating x, we can solve without resorting to long division beyond a single clean step. Given Data / Assumptions: Equation: (17.28 ÷ x) / (3.6 × 0.2) = 2. All numbers are decimals; aim to minimize decimal complexity. Concept / Approach: First evaluate the denominator product. Then convert the compound fraction into a simple equation for 17.28 ÷ x. Finally, invert to find x. Keeping exact decimals prevents rounding errors. Step-by-Step Solution: Compute denominator: 3.6 × 0.2 = 0.72.Set (17.28 ÷ x) / 0.72 = 2 ⇒ 17.28 ÷ x = 2 × 0.72 = 1.44.Solve for x: x = 17.28 ÷ 1.44 = 12. Verification / Alternative check: Plug back: (17.28/12) = 1.44; 1.44 / 0.72 = 2. The equality holds exactly. Why Other Options Are Wrong: 120 and 0.12 are order-of-magnitude errors; 1.20 and 0.72 come from misplacing the factor 2 or stopping mid-simplification. Common Pitfalls: Multiplying 3.6 by 0.2 incorrectly (should be 0.72); dividing by 2 instead of multiplying; rounding 17.28/1.44 inaccurately. Final Answer: 12

Simplification Questions