Question 1

Interpreting a Proportional Statement — Percent Greater If dividing a first number by 12 gives one-fourth of a second number (i.e., first/12 = second/4), by what percent is the first number greater than the second number?

Accepted Answer

Introduction / Context:This question requires careful interpretation of a verbal proportional statement and conversion to an equation. Once the relationship between the two numbers is clarified, computing the percentage difference is straightforward and relies on a basic percent-change formula. Given Data / Assumptions: Interpretation (Recovery-First): “When a number is divided by 12 then [the result] becomes 1/4 of the other number,” i.e., first/12 = second/4. Let the first number be A and the second be B. We seek percent by which A is greater than B. Concept / Approach:From A/12 = B/4, multiply both sides by 12 to get A = 3B. Percent greater is computed as ((A − B)/B) * 100%. Substitute A = 3B to obtain a numerical percentage independent of specific values of A and B. Step-by-Step Solution:Start: A/12 = B/4.Multiply by 12: A = 3B.Compute percent greater: ((A − B)/B) * 100% = ((3B − B)/B) * 100% = (2) * 100%.Therefore, A is 200% greater than B. Verification / Alternative check:Example: Let B = 10 → A = 30. Then A/12 = 2.5 and B/4 = 2.5, satisfying the condition. A is 200% greater than B since (30 − 10)/10 * 100% = 200%. Why Other Options Are Wrong:150% and 300% misrepresent the 3:1 ratio; “Data inadequate” is incorrect because the proportional statement fully determines A/B; 100% would only hold for A = 2B. Common Pitfalls:Misreading the phrase and setting A/12 = 1/4 (numberless), or mixing dividend/divisor terminology; computing “percent of” instead of “percent greater than.” Final Answer:200

Question 2

Fractions of a Number — Sum Exceeds the Number by 4 The sum of one-half, one-third, and one-fourth of a certain number exceeds the number itself by 4. Determine the number.

Accepted Answer

Introduction / Context:This problem is a straightforward algebraic setup involving fractions of an unknown. The key is to carefully translate “sum exceeds the number by 4” into an equation and then solve for the unknown using common denominators. It is representative of quick-reasoning number system questions. Given Data / Assumptions: Let the number be x. Sum of fractions: x/2 + x/3 + x/4. This sum is equal to x + 4. Find x and match with the options. Concept / Approach:Combine the fractions on the left using the common denominator 12. The coefficients become 6/12, 4/12, and 3/12, summing to 13/12. Then equate 13x/12 to x + 4 and solve. Finally, check whether the calculated value is present among the options; if not, select “None of these.” Step-by-Step Solution:Compute LHS: x/2 + x/3 + x/4 = (6x + 4x + 3x)/12 = 13x/12.Set equation: 13x/12 = x + 4.Subtract x: (13x/12 − x) = 4 → (x/12) = 4.Solve: x = 4 * 12 = 48. Verification / Alternative check:Plug back: 48/2 + 48/3 + 48/4 = 24 + 16 + 12 = 52; the number itself is 48; 52 exceeds 48 by 4. Condition satisfied. Why Other Options Are Wrong:24, 36, and 72 do not satisfy the equation; the correct value 48 is absent from the provided options, so “None of these” is correct. Common Pitfalls:Adding fractional coefficients incorrectly; setting “exceeds by 4” as equals 4 instead of equals x + 4; forgetting to scan choices for the exact computed value. Final Answer:None of these

Question 3

Number system — two-number word problem with proportions and a linear condition: 1/5 of the first number equals 5/8 of the second number. If 35 is added to the first number, the result becomes 4 times the second number. Find the exact value of the s…

Accepted Answer

Introduction / Context: This problem combines a ratio (proportional) relationship between two unknown integers with a linear shift condition. Such questions commonly appear in aptitude tests to assess fluency with translating words into equations and solving a small system consistently. Given Data / Assumptions: Let the first number be x and the second number be y. Condition 1: (1/5) * x = (5/8) * y. Condition 2: x + 35 = 4 * y. All quantities are real numbers; we anticipate integer answers from tidy ratios. Concept / Approach: Convert each sentence into an equation and solve the two equations in two unknowns. From the proportional statement we isolate x in terms of y, then substitute into the linear shift condition. This avoids handling two variables simultaneously in multiple places and reduces arithmetic errors. Step-by-Step Solution:From (1/5) * x = (5/8) * y, multiply both sides by 5: x = (25/8) * y.Use the second condition: x + 35 = 4y.Substitute x: (25/8) * y + 35 = 4y.Move terms: 35 = 4y − (25/8) * y = (32/8 − 25/8) * y = (7/8) * y.Solve for y: y = 35 * 8 / 7 = 40. Verification / Alternative check: Compute x from x = (25/8) * y = (25/8) * 40 = 125. Check the second condition: 125 + 35 = 160 and 4 * y = 4 * 40 = 160, which matches perfectly. Why Other Options Are Wrong:125: That is the first number x, not the second number y.70 or 25: Substituting either breaks the proportional or the linear condition; they do not satisfy both simultaneously. Common Pitfalls: Mixing up which variable is the first or second number; forgetting to multiply both sides evenly when clearing denominators; skipping simplification of fractional coefficients leading to arithmetic slips. Final Answer: 40

Question 4

Fast multiplication using 10,000 − 1 trick: Evaluate 469157 × 9999 accurately without long multiplication.

Accepted Answer

Introduction / Context: Products involving 9999 suggest a classic shortcut: 9999 = 10000 − 1. This allows a quick calculation using place-value shifting and a single subtraction instead of full long multiplication. Given Data / Assumptions: a = 469157 Compute a * 9999. Use the identity 9999 = 10000 − 1. Concept / Approach: Rewrite a * 9999 as a * 10000 − a. Multiplication by 10000 shifts the number four places to the left (because 10000 = 10^4). Then subtract the original number once. This avoids cumbersome partial products and reduces error probability. Step-by-Step Solution:Compute a * 10000 = 469157 * 10000 = 4,691,570,000.Subtract a: 4,691,570,000 − 469,157 = 4,691,100,843.Match with options: 4,691,100,843 corresponds to 4691100843. Verification / Alternative check: Reverse check by adding back: 4,691,100,843 + 469,157 = 4,691,570,000, which equals 469157 * 10000. Therefore the transformation using 10000 − 1 is validated. Why Other Options Are Wrong:Values close in magnitude but with swapped inner digits indicate common subtraction misalignments.Any option not ending with 0843 is suspect because subtracting 469,157 from an exact hundred-million multiple naturally yields the final four digits 0843. Common Pitfalls: Misaligning place values during subtraction; forgetting that multiplying by 10000 appends four zeros; performing a − b as b − a by mistake. Final Answer: 4691100843

Question 5

Find the missing digit using division equality: 2?63 ÷ 11 = 233. Determine the digit replacing the question mark.

Accepted Answer

Introduction / Context: Here a four-digit number has one missing digit. You are told the exact quotient when dividing by 11. The task is to recover the unknown digit by reversing the division, a staple skill in number-system puzzles. Given Data / Assumptions: Number form: 2?63. 2?63 / 11 = 233. All values are integers; division is exact with no remainder. Concept / Approach: If N / 11 = 233, then N = 233 * 11. Compute the product, then compare digits with the pattern 2?63 to read off the missing digit. This avoids long division and reduces computational effort. Step-by-Step Solution:Compute 233 * 11 = 233 * (10 + 1) = 2330 + 233 = 2563.Match 2563 with 2?63: clearly, ? = 5.Therefore the missing digit is 5. Verification / Alternative check: Divide 2563 by 11: 2563 / 11 = 233 exactly, confirming correctness and aligning with the given quotient. Why Other Options Are Wrong:Digits 4, 3, or 6 would produce 2463, 2363, or 2663, none of which are multiples of 11 equaling 233. Common Pitfalls: Trying to apply the alternating-sum test for divisibility by 11 to guess the digit, which can be slower than simply multiplying 233 * 11 for this exact quotient scenario. Final Answer: 5

Question 6

Recover the missing tens digit in a product: 2013 × _1 = 62403. Identify the tens digit so that the multiplier becomes a two-digit number ending in 1.

Accepted Answer

Introduction / Context: The multiplier is a two-digit number ending with 1. We are asked to find its tens digit. This is a reverse-multiplication problem where recognizing nearby multiples simplifies the task dramatically. Given Data / Assumptions: Known multiplicand: 2013. Product: 62403. Multiplier format: _1 (a two-digit number whose ones digit is 1). Concept / Approach: Check 2013 × 31, since 31 is a common candidate (30 + 1). Multiplying by 30 then adding one extra 2013 is a quick mental method. If it matches the target product, the tens digit is 3; otherwise try 21, 41, etc. The structure strongly suggests 31 due to the product’s size relative to 2013 × 30. Step-by-Step Solution:Compute 2013 × 30 = 60390.Add one more 2013 (for ×31): 60390 + 2013 = 62403.Hence, multiplier = 31, so the tens digit is 3. Verification / Alternative check: Try 2013 × 21 = 2013 × (20 + 1) = 40260 + 2013 = 42273, which is too small. Try 41: 2013 × 41 = 82533, too large. Therefore 31 is uniquely correct. Why Other Options Are Wrong:Tens digits 1, 2, or 4 correspond to 11, 21, or 41, none of which produce 62403. Common Pitfalls: Misplacing zeros during the ×30 step; forgetting to add the extra 2013 for the +1 part; reading the question as asking for the whole multiplier rather than only the tens digit. Final Answer: 3

Question 7

Compute an exact division by 11 quickly: Evaluate 777777 ÷ 11 using an efficient mental strategy.

Accepted Answer

Introduction / Context: Dividing a repeating-digit integer by 11 can often be done mentally. Recognizing patterns or verifying with quick multiplication helps avoid long division while ensuring accuracy. Given Data / Assumptions: N = 777777. We require N ÷ 11. All operations are exact with no remainder expected. Concept / Approach: A direct approach is to test the most plausible quotient and confirm by multiplication. A helpful pattern: 70707 × 11 = 70707 × (10 + 1) = 707070 + 70707 = 777777, matching the dividend exactly. This confirms the quotient efficiently. Step-by-Step Solution:Assume candidate q = 70707.Compute q × 11 = q × 10 + q = 707070 + 70707 = 777777.Therefore, 777777 ÷ 11 = 70707. Verification / Alternative check: Perform standard long division or apply the 11-divisibility alternating-sum technique to reassure yourself the number is divisible by 11 and then back-multiply to confirm exactness. Why Other Options Are Wrong:7077, 7707, 7007 produce products with 11 that are far from 777777, revealing incorrect magnitude. Common Pitfalls: Mixing up the number of digits in the quotient; attempting quick mental division but dropping carries. The back-multiplication check is the safest guarantee. Final Answer: 70707

Question 8

Solve for the missing factor: Find the exact integer that satisfies ? × 11 = 555555.

Accepted Answer

Introduction / Context: This is a reverse multiplication problem. Instead of multiplying, you are asked to divide a patterned number by 11 to recover the exact integer factor. Recognizing repeating-digit structures can speed up calculation. Given Data / Assumptions: Equation: ? × 11 = 555555. All values are integers. No remainder is implied; an exact integer solution is expected. Concept / Approach: Since 11 × 50505 = 50505 × (10 + 1) = 505050 + 50505 = 555555, the clean repeating pattern suggests 50505 is the factor. This avoids long division and relies on the structure of base-10 multiplication by 11. Step-by-Step Solution:Compute 50505 × 10 = 505050.Add 50505 (for the +1 part): 505050 + 50505 = 555555.Hence the missing factor is 50505. Verification / Alternative check: Divide directly: 555555 ÷ 11 = 50505. The quotient is an integer with no remainder, confirming the choice. Why Other Options Are Wrong:505, 5050, 5005 produce significantly smaller products with 11 and cannot reach 555555. Common Pitfalls: Guessing small factors due to the compact look of 11; forgetting that multiplying by 11 roughly scales by ten-plus-one, so the factor should have a similar number of digits as the quotient pattern. Final Answer: 50505

Question 9

Single-step equation from a product: Determine the exact integer that satisfies 87 × ? = 3393.

Accepted Answer

Introduction / Context: This problem is a direct linear equation in multiplicative form. The unknown is a whole number. Dividing the known product by the known factor quickly yields the missing integer if computed carefully. Given Data / Assumptions: a = 87, P = 3393. Find q such that a × q = P. Expect integer division: q = P / a. Concept / Approach: Use exact division: q = 3393 / 87. A quick estimate helps: 87 × 40 = 3480 which is slightly high, so try 39. If 87 × 39 equals 3393, the guess is perfect. Step-by-Step Solution:Compute 87 × 40 = 3480.Subtract one 87 to check 39: 3480 − 87 = 3393.Hence q = 39. Verification / Alternative check: Perform direct division: 3393 ÷ 87 = 39 exactly, confirming the result without any remainder. Why Other Options Are Wrong:49 or 67 would overshoot the product; 27 is far too small. Only 39 fits precisely. Common Pitfalls: Estimating only and not verifying; forgetting to check by multiplication after a guess; arithmetic slips when subtracting 87 to move from 40 down to 39. Final Answer: 39

Question 10

Reverse a product to find the missing factor: Find the number that satisfies ? × 147 = 6909.

Accepted Answer

Introduction / Context: Another single-step multiplication equation: recover the unknown integer by dividing the product by the known factor. Efficient estimation narrows the candidate before confirming exactly. Given Data / Assumptions: Known factor: 147. Product: 6909. We seek q = 6909 / 147. Concept / Approach: Approximate: 147 × 50 = 7350, which is too high. Reduce by 3 × 147 = 441 to test 47: 7350 − 441 = 6909. This mental adjustment confirms the exact multiplier without long division. Step-by-Step Solution:Compute 147 × 50 = 7350.Subtract 3 × 147 = 441 to try 47: 7350 − 441 = 6909.Thus q = 47. Verification / Alternative check: Direct division gives 6909 ÷ 147 = 47 exactly, validating the mental adjustment approach. Why Other Options Are Wrong:37 and 27 yield products far below 6909; 67 overshoots substantially. Only 47 matches the target. Common Pitfalls: Estimating 147 as 150 and then failing to correct; not adjusting by the proper multiple when stepping down from 50 to the correct factor. Final Answer: 47

Question 11

Apply factoring to simplify the computation: Evaluate 356 × 936 − 356 × 836 by extracting the common factor.

Accepted Answer

Introduction / Context: Expressions of the form a*b − a*c are tailor-made for the distributive property. Recognizing and factoring out the common term avoids heavy multiplication and reduces the chance of arithmetic mistakes. Given Data / Assumptions: Expression: 356 × 936 − 356 × 836. Common factor: 356. We want an exact numeric result. Concept / Approach: Use the distributive law: a*b − a*c = a*(b − c). This converts two large products and a subtraction into a single subtraction and one manageable multiplication, providing a neat and quick result. Step-by-Step Solution:Factor 356: 356 × (936 − 836).Compute the inner difference: 936 − 836 = 100.Multiply: 356 × 100 = 35600.Therefore, the value is 35600. Verification / Alternative check: If you compute 356 × 936 and 356 × 836 separately and then subtract, you will also reach 35600, confirming the factorization approach. Why Other Options Are Wrong:93600 and 34500 come from misreading the difference or misapplying the common factor.49630 resembles a random product and does not honor the factorization identity. Common Pitfalls: Forgetting to subtract the inner numbers first; multiplying 356 by 936 and 836 fully (time-consuming) and making a slip; swapping the terms and getting a sign error. Final Answer: 35600

Question 12

Proportion equivalence — solve for the missing term: 24 / 18 = ? / 6. Compute the exact value that balances the proportion.

Accepted Answer

Introduction / Context: Proportions equate two ratios. Solving for a missing term often boils down to either cross-multiplication or simplifying the known ratio and scaling it to the comparable denominator (or numerator). This reinforces fraction manipulation skills. Given Data / Assumptions: 24 / 18 = x / 6. All terms are positive real numbers; an integer solution is expected due to clean factors. Goal: find x. Concept / Approach: Two efficient routes exist. Route 1: Reduce 24 / 18 first. Route 2: Cross-multiply directly. Reducing first often clarifies the necessary scaling and helps check reasonableness instantly. Step-by-Step Solution:Simplify 24 / 18 by dividing numerator and denominator by 6: 24 / 18 = 4 / 3.Set 4 / 3 = x / 6.Cross-multiply: 4 * 6 = 3 * x ⇒ 24 = 3x.Solve: x = 24 / 3 = 8. Verification / Alternative check: Compute decimals: 24 / 18 = 1.333..., and x / 6 with x = 8 gives 8 / 6 = 1.333..., confirming equality. Why Other Options Are Wrong:12 or 10 do not keep the ratios equal; 9 yields 9/6 = 1.5, not 4/3. Common Pitfalls: Forgetting to reduce the left fraction; mixing up cross-multiplication; dividing by the wrong factor when simplifying 24/18. Final Answer: 8

Question 13

Number System — Solve for the unknown value (?) given the proportion:  (sqrt(?)) / 196 = 72 / 56.  Compute the exact value of ? that makes the equality true (do not approximate square roots).

Accepted Answer

Introduction / Context:This question checks comfort with proportional reasoning and square roots. Instead of expanding into large products, we manipulate the given ratio to isolate sqrt(?) and then square to recover ?. Careful simplification prevents arithmetic errors and makes the computation fast and exact. Given Data / Assumptions: (sqrt(?)) / 196 = 72 / 56. All quantities are real and positive in the context of basic number system problems. No approximation is needed; the task asks for the exact value of ?. Concept / Approach:Use equivalence of ratios and simplify the right-hand fraction before solving. Once sqrt(?) is isolated, square both sides to remove the square root. Keep arithmetic exact to avoid rounding mistakes. Recognizing common factors in 72 and 56 simplifies the process immediately. Step-by-Step Solution:1) Simplify 72 / 56 by dividing numerator and denominator by 8: 72/56 = 9/7.2) Set up equality: (sqrt(?)) / 196 = 9 / 7.3) Multiply both sides by 196: sqrt(?) = 196 * (9/7) = 28 * 9 = 252.4) Square both sides to solve for ?: ? = 252^2.5) Compute 252^2: 252 × 252 = 63504. Verification / Alternative check:Back-substitute: sqrt(63504) = 252, then 252/196 = 9/7, which equals 72/56 after simplification. Everything is consistent, confirming 63504 is correct. Why Other Options Are Wrong:62504 and 63540 are near-miss digit transpositions; 252 confuses sqrt(?) with ?; 324 is 18^2 and unrelated to the derived 252. Common Pitfalls:Simplifying 72/56 incorrectly; forgetting to square after isolating sqrt(?); computing 252^2 with digit slips; treating 252 (the root) as the final answer instead of squaring back to ?. Final Answer:63504

Question 14

Number System — Given p/q = 4/5, evaluate the expression: 4/7 + (2q - p) / (2q + p). Provide the exact simplified value.

Accepted Answer

Introduction / Context:This problem tests substitution with ratios and algebraic simplification. Rather than solving for explicit p and q, we can set p and q to proportional values consistent with p/q = 4/5 and then simplify the target expression directly. Given Data / Assumptions: p/q = 4/5 with p, q nonzero and of the same sign. Expression: 4/7 + (2q - p) / (2q + p). We seek an exact rational number. Concept / Approach:Let p = 4k and q = 5k for some nonzero k. This preserves p/q = 4/5 while keeping algebra simple. Substitute into (2q - p)/(2q + p) and simplify before adding 4/7. Work with common denominators carefully to avoid mistakes. Step-by-Step Solution:1) Put p = 4k and q = 5k.2) Compute numerator: 2q - p = 10k - 4k = 6k.3) Compute denominator: 2q + p = 10k + 4k = 14k.4) Thus (2q - p)/(2q + p) = 6k/14k = 3/7.5) Add 4/7: 4/7 + 3/7 = 7/7 = 1. Verification / Alternative check:Choose a concrete pair (p, q) such as (4, 5). Then (2*5 - 4)/(2*5 + 4) = (10 - 4)/(10 + 4) = 6/14 = 3/7. Adding 4/7 again yields 1, confirming the result. Why Other Options Are Wrong:3/7 omits adding 4/7; 34 and 2 are not reasonable with a small rational expression; 11/7 arises if someone adds 4/7 to 1 incorrectly or mishandles denominators. Common Pitfalls:Forgetting to maintain proportionality (using arbitrary p, q that do not satisfy 4/5); mismanaging the common factor k; arithmetic slips when adding rational numbers. Final Answer:1

Question 15

Number System — If 3^n = 81, determine the value of n. Show the reasoning using powers of 3 and exact equality (no logarithms needed).

Accepted Answer

Introduction / Context:This is a direct exponent matching problem. The number 81 is a well-known power of 3, so recognizing the relevant exponent avoids longer calculations and ensures an exact answer quickly. Given Data / Assumptions: 3^n = 81. n is a real number; we look for the integer that satisfies the equality. Basic exponent rules apply. Concept / Approach:Rewrite 81 as a power of 3. Using the small powers of 3 helps: 3^2 = 9, 3^3 = 27, 3^4 = 81. Matching the base and comparing exponents yields the solution immediately without logarithms. Step-by-Step Solution:1) Recall powers of 3: 3^2 = 9, 3^3 = 27, 3^4 = 81.2) Since 81 = 3^4, set 3^n = 3^4.3) With equal positive bases, exponents are equal: n = 4. Verification / Alternative check:Compute 3^4 explicitly: 3*3*3*3 = 9*9 = 81. The equality holds exactly, confirming n = 4. Why Other Options Are Wrong:2 and 3 are too small (3^2 = 9, 3^3 = 27); 6 and 8 are too large (3^6 = 729, 3^8 = 6561). None of these equal 81. Common Pitfalls:Confusing 81 with 3^3; mixing up 9 and 81; attempting unnecessary logarithms that increase the chance of arithmetic slips. Final Answer:4

Question 16

Number System — If a/b = 4/3, evaluate (3a + 2b) / (3a − 2b) exactly by substituting proportional values and simplifying.

Accepted Answer

Introduction / Context:Given a ratio between a and b, it is often simplest to set a and b to proportional values and evaluate the target expression directly. This prevents algebraic complexity and ensures exact simplification without solving separate equations. Given Data / Assumptions: a/b = 4/3 with a, b ≠ 0. Expression to evaluate: (3a + 2b) / (3a − 2b). Proportional substitution is valid for ratios. Concept / Approach:Let a = 4k and b = 3k. Substitute into the expression and simplify. Because k cancels, the result is independent of the particular proportional values chosen, giving an exact rational value. Step-by-Step Solution:1) Put a = 4k, b = 3k.2) Compute numerator: 3a + 2b = 3*4k + 2*3k = 12k + 6k = 18k.3) Compute denominator: 3a − 2b = 12k − 6k = 6k.4) Divide: (18k)/(6k) = 3. Verification / Alternative check:Pick a concrete pair, e.g., a = 4, b = 3: (3*4 + 2*3)/(3*4 − 2*3) = (12 + 6)/(12 − 6) = 18/6 = 3. Matches the general result. Why Other Options Are Wrong:6 and 5 come from arithmetic slips in combining terms; −1 requires numerator and denominator to be negatives of each other, which they are not; 2 would require 18k/9k, not present here. Common Pitfalls:Failing to distribute coefficients correctly; forgetting that the common factor k cancels; sign errors in the denominator. Final Answer:3

Question 17

Number System — A 192 m long rod is cut into equal pieces of length 3.2 m each. How many pieces are obtained? Provide the exact count.

Accepted Answer

Introduction / Context:This is a unit-consistent division problem: total length divided by length per piece gives the number of pieces. Converting to decimals and dividing carefully yields the exact integer count when the division is exact. Given Data / Assumptions: Total rod length = 192 m. Each piece length = 3.2 m. All cuts are equal; waste is ignored. Concept / Approach:Pieces = total length / piece length. Because 3.2 = 32/10, the division can be simplified by scaling both numerator and denominator to remove the decimal, which keeps arithmetic clean and reduces error risk. Step-by-Step Solution:1) Compute number of pieces: 192 / 3.2.2) Multiply numerator and denominator by 10 to clear the decimal: (1920) / 32.3) Divide: 1920 / 32 = 60 (since 32 × 60 = 1920).4) Therefore, 60 pieces are obtained. Verification / Alternative check:Multiply back: 60 × 3.2 m = 192 m, matching the original length exactly, so the count is confirmed. Why Other Options Are Wrong:52, 62, and 68 correspond to misdivisions (for example, using 3 or 3.1 instead of 3.2) or arithmetic slips. “None of these” is incorrect since an exact correct option is provided. Common Pitfalls:Dividing by 3 instead of 3.2; dropping the decimal when scaling; rounding prematurely; forgetting to verify by multiplication. Final Answer:60

Question 18

Number System — A 13 m long rod weighs 23.4 kg. Assuming uniform density, find the weight of a 6 m length of the same rod.

Accepted Answer

Introduction / Context:Linear scaling applies to uniform rods: weight is proportional to length when cross-section and material are constant. Therefore, finding the weight of another length simply requires a ratio or unit rate approach. Given Data / Assumptions: Weight of 13 m rod = 23.4 kg. Uniform cross-section and material (constant mass per meter). Find weight for 6 m of the same rod. Concept / Approach:Compute mass per meter first, then multiply by the target length. This method avoids proportionality mistakes and keeps numbers small and manageable. Step-by-Step Solution:1) Mass per meter = 23.4 kg / 13 m = 1.8 kg/m.2) Weight for 6 m = 1.8 kg/m * 6 m = 10.8 kg.3) Therefore, the 6 m length weighs 10.8 kg. Verification / Alternative check:Proportion method: 23.4 : 13 = x : 6 → x = 23.4 * 6 / 13 = 140.4 / 13 = 10.8 kg. Matches the unit-rate method. Why Other Options Are Wrong:7.2 kg and 9.6 kg undercount by using 1.2 or 1.6 kg/m; 12.4 kg and 18.0 kg overcount; only 10.8 kg matches the uniform-density assumption. Common Pitfalls:Rounding mass per meter too early; mixing meters and centimeters; forgetting proportional scaling. Final Answer:10.8 kg

Question 19

Number System — Costing problem: A chair costs ₹214 and a table costs ₹937. Estimate (to the nearest reasonable thousand) the total cost of 6 dozen chairs and 4 dozen tables.

Accepted Answer

Introduction / Context:This applied arithmetic question mixes unit price, dozens, and rounding. The task is to compute an exact total first and then select the nearest appropriate option (an approximation) among the choices provided, which are rounded to convenient thousands. Given Data / Assumptions: Chair price = ₹214 each; quantity = 6 dozen = 72 chairs. Table price = ₹937 each; quantity = 4 dozen = 48 tables. Select an approximate total matching the closest option. Concept / Approach:Calculate each subtotal exactly, add them, then compare to the rounded options. Keeping arithmetic organized avoids digit transposition. Because the options are rounded, pick the nearest value to the exact total. Step-by-Step Solution:1) Chairs: 72 × 214 = (70 × 214) + (2 × 214) = 14980 + 428 = 15408.2) Tables: 48 × 937 = (50 × 937) − (2 × 937) = 46850 − 1874 = 44976.3) Total exact cost = 15408 + 44976 = 60384.4) Compare with options: nearest thousand is ≈ ₹60,000. Verification / Alternative check:Quick rounding check: 214 ≈ 200, 937 ≈ 940 → 72*200 = 14400; 48*940 = 45120; sum ≈ 59520, still closest to ₹60,000. Why Other Options Are Wrong:₹58,000 and ₹55,000 are too low; ₹62,000 and ₹65,000 are slightly high compared to 60,384. ₹60,000 is the best approximation. Common Pitfalls:Forgetting that a dozen is 12; misplacing zeros when multiplying by near-1000 numbers; picking a rounded option without computing the exact base first. Final Answer:Rs. 60000

Question 20

Number System — Five dozen toys are packed per box, and 98 boxes fit in one tempo. How many tempos are needed to transport 29400 toys in a single round?

Accepted Answer

Introduction / Context:This logistics arithmetic problem requires converting dozens to units, computing capacity per tempo, and then dividing the total demand by that capacity. Clear unit handling (dozen = 12) prevents mistakes and leads to an exact count of tempos required. Given Data / Assumptions: 5 dozen toys per box → 60 toys per box. 98 boxes per tempo. Total toys to move = 29400. Concept / Approach:Capacity per tempo = (toys per box) * (boxes per tempo). Then tempos needed = total toys / capacity per tempo. Exact division indicates no partial tempos are required; if a remainder existed, we would round up to the next integer. Step-by-Step Solution:1) Convert: 5 dozen = 5 * 12 = 60 toys per box.2) Capacity per tempo = 60 * 98 = 5880 toys.3) Required tempos = 29400 / 5880 = 5.4) Therefore, 5 tempos are needed for one round. Verification / Alternative check:Reverse check: 5 tempos * 5880 toys/tempo = 29400 toys, matching the requirement exactly. Why Other Options Are Wrong:4 and 6 would give 23520 or 35280 toys respectively, not 29400; 7 and 8 overshoot capacity even more. Common Pitfalls:Using 10 instead of 12 for a dozen; mixing up boxes per tempo; forgetting to multiply both factors before dividing total toys. Final Answer:5

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