Question 1

Place-value basics — face value query: What is the face value of the digit 8 in the numeral 458926?

Accepted Answer

Introduction / Context: Place-value problems often distinguish between a digit’s face value and its place value. The face value is simply the digit itself, regardless of its position in the number. This question checks that you can separate the two concepts without confusion. Given Data / Assumptions: Number: 458926 Digit of interest: 8 Definition: face value equals the digit itself; place value equals the digit times its positional weight. Concept / Approach: Identify the requested type of value. For face value, ignore where the digit sits; you do not multiply by any power of 10. Only the raw digit matters. This is different from place (local) value, which would multiply by a power of 10 based on position. Step-by-Step Solution:1) Read the term carefully: “face value.”2) Extract the digit itself, which is 8.3) Do not apply any positional multiplier.4) Therefore, the face value is 8. Verification / Alternative check: Contrast with place value here: if we asked for the place value of the 8 in 458926, we would compute 8000 because the 8 is in the thousands place. That contrast confirms the interpretation. Why Other Options Are Wrong:8000: That is the place (local) value, not the face value.1000 and 458000: Unrelated to the requested face value definition. Common Pitfalls: Mixing face and place values, especially when the digit sits in a high place-value position. Always read whether the question asks for face value or place value. Final Answer: 8

Question 2

Place-value basics — local (place) value query: What is the local value of the digit 7 in the numeral 5679032?

Accepted Answer

Introduction / Context: This question focuses on local (place) value, which equals a digit multiplied by its positional weight. Unlike face value (the digit itself), local value depends entirely on where the digit sits within the numeral. Given Data / Assumptions: Number: 5679032 Digit of interest: 7 We apply standard base-10 place values: ones, tens, hundreds, thousands, ten-thousands, hundred-thousands, millions, etc. Concept / Approach: Write the number with place markers to identify the position of the 7. In 5,679,032 the digits correspond to millions (5), hundred-thousands (6), ten-thousands (7), thousands (9), hundreds (0), tens (3), ones (2). Therefore, the 7 is in the ten-thousands place, giving a local value of 7 * 10000. Step-by-Step Solution:1) Position mapping: 5 (millions), 6 (hundred-thousands), 7 (ten-thousands), 9 (thousands), 0 (hundreds), 3 (tens), 2 (ones).2) Identify the 7 as ten-thousands place.3) Compute local value: 7 * 10000 = 70000.4) Conclude that the local value is 70000. Verification / Alternative check: Expand the number: 5*10^6 + 6*10^5 + 7*10^4 + 9*10^3 + 0*10^2 + 3*10 + 2. The contribution from the 7 term is 70000, confirming the reasoning. Why Other Options Are Wrong:7: That is the face value, not the place value.10000: Misses the multiplier 7; it is only the positional weight.5070000: Combines multiple digits and positions incorrectly. Common Pitfalls: Confusing the thousand and ten-thousand positions; overlooking the commas that separate groups; or mistaking face value for place value. Final Answer: 70000

Question 3

Number System — Solve for the missing factor: ? × 48 = 173 × 240. Compute the exact numerical value that must replace the question mark so the equality holds.

Accepted Answer

Introduction / Context:This question checks fluency with proportional reasoning and factor manipulation. Instead of multiplying out large numbers on both sides, we isolate the unknown factor by using basic properties of equality and cancellation. This avoids arithmetic overflow and dramatically speeds up the solution. Given Data / Assumptions: An equation: ? × 48 = 173 × 240. All quantities are integers; we need the exact value of the missing factor. Division by a nonzero number preserves equality and is allowed. Concept / Approach:When a product equals another product, we can solve for the unknown by dividing both sides by the known multiplier. Here, dividing both sides by 48 isolates the missing factor. Reducing fractions first avoids large intermediate numbers and keeps the computation clean. Step-by-Step Solution:1) Start with ? × 48 = 173 × 240.2) Divide both sides by 48 to isolate ?: ? = (173 × 240) / 48.3) Simplify 240 / 48 = 5 (since 48 × 5 = 240).4) Therefore, ? = 173 × 5 = 865. Verification / Alternative check:Compute the right-hand product using the simplified result: 865 × 48 = (800 + 60 + 5) × 48 = 800×48 + 60×48 + 5×48 = 38400 + 2880 + 240 = 41520. Also, 173 × 240 = 173 × (24 × 10) = (173 × 24) × 10 = (4152) × 10 = 41520. Both sides match exactly. Why Other Options Are Wrong:545, 685, 495, and 735 each, when multiplied by 48, fail to produce 173 × 240. They deviate by hundreds or thousands once checked against the true product 41520. Common Pitfalls:Multiplying 173 × 240 fully before simplifying; forgetting to reduce by 48 early; arithmetic slips when expanding large products; misreading the equation as addition rather than multiplication. Final Answer:865

Question 4

Number System — Evaluate the arithmetic expression exactly: 160 × 106 + 94 × 94. Choose the correct total from the options.

Accepted Answer

Introduction / Context:This problem requires accurate computation with mixed products and a square term. Precision with place-value multiplication and addition is the key. Recognizing that 94 × 94 is a square can help you compute it cleanly before combining with the other product. Given Data / Assumptions: We must compute 160 × 106 + 94 × 94 exactly. Standard integer arithmetic applies. No approximations are needed; the goal is the exact sum. Concept / Approach:Break each term into manageable parts. For the product 160 × 106, use distributive multiplication. For 94 × 94, compute the square directly (or use a near-100 adjustment if preferred), then add the two results. Double-check intermediate totals to avoid digit slips. Step-by-Step Solution:1) Compute 160 × 106 = 160 × (100 + 6) = 16000 + 960 = 16960.2) Compute 94 × 94. One way: 90 × 94 = 8460; add 4 × 94 = 376; total = 8460 + 376 = 8836.3) Add results: 16960 + 8836 = 25796.4) Therefore, the exact value is 25796. Verification / Alternative check:Alternative square check: (100 − 6)^2 = 10000 − 2*100*6 + 6^2 = 10000 − 1200 + 36 = 8836. Adding to 16960 again gives 25796, confirming consistency. Why Other Options Are Wrong:25876 and 25716 are common digit-transposition errors after addition; 24796 undercounts by 1000 (missed carry); 20032 ignores most of the second term or miscomputes both products. Common Pitfalls:Dropping a zero in 160 × 106; mixing up 94 × 94 with 94 × 90; forgetting to carry during addition; confusing (a − b)^2 with a^2 − b^2. Final Answer:25796

Question 5

Number System — Compute the exact product: 5358 × 51. Select the correct result.

Accepted Answer

Introduction / Context:Multiplying by 51 can be simplified using the identity 51 = 50 + 1. This reduces the problem to one large product plus the original number. Such decomposition helps prevent mistakes and keeps the arithmetic organized. Given Data / Assumptions: We need the exact value of 5358 × 51. Use distributive property to simplify mental math. All calculations are with integers. Concept / Approach:Use 51 = 50 + 1. Then 5358 × 51 = 5358 × 50 + 5358. Compute each term separately and add. This minimizes errors compared to long multiplication done all at once. Step-by-Step Solution:1) Compute 5358 × 50 = 5358 × (5 × 10) = (5358 × 5) × 10.2) 5358 × 5 = 26790; then × 10 → 267900.3) Add the remaining 5358: 267900 + 5358 = 273258.4) Therefore, 5358 × 51 = 273258. Verification / Alternative check:Long multiplication: 5358 × 51 = 5358 × (1) + 5358 × (50) = 5358 + 267900 = 273258. Matches the distributive method result exactly. Why Other Options Are Wrong:273358, 273348, 273268, and 273308 are near-miss transpositions or off-by-10/50 errors that typically arise from misplacing zeros or mis-adding. Common Pitfalls:Forgetting the extra zero when multiplying by 50; arithmetic slips when adding the two parts; swapping digits in the thousands and hundreds places. Final Answer:273258

Question 6

Number System — Evaluate the square exactly: 1307 × 1307. Choose the correct value.

Accepted Answer

Introduction / Context:Squaring a four-digit number can be handled accurately using algebraic expansion. Noticing that 1307 is near 1300 enables a near-base method to reduce effort and organize calculations cleanly. Given Data / Assumptions: We must compute 1307^2 exactly. Use place-value or near-base expansions for reliability. All arithmetic steps should be double-checked to avoid transposition errors. Concept / Approach:Write 1307 = 1300 + 7. Then (1300 + 7)^2 = 1300^2 + 2*1300*7 + 7^2. This standard expansion separates the problem into manageable parts and minimizes mental load. Step-by-Step Solution:1) Compute 1300^2 = (13^2) × 100^2 = 169 × 10000 = 1,690,000.2) Compute 2*1300*7 = 2600*7 = 18,200.3) Compute 7^2 = 49.4) Sum: 1,690,000 + 18,200 + 49 = 1,708,249. Verification / Alternative check:Direct multiplication (1307 × 1307) will yield the same 1,708,249. Checking the last two digits: 07 × 07 ends with 49, matching the result's ending 49, which is a good quick sanity check. Why Other Options Are Wrong:1601249, 1607249, 1609209 are too small (they ignore the growth from adding 7 to 1300 twice). 1701249 is short by 7000 compared to the correct middle term contribution. Common Pitfalls:Forgetting the 2ab middle term; miscomputing 1300^2; dropping zeros when handling 10000 multipliers; transposing digits when adding the parts. Final Answer:1708249

Question 7

Number System — Evaluate the square exactly: 1299 × 1299. Select the correct answer.

Accepted Answer

Introduction / Context:Squaring a near-1000 or near-1300 number can be made efficient with a near-base approach. Here, 1299 is one less than 1300, so the binomial (a − b)^2 expansion is especially convenient and reduces risk of arithmetic errors. Given Data / Assumptions: Compute 1299^2 exactly. Use (a − b)^2 = a^2 − 2ab + b^2 with a = 1300 and b = 1. Carefully handle large-place multipliers to avoid dropped zeros. Concept / Approach:Let 1299 = 1300 − 1. Then (1300 − 1)^2 = 1300^2 − 2*1300*1 + 1^2. This avoids full long multiplication while remaining exact and transparent. Step-by-Step Solution:1) 1300^2 = 1,690,000.2) 2*1300*1 = 2600; subtract: 1,690,000 − 2,600 = 1,687,400.3) Add 1^2 = 1 → 1,687,401.4) Therefore, 1299^2 = 1,687,401. Verification / Alternative check:Check last two digits: 99 × 99 ends with 01, matching the result's ending 01. The magnitude also aligns with being slightly less than 1300^2 (1,690,000), confirming plausibility. Why Other Options Are Wrong:1684701 and 1685401 are off by hundreds or thousands due to middle-term mistakes; 1692601 exceeds 1300^2; 1585301 is far too small, indicating a base or place-value error. Common Pitfalls:Forgetting to subtract 2ab; misplacing zeros in 1300^2; arithmetic slips when subtracting 2600 from 1,690,000. Final Answer:1687401

Question 8

Number System — Compute the exact product: 1014 × 986. Use any reliable method (near-1000 identity or standard multiplication) and choose the correct value.

Accepted Answer

Introduction / Context:This multiplication is tailor-made for the identity (1000 + a) × (1000 − a) = 1000^2 − a^2. Spotting symmetry around 1000 not only speeds up the computation but also reduces the chance of digit errors compared to full long multiplication. Given Data / Assumptions: We must compute 1014 × 986 exactly. Notice 1014 = 1000 + 14 and 986 = 1000 − 14. Apply the difference of squares with a = 14. Concept / Approach:Use (1000 + 14) × (1000 − 14) = 1000^2 − 14^2. This uses fewer steps and produces an exact result quickly. Calculating 14^2 is simple and ensures accuracy in the final subtraction. Step-by-Step Solution:1) Recognize the pattern: (1000 + 14)(1000 − 14) = 1000000 − 14^2.2) Compute 14^2 = 196.3) Subtract: 1,000,000 − 196 = 999,804.4) Therefore, 1014 × 986 = 999,804. Verification / Alternative check:Rough magnitude check: 1014 × 986 is slightly less than 1014 × 1000 (≈ 1,014,000) and slightly less than 1000 × 1000. The exact identity confirms the precise value 999,804. Why Other Options Are Wrong:998904, 998814, 998804, and 998744 all subtract too much from 1,000,000 (larger than 196), reflecting errors in computing 14^2 or in the final subtraction. Common Pitfalls:Using 14^2 = 194 or 200 by mistake; dropping zeros when handling 1,000,000; confusing (a + b)(a − b) with a^2 − b^2 and reversing terms. Final Answer:999804

Question 9

Number System — Evaluate: 387 × 387 + 114 × 114 + 2 × 387 × 114. Identify the exact value.

Accepted Answer

Introduction / Context:Recognize the classic binomial identity embedded in the expression. The sum of two squares plus twice their product equals the square of the sum. Spotting this identity saves time and eliminates arithmetic burden. Given Data / Assumptions: Expression: 387^2 + 114^2 + 2 × 387 × 114. Numbers are integers; exact evaluation is required. Binomial identity applies directly. Concept / Approach:Use (x + y)^2 = x^2 + y^2 + 2xy. Here, x = 387 and y = 114. Therefore, the entire expression collapses neatly to (387 + 114)^2, which is much faster to compute accurately. Step-by-Step Solution:1) Identify x = 387, y = 114.2) Compute x + y = 387 + 114 = 501.3) Evaluate (x + y)^2 = 501^2.4) 501^2 = (500 + 1)^2 = 500^2 + 2*500*1 + 1^2 = 250000 + 1000 + 1 = 251001. Verification / Alternative check:Direct calculation of the original expression using the identity steps confirms the same result. The near-500 square check (500^2 = 250000) ensures the magnitude is sensible, with the extra 1001 added giving 251001. Why Other Options Are Wrong:250001 omits the 2xy and most of the linear term; 260101 and 261001 overshoot the true square; 251121 is a common mis-square of 501 (it would correspond to 511^2 ≈ 261121, not applicable). Common Pitfalls:Forgetting the 2xy term; computing 501^2 as 500^2 + 1^2 only; arithmetic slips when adding 1000 + 1 to 250000. Final Answer:251001

Question 10

Number System — If 11 times a number equals 132, find the number. Provide the exact integer value.

Accepted Answer

Introduction / Context:This is a straightforward linear equation in one variable. Interpreting the statement “11 times a number gives 132” translates directly into an equation that can be solved by division. Accuracy relies on correct parsing and clean arithmetic. Given Data / Assumptions: Let the unknown be x (a real number). Equation: 11 * x = 132. We seek the exact value of x. Concept / Approach:To isolate x, divide both sides of the equation by 11. Division by a nonzero constant preserves equality. This is a prototypical example of solving linear equations with a single operation. Step-by-Step Solution:1) Start with 11 * x = 132.2) Divide both sides by 11: x = 132 / 11.3) Compute 132 / 11 = 12 (since 11 × 12 = 132).4) Therefore, the number is 12. Verification / Alternative check:Multiply the result by 11 to confirm: 12 × 11 = 132, matching the given statement exactly. Why Other Options Are Wrong:11 is too small (11 × 11 = 121); 13.2 would imply 11 × 13.2 = 145.2; “None of these” is unnecessary since a valid choice exists; 10.2 is far from satisfying the equation. Common Pitfalls:Misreading “11 times” as “11 plus”; arithmetic mistakes when dividing 132 by 11; forgetting to check by back-substitution. Final Answer:12

Question 11

Number System — A number is as much greater than 31 as it is less than 55. Determine the number (the value exactly halfway between 31 and 55).

Accepted Answer

Introduction / Context:The phrase “as much greater than 31 as it is less than 55” means the number is equidistant from the two given values on the number line. Therefore, it must be the midpoint of 31 and 55. Finding midpoints is a common technique in arithmetic and algebra. Given Data / Assumptions: The unknown number x satisfies x − 31 = 55 − x. All values are integers; solution can be non-integer in general, but here it turns out integral. Midpoint formula applies: midpoint = (a + b) / 2. Concept / Approach:Set the “greater than 31” difference equal to the “less than 55” difference. Solving this linear equation gives the midpoint directly. Alternatively, use the midpoint formula between 31 and 55 to reach the same result instantly. Step-by-Step Solution:1) Translate: x − 31 = 55 − x.2) Add x to both sides: 2x − 31 = 55.3) Add 31: 2x = 86.4) Divide by 2: x = 86 / 2 = 43. Verification / Alternative check:Midpoint method: (31 + 55) / 2 = 86 / 2 = 43. Differences check: 43 − 31 = 12 and 55 − 43 = 12, equal as required. Why Other Options Are Wrong:47 and 52 are not centered: their distances to 31 and 55 differ; 39 is closer to 31 than to 55; 42 is 11 away from 31 and 13 away from 55, not equal. Common Pitfalls:Subtracting the numbers instead of averaging; misreading the phrase and setting x − 31 = x − 55; arithmetic slips when dividing by 2. Final Answer:43

Question 12

Number System — One fourth of a number subtracted from one third of the same number equals 12. Find the original number.

Accepted Answer

Introduction / Context:This problem involves forming and solving a linear equation with fractional coefficients. Careful handling of fractions and finding a common denominator streamlines the solution. The task assesses basic algebraic manipulation and fraction operations. Given Data / Assumptions: Let the number be x. Equation from words: (1/3)*x − (1/4)*x = 12. We need the exact value of x. Concept / Approach:Combine like terms with fractional coefficients by finding a common denominator (12). The expression (1/3 − 1/4) simplifies to 1/12. Then solve the resulting simple linear equation. This avoids mistakes from prematurely converting to mixed numbers or decimals. Step-by-Step Solution:1) Start: (1/3)*x − (1/4)*x = 12.2) Compute the coefficient: 1/3 − 1/4 = (4 − 3) / 12 = 1/12.3) So (1/12)*x = 12.4) Multiply both sides by 12: x = 12 * 12 = 144. Verification / Alternative check:Plug back: one third of 144 is 48; one fourth is 36; 48 − 36 = 12. The condition is satisfied exactly. Why Other Options Are Wrong:120 gives (40 − 30) = 10; 72 gives (24 − 18) = 6; 63 gives (21 − 15.75) = 5.25; 96 gives (32 − 24) = 8. None produce 12. Common Pitfalls:Adding fractions instead of subtracting; computing 1/3 − 1/4 as 1/7; converting to decimals and rounding prematurely; forgetting to multiply both sides by the common denominator. Final Answer:144

Question 13

Number system — If three-fourths (3/4) of a number is 19 less than the original number, find the value of that number. Show clear reasoning using a simple linear equation.

Accepted Answer

Introduction / Context:This is a classic one-variable linear equation problem from number systems. The phrase “three-fourths of a number is 19 less than the original number” translates directly into an equation that can be solved with basic algebra. These questions test your ability to convert verbal statements into symbolic form and manipulate fractions confidently. Given Data / Assumptions: Let the unknown number be n. Three-fourths of the number means (3/4) * n. “Is 19 less than the original number” means equals n - 19. We assume n is a real number (solution will be an integer here). Concept / Approach:Translate words to an equation and isolate n. Fractional coefficients are handled by moving terms to one side and using basic operations. Keep track of units and signs carefully to avoid simple mistakes. Step-by-Step Solution:Form the equation: (3/4)*n = n - 19.Move terms: n - (3/4)*n = 19.Compute left side: (1/4)*n = 19.Solve for n: n = 19 * 4 = 76. Verification / Alternative check:Check n = 76. Three-fourths of 76 is 57. Then n - 19 = 76 - 19 = 57. Both sides match (57), so the solution is consistent. Why Other Options Are Wrong: 84/64/72/68: These values do not satisfy (3/4)*n = n - 19 when substituted; the two sides will differ. Common Pitfalls:Forgetting that “19 less than n” is n - 19 (not 19 - n); mishandling fractions when moving terms; multiplying by 4 incorrectly. Keep the algebraic steps neat to avoid sign errors. Final Answer:76

Question 14

Linear equation with fractions — If one-fifth of a number decreased by 5 equals 5, find the original number. Model the statement precisely and solve.

Accepted Answer

Introduction / Context:Verbal equations involving fractions occur frequently in aptitude tests. Converting “one-fifth of a number decreased by 5 is 5” into algebra allows you to compute the unknown quickly and reliably. Given Data / Assumptions: Let the number be n. One-fifth of n is n/5. “Decreased by 5” means subtract 5. Equation: n/5 - 5 = 5. Concept / Approach:Use the standard method for linear equations with fractions: isolate the fraction term, then clear denominators by multiplying both sides appropriately. Keep arithmetic tidy to avoid errors. Step-by-Step Solution:Start with n/5 - 5 = 5.Add 5 to both sides: n/5 = 10.Multiply both sides by 5: n = 10 * 5 = 50. Verification / Alternative check:Check n = 50: one-fifth is 10; 10 - 5 = 5, which matches the condition exactly. Therefore, the solution is correct. Why Other Options Are Wrong: 25/60/75/40: Substituting any of these does not satisfy n/5 - 5 = 5; you will get a value different from 5. Common Pitfalls:Misreading “decreased by 5” as division; forgetting to add 5 to both sides; arithmetic slips when clearing denominators. Always verify by substitution. Final Answer:50

Question 15

Fraction comparison equation — A number is 25 more than its two-fifths. Form the equation and determine the number.

Accepted Answer

Introduction / Context:This problem uses a linear relationship between a number and a fraction of itself. Such problems test your ability to express relationships precisely and handle fractional coefficients in equations. Given Data / Assumptions: Unknown number: n. Statement: n is 25 more than (2/5)*n. Equation: n = (2/5)*n + 25. Concept / Approach:Move all n-terms to one side, constants to the other. Combine like terms and solve. Since the coefficient leads to a fractional result, the final answer may be a rational number rather than an integer. Step-by-Step Solution:Start: n = (2/5)*n + 25.Subtract (2/5)*n from both sides: n - (2/5)*n = 25.Compute: (3/5)*n = 25.Solve: n = 25 * (5/3) = 125/3. Verification / Alternative check:Compute two-fifths of 125/3: (2/5)*(125/3) = 250/15 = 50/3. Now 25 more than 50/3 is 50/3 + 25 = 50/3 + 75/3 = 125/3, which equals n, confirming correctness. Why Other Options Are Wrong: 60/80/75/125/7: None satisfy n = (2/5)*n + 25 when substituted; the equality fails. Common Pitfalls:Incorrectly adding 25 to the fraction without aligning denominators; moving terms to the wrong side; assuming the answer must be an integer. Rational results are perfectly valid. Final Answer:125/3

Question 16

Nested fractions — Three-fourths of one-fifth of a number equals 60. Translate and solve for the number.

Accepted Answer

Introduction / Context:Problems with nested fractions require careful reading. “Three-fourths of one-fifth of a number” means multiply the number by both fractions. Mastery of fraction multiplication and equation solving is the key here. Given Data / Assumptions: Unknown number: n. (3/4)*(1/5)*n = 60. All operations are standard arithmetic over real numbers. Concept / Approach:Combine the fractions first, then invert the coefficient to isolate n. Keep fractions exact to avoid rounding errors and convert to integers only at the end if desired. Step-by-Step Solution:Compute combined fraction: (3/4)*(1/5) = 3/20.Equation: (3/20)*n = 60.Multiply both sides by 20/3: n = 60 * (20/3).Calculate: 60 * 20 = 1200; 1200 / 3 = 400 → n = 400. Verification / Alternative check:Check: one-fifth of 400 is 80; three-fourths of 80 is 60. The statement is satisfied exactly, confirming the solution. Why Other Options Are Wrong: 300/450/1200/360: Substituting these into (3/20)*n does not yield 60; outcomes are different. Common Pitfalls:Misreading “of” as addition; multiplying by 4/3 instead of 20/3; arithmetic slips with 60 * 20. Work systematically to avoid mistakes. Final Answer:400

Question 17

Consecutive integers — The difference between the squares of two consecutive numbers is 35. Identify the two numbers.

Accepted Answer

Introduction / Context:For consecutive integers n and n+1, the difference of their squares has a well-known identity that makes the calculation immediate. Recognizing this identity avoids unnecessary computation and speeds up problem solving. Given Data / Assumptions: Two numbers are consecutive: n and n + 1. Given: (n + 1)^2 - n^2 = 35. We solve for integer n. Concept / Approach:Use the difference of squares identity: (a^2 - b^2) = (a - b) * (a + b). For a = n + 1 and b = n, we get (n + 1)^2 - n^2 = (1) * (2n + 1) = 2n + 1. Set 2n + 1 equal to 35 and solve. Step-by-Step Solution:Write identity: (n + 1)^2 - n^2 = 2n + 1.Set equal to 35: 2n + 1 = 35.Solve: 2n = 34 → n = 17.Therefore, the numbers are 17 and 18. Verification / Alternative check:Compute directly: 18^2 - 17^2 = 324 - 289 = 35, exactly as required. This confirms the identity-based solution. Why Other Options Are Wrong: 14,15 → 225 - 196 = 29; 15,16 → 256 - 225 = 31; 18,19 → 361 - 324 = 37; 16,17 → 289 - 256 = 33. None equals 35. Common Pitfalls:Squaring both numbers and subtracting without using the identity (slower and error-prone); forgetting that for consecutive numbers, the difference of squares simplifies to 2n + 1, an odd number. Final Answer:17, 18

Question 18

Ratio and sum — Two numbers are in the ratio 3:4 and their sum is 420. Compute the greater number using the parts method.

Accepted Answer

Introduction / Context:Ratio problems are efficiently solved by interpreting ratios as parts of a whole. Given both the ratio and the total sum, you can find the value of one part and then scale each term to obtain the actual numbers. Given Data / Assumptions: Ratio of two numbers: 3:4. Sum of the numbers: 420. We seek the greater number corresponding to the 4-part share. Concept / Approach:If the numbers are 3k and 4k, then their sum is 7k. Solving for k from the sum and multiplying yields each number. This method generalizes to any ratio-sum combination and avoids trial and error. Step-by-Step Solution:Let numbers be 3k and 4k.Sum: 3k + 4k = 7k = 420.Solve for k: k = 420 / 7 = 60.Greater number: 4k = 4 * 60 = 240. Verification / Alternative check:Add the two computed values: 3*60 + 4*60 = 180 + 240 = 420, matching the given sum; the ratio 180:240 simplifies to 3:4, confirming correctness. Why Other Options Are Wrong: 175/200/315/180: These either do not align with a 3:4 split of 420 or correspond to the smaller number, not the greater one. Common Pitfalls:Dividing 420 by 3 or 4 prematurely; forgetting to divide by the total number of parts (7); mistakenly reporting the smaller number instead of the greater one. Final Answer:240

Question 19

Algebraic identity application — The sum of two numbers is 100 and their difference is 37. What is the difference of their squares? Use (a^2 - b^2) = (a - b)(a + b).

Accepted Answer

Introduction / Context:Many problems become trivial once you recall the right algebraic identity. Here, knowing that the difference of squares factors into product of sum and difference allows instant calculation without finding the individual numbers first. Given Data / Assumptions: a + b = 100. a - b = 37. We need a^2 - b^2. Concept / Approach:Use the identity a^2 - b^2 = (a - b) * (a + b). Substitute the given sum and difference directly to compute the result in one step. This is both faster and less error-prone than solving for a and b first. Step-by-Step Solution:Write the identity: a^2 - b^2 = (a - b)(a + b).Substitute: (a - b) = 37 and (a + b) = 100.Compute product: 37 * 100 = 3700.Therefore, a^2 - b^2 = 3700. Verification / Alternative check:If desired, solve a = (100 + 37)/2 = 68.5 and b = (100 - 37)/2 = 31.5, then compute 68.5^2 - 31.5^2; it will equal 3700 due to the identity, confirming the result. Why Other Options Are Wrong: 37/100/63/2963: None equals (a - b)(a + b) with the given values. Only 3700 is consistent with the identity. Common Pitfalls:Adding or subtracting the numbers first to get individual values unnecessarily; forgetting the identity and attempting long multiplication; mixing up sum and difference positions. Final Answer:3700

Question 20

System of equations — The difference of two numbers is 8 and one-eighth (1/8) of their sum is 35. Determine the two numbers.

Accepted Answer

Introduction / Context:When given relationships for sum and difference, you can solve for two numbers directly using linear equations. Recognizing that “one-eighth of their sum is 35” gives the sum immediately is the key step here. Given Data / Assumptions: Let the numbers be a and b with a > b. a - b = 8. (1/8) * (a + b) = 35 → a + b = 280. We seek ordered pair (b, a) or (a, b) as presented in options. Concept / Approach:Use the standard transformations: a = (sum + difference)/2 and b = (sum - difference)/2. This avoids solving two equations separately and guarantees integer results when sum and difference share parity. Step-by-Step Solution:From (1/8)(a + b) = 35, compute sum: a + b = 35 * 8 = 280.Given difference: a - b = 8.Compute a = (280 + 8)/2 = 288/2 = 144.Compute b = (280 - 8)/2 = 272/2 = 136. Verification / Alternative check:Check: a - b = 144 - 136 = 8; (1/8)(a + b) = (1/8)(280) = 35. Both conditions are satisfied, confirming the pair 136 and 144. Why Other Options Are Wrong: 132,140 / 128,136 / 124,132 / 140,148: These pairs do not simultaneously yield difference 8 and sum 280. Common Pitfalls:Forgetting to multiply 35 by 8 to get the sum; swapping sum and difference in the formulas; arithmetic slips when halving the totals. Final Answer:136, 144

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