Question 1

When one-seventh of a number is subtracted from the number itself, the result equals the sum of the angles in a triangle (180 degrees). What is the number?

Accepted Answer

Introduction / Context:This is a linear equation in one variable with a familiar constant (the angle sum of a triangle). Converting the verbal statement into an algebraic equation and solving it yields the unique value of the number. Given Data / Assumptions: Let the number be N. The sum of angles in a triangle is 180 degrees. Condition: N − (1/7)N = 180. Concept / Approach:Combine like terms on the left: N − (1/7)N = (6/7)N. Then isolate N by dividing 180 by the fractional coefficient. This is a direct application of fraction arithmetic. Step-by-Step Solution: Write the equation: N − N/7 = 180.Compute the coefficient: (6/7)N = 180.Isolate N: N = 180 * (7/6) = 30 * 7 = 210. Verification / Alternative check:Check: 210 − (1/7)*210 = 210 − 30 = 180, which satisfies the given relation. Why Other Options Are Wrong:224, 140, 350, and 196 do not produce 180 after subtracting one-seventh of themselves. Common Pitfalls:Subtracting 1/7 incorrectly (e.g., taking N − 1/7 instead of N − N/7), or miscomputing 180 * 7/6. Final Answer:210

Question 2

In a division operation, the divisor equals ten times the quotient and also equals five times the remainder. If the remainder is 46, find the dividend.

Accepted Answer

Introduction / Context:This problem connects the four elements of a division algorithm: dividend, divisor, quotient, and remainder. Using the given proportional relationships, you can determine the divisor and quotient, then reconstruct the dividend through the standard identity Dividend = Divisor * Quotient + Remainder. Given Data / Assumptions: Remainder R = 46. Divisor D = 5 * R. Divisor D = 10 * Quotient Q. Division identity: Dividend N = D * Q + R. Concept / Approach:First compute D from the remainder. Then use D = 10Q to find Q. Finally, apply N = DQ + R. This sequence avoids guesswork and keeps arithmetic simple and exact. Step-by-Step Solution: Compute D: D = 5 * 46 = 230.Find Q from D = 10Q → Q = D / 10 = 23.Compute N: N = D * Q + R = 230 * 23 + 46.Evaluate: 230 * 23 = 230 * (20 + 3) = 4600 + 690 = 5290; then N = 5290 + 46 = 5336. Verification / Alternative check:Check proportions: D = 230 is five times 46 and ten times 23. Substituting into N = DQ + R reproduces the given remainder exactly. Why Other Options Are Wrong:5388, 5343, 5391, and 5328 do not satisfy the division identity with D = 230, Q = 23, and R = 46. Common Pitfalls:Confusing which term is 10 times which (mistaking Q = 10D), or omitting the remainder when reconstructing the dividend. Final Answer:5336

Question 3

Consecutive odd integers — product given The product of two consecutive odd numbers is 6723. Determine the greater of the two odd integers.

Accepted Answer

Introduction / Context:This question checks understanding of consecutive odd numbers and converting a word statement into a simple quadratic equation. Consecutive odd integers differ by 2, and their product is supplied. The task is to retrieve the larger integer directly and verify it. Given Data / Assumptions: The two odd integers are consecutive: let them be n and n + 2. Their product equals 6723: n(n + 2) = 6723. We seek the greater integer (n + 2). Concept / Approach:Form a quadratic equation from the product relation and solve it using the discriminant method. Because the numbers are integers and consecutive odds, the quadratic should factor nicely or yield an integer root after applying the square root to the discriminant. Step-by-Step Solution:Start: n(n + 2) = 6723 → n^2 + 2n − 6723 = 0.Compute discriminant: D = 2^2 + 4*6723 = 4 + 26892 = 26896.Recognize 26896 = 164^2, so roots are integers.n = (−2 + 164)/2 = 81 (take positive root).Consecutive odd numbers: 81 and 83 → greater = 83. Verification / Alternative check:Multiply 81 * 83 = 81*(80 + 3) = 6480 + 243 = 6723, which matches the given product. Therefore, 83 is correct. Why Other Options Are Wrong:89, 85, and 91 are not paired with a neighboring odd to give 6723; 81 is the smaller integer, not the greater one requested. Common Pitfalls:Using (n, n + 1) instead of odd spacing of 2; arithmetic errors in discriminant; forgetting to extract the greater value after solving for n. Final Answer:83

Question 4

Sum of four consecutive even integers The sum of four consecutive even numbers is 284. What is the smallest of these even integers?

Accepted Answer

Introduction / Context:Consecutive even integers increase by 2 each time. Summation of such a group forms a quick linear equation. This tests modeling skills and careful arithmetic. Given Data / Assumptions: The four even numbers are consecutive: a, a + 2, a + 4, a + 6. Their total sum is 284. We must find the smallest, a. Concept / Approach:Write the sum in terms of a and equate to 284. Because the common difference is 2, the sum is 4a + 12. Solve the resulting linear equation for a. Step-by-Step Solution:Sum: a + (a + 2) + (a + 4) + (a + 6) = 284.Combine like terms: 4a + 12 = 284.Subtract 12: 4a = 272.Divide by 4: a = 68. Verification / Alternative check:Numbers are 68, 70, 72, 74. Sum = 68 + 70 + 72 + 74 = (68 + 74) + (70 + 72) = 142 + 142 = 284. Checks out. Why Other Options Are Wrong:66 and 70 are near misses that give totals not equal to 284; 72 and 74 are not the smallest in the valid set. Common Pitfalls:Using odd increments of 1 instead of 2; arithmetic slips when subtracting 12; misidentifying the smallest number after solving. Final Answer:68

Question 5

Two-digit number with digit sum and digit difference A two-digit number has digits that sum to 14 and differ by 2. What is the product of the two digits of the number?

Accepted Answer

Introduction / Context:Digit problems often give the sum and difference of two digits. This makes it straightforward to solve for the individual digits using simultaneous linear equations and then compute the desired product. Given Data / Assumptions: Let the digits be L (larger) and S (smaller). L + S = 14. L − S = 2. We want L * S. Concept / Approach:Add and subtract the equations to isolate L and S. This is a standard method for two equations in two unknowns. After finding the digits, multiply them for the requested product. Step-by-Step Solution:Add equations: (L + S) + (L − S) = 14 + 2 → 2L = 16 → L = 8.Find S: S = 14 − L = 14 − 8 = 6.Product: L * S = 8 * 6 = 48. Verification / Alternative check:Digits 8 and 6 satisfy both conditions: sum 14, difference 2. The product 48 follows directly. Why Other Options Are Wrong:56 and 45 arise from mixing digits or arithmetic slips; 40 is another near miss; “Couldn't be determined” is incorrect because the system has a unique solution. Common Pitfalls:Interchanging sum and difference equations or assuming the digits can be in any order without checking the difference sign. Final Answer:48

Question 6

Simplify: $8888 + 888 + 88 + 8$

Accepted Answer

### Concept & Strategy The most efficient way to add a repetitive sequence of a single digit is by factoring out the common digit or using the column addition multiplication shortcut. ### Step-by-Step Solution * **Calculation:** Instead of standard addition, factor out the common number 8: $$ 8888 + 888 + 88 + 8 = 8 	imes (1111 + 111 + 11 + 1) $$ * Add the sequence of ones: $$ 1111 + 111 + 11 + 1 = 1234 $$ * Multiply by the factored digit: $$ 8 	imes 1234 = 9872 $$ ### Exam Strategy & Shortcut For repetitive additions of the same digit, simply multiply the digit by the descending sequence of its count. Here, 8 appears 4 times, then 3, then 2, then 1. Simply calculate $8 	imes 1234 = 9872$ to get the answer in seconds. ### Common Pitfall When doing traditional column addition under time pressure, students often misalign the place values, leading to incorrect sums in the hundreds and thousands places. Always align right-to-left if writing it out. ### Final Answer Therefore, the correct answer is **9872**.

Question 7

Simplify: $715632 - 631104 - 9874 - 999$

Accepted Answer

### Concept & Logic When an expression contains one positive number and multiple negative numbers, it is mathematically safer and faster to sum all the negative numbers first, then subtract that combined total from the positive number. ### Step-by-Step Solution * **Calculation:** Group all the negative terms together: $$ -(631104 + 9874 + 999) $$ * Sum the grouped negative values: $$ 631104 + 9874 + 999 = 641977 $$ * Subtract this combined total from the initial positive number: $$ 715632 - 641977 = 73655 $$ ### Exam Strategy & Shortcut Use the "Last Two Digits" strategy to quickly eliminate options. The sum of the last two digits of the negatives is $04 + 74 + 99 = 177$ (so the total ends in $77$). Subtracting $77$ from the last two digits of the positive number ($32$ becomes $132$ after a borrow): $132 - 77 = 55$. The correct option must end in $55$. ### Common Pitfall Attempting sequential left-to-right subtraction (e.g., subtracting $631104$, then subtracting $9874$ from that result) vastly increases the chance of a minor arithmetic error and consumes significantly more time. ### Final Answer Therefore, the correct answer is **73655**.

Question 8

What value will replace the question mark in the following equation? $x - 1936248 = 1635773$

Accepted Answer

### Concept & Formula This is a fundamental linear equation in one variable. To isolate the unknown variable $x$, you transpose the known constant to the opposite side of the equation, which inverts its mathematical sign (subtraction becomes addition). ### Step-by-Step Solution * **Given:** $x - 1936248 = 1635773$ * **Calculation:** Transpose the negative term to the right side of the equation to isolate $x$: $$ x = 1635773 + 1936248 $$ * Add the two values: $$ x = 3572021 $$ ### Exam Strategy & Shortcut To quickly verify or eliminate options without doing the full addition, check the last two digits. The last two digits of the addends are $73$ and $48$. $73 + 48 = 121$. The correct answer must end in **21**. If multiple options end in 21, check the first few digits: $16 + 19 = 35$. The answer is roughly 3.5 million. ### Common Pitfall A frequent mistake under time pressure is subtracting the numbers instead of adding them when moving the negative value across the equals sign, incorrectly leading to $1936248 - 1635773$. ### Final Answer Therefore, the correct answer is **3572021**.

Question 9

What value will replace the question mark in the following equation? $9587 - x = 7429 - 4358$

Accepted Answer

### Concept & Strategy When an equation contains operations on both sides, always simplify the side with purely known values first. Then, rearrange the equation to isolate the unknown variable, ensuring you manage the negative sign carefully. ### Step-by-Step Solution * **Given:** $9587 - x = 7429 - 4358$ * **Calculation:** First, simplify the right side of the equation by subtracting the values: $$ 7429 - 4358 = 3071 $$ * Substitute this back into the equation: $$ 9587 - x = 3071 $$ * Rearrange to isolate $x$. Add $x$ to both sides and subtract $3071$ from both sides: $$ x = 9587 - 3071 $$ * Perform the final subtraction: $$ x = 6516 $$ ### Exam Strategy & Shortcut Use the "Unit Digit" method for split-second option elimination. Calculate the unit digit of the right side: $9 - 8 = 1$. So, $9587 - x$ must end in $1$. Looking at the unit digit of the left side: $7 - (	ext{unit digit of } x) = 1$. The unit digit of $x$ MUST be $6$. If only one option ends in 6, mark it and move on. ### Common Pitfall Students often fumble the algebraic signs during transposition. A common error is writing $x = 3071 - 9587$, resulting in a negative number, or mistakenly adding the numbers instead of subtracting. Always move the negative $x$ to make it positive. ### Final Answer Therefore, the correct answer is **6516**.

Question 10

What could be the maximum value of $Q$ in the following equation? $$5P9 + 3R7 + 2Q8 = 1114$$

Accepted Answer

### Concept & Logic This is a cryptarithmetic puzzle based on column addition. To find the maximum value of one digit variable ($Q$) in a sum, you must determine the combined sum of all variables and then set the other unknown variables ($P$ and $R$) to their minimum possible integer value, which is $0$. ### Step-by-Step Solution * **Deduction:** Analyze the addition column by column, starting from the units place (right to left). * **Units Column:** Add the known digits: $9 + 7 + 8 = 24$. The result writes down $4$ and carries over $2$ to the tens column. * **Tens Column:** The sum of the tens column includes the carry-over ($2$) plus the unknown digits ($P, R, Q$). We know the final total $1114$ has a $1$ in the tens place. * **Hundreds Column Check:** The hundreds digits are $5 + 3 + 2 = 10$. However, the final sum is $11$ (from $1114$). This confirms the tens column generated a carry-over of exactly $1$. * **Formulating the Equation:** Since the tens column ends in $1$ and carries over $1$, its total sum is exactly $11$. $$ 2 	ext{ (carry)} + P + R + Q = 11 $$ $$ P + R + Q = 9 $$ * **Maximizing Q:** To make $Q$ as large as possible, $P$ and $R$ must be as small as possible. Since $P$, $Q$, and $R$ are digits, their minimum value is $0$. * Setting $P = 0$ and $R = 0$: $$ 0 + 0 + Q = 9 \Rightarrow Q = 9 $$ ### Exam Strategy & Shortcut Immediately sum the units: $9+7+8 = 24$. Note the carry $2$. The total tens sum must be $11$ because the hundreds sum ($5+3+2=10$) needs $+1$ to reach the $11$ in $1114$. Therefore, $2 + P + Q + R = 11$, so $P+Q+R = 9$. To maximize one variable in a sum of non-negative integers, make the others zero. Max $Q = 9$. ### Common Pitfall The most common mistake is forgetting to add the carry-over from the units place ($2$) to the tens place equation. This leads to the incorrect assumption that $P + R + Q = 11$, which makes students wrongly assume a digit could theoretically be greater than 9 if the question asked for a combined total. ### Final Answer Therefore, the correct answer is **9**.

Question 11

Simplify: $5793405 	imes 9999$

Accepted Answer

### Concept & Strategy When multiplying a number by a series of 9s (like $99$, $999$, $9999$), use the distributive property by representing the multiplier as a power of 10 minus 1. ### Step-by-Step Solution * **Calculation:** Rewrite $9999$ as $(10000 - 1)$: $$ 5793405 	imes (10000 - 1) $$ * Distribute the multiplication: $$ (5793405 	imes 10000) - (5793405 	imes 1) $$ $$ 57934050000 - 5793405 $$ * Perform the final subtraction: $$ 57934050000 - 5793405 = 57928256595 $$ ### Exam Strategy & Shortcut The "Append and Subtract" shortcut is the fastest method here. Since $9999$ has four 9s, simply append four zeros to the original number ($5793405 ightarrow 57934050000$) and then subtract the original number ($5793405$) from it. ### Common Pitfall Attempting to multiply a 7-digit number by a 4-digit number using standard long multiplication will consume minutes of exam time and almost certainly lead to a calculation error. Always look for base-10 properties. ### Final Answer Therefore, the correct answer is **57928256595**.

Question 12

Simplify: $839478 	imes 625$

Accepted Answer

### Concept & Formula Multiplying by powers of 5 (like $5, 25, 125, 625$) can be simplified by converting the multiplier into a fraction based on powers of 10. We know that $625 = 5^4$. We can rewrite this as $(\frac{10}{2})^4 = \frac{10000}{16}$. ### Step-by-Step Solution * **Calculation:** Substitute $625$ with its equivalent fraction $\frac{10000}{16}$: $$ 839478 	imes \left(\frac{10}{2}ight)^4 $$ $$ 839478 	imes \frac{10000}{16} $$ * Multiply the numerator first (append four zeros): $$ \frac{8394780000}{16} $$ * Divide by $16$ (or divide by $2$ four consecutive times): $$ \frac{8394780000}{16} = 524673750 $$ ### Exam Strategy & Shortcut Whenever multiplying by $625$, append four zeros to the end of the number and divide the result by 16. Dividing by 16 is often much faster and less error-prone than doing a 3-digit multiplication, especially if you halve the number four times in your head. ### Common Pitfall Students often miss the power-of-10 conversion and try to multiply by 625 directly. While doable, it creates three separate massive lines of addition which eats up valuable time in competitive exams. ### Final Answer Therefore, the correct answer is **524673750**.

Question 13

Evaluate: $986 	imes 137 + 986 	imes 863$

Accepted Answer

### Concept & Logic This problem relies on the Distributive Property of Multiplication over Addition. The formula is $a 	imes b + a 	imes c = a 	imes (b + c)$. Identifying a common factor allows you to simplify complex arithmetic into simple base-10 multiplication. ### Step-by-Step Solution * **Given:** $986 	imes 137 + 986 	imes 863$ * **Calculation:** Identify the common factor, which is $986$. Factor it out of the expression: $$ 986 	imes (137 + 863) $$ * Simplify the addition inside the parentheses first: $$ 137 + 863 = 1000 $$ * Multiply by the factored number: $$ 986 	imes 1000 = 986000 $$ ### Exam Strategy & Shortcut Scan expressions for repeating numbers before doing any math. If you see the same large number being multiplied in separate terms, always factor it out. The non-common numbers (here, $137$ and $863$) are almost always designed by the examiner to sum perfectly to $10$, $100$, or $1000$. ### Common Pitfall A strict, blind application of the BODMAS rule without looking for algebraic simplifications first will cause a student to multiply $986 	imes 137$ and $986 	imes 863$ separately, taking minutes instead of seconds. ### Final Answer Therefore, the correct answer is **986000**.

Question 14

Evaluate: $983 	imes 207 - 983 	imes 107$

Accepted Answer

### Concept & Logic This question tests the Distributive Property of Multiplication over Subtraction, represented by the formula $a 	imes b - a 	imes c = a 	imes (b - c)$. By factoring out the common multiplier, you turn a tedious calculation into mental math. ### Step-by-Step Solution * **Given:** $983 	imes 207 - 983 	imes 107$ * **Calculation:** Extract the common multiplier, $983$: $$ 983 	imes (207 - 107) $$ * Perform the subtraction inside the parentheses: $$ 207 - 107 = 100 $$ * Multiply the common factor by the result: $$ 983 	imes 100 = 98300 $$ ### Exam Strategy & Shortcut Just like with addition, look for identical numbers separated by a minus sign. Factor out the $983$, and quickly subtract the remaining terms ($207 - 107 = 100$). Simply add two zeros to $983$ to arrive at the answer instantly. ### Common Pitfall Failing to spot the common factor and attempting to brute-force the multiplication of both terms individually. Examiners specifically write these numbers so that their difference is a neat power of 10 to reward students who check for patterns first. ### Final Answer Therefore, the correct answer is **98300**.

Question 15

Simplify $1605 	imes 1605$

Accepted Answer

### Concept & Formula The problem requires finding the square of a number close to a base of 1600. We can simplify the calculation by using the algebraic identity for the square of a sum: $$ (a + b)^2 = a^2 + b^2 + 2ab $$ ### Step-by-Step Solution * We can write $1605$ as $(1600 + 5)$. * Therefore, $1605 	imes 1605 = (1600 + 5)^2$. * Applying the formula where $a = 1600$ and $b = 5$: $$ (1600 + 5)^2 = (1600)^2 + (5)^2 + 2 	imes 1600 	imes 5 $$ * Calculate the individual terms: $$ (1600)^2 = 2560000 $$ $$ (5)^2 = 25 $$ $$ 2 	imes 1600 	imes 5 = 10 	imes 1600 = 16000 $$ * Add them together: $$ 2560000 + 25 + 16000 = 2576025 $$ ### Exam Strategy & Shortcut When squaring a number ending in 5, the last two digits of the answer will always be 25. For numbers near a round base like 1600, breaking it into $(1600 + 5)$ allows you to calculate entirely using mental math and simple additions, bypassing tedious multi-digit multiplication. ### Common Pitfall A frequent mistake is applying the distributive property incorrectly by forgetting the $2ab$ term, calculating only $1600^2 + 5^2$ to get 2560025. Always remember the middle term when expanding binomial squares. ### Final Answer Therefore, the correct answer is **2576025**.

Question 16

Simplify $1398 	imes 1398$

Accepted Answer

### Concept & Formula The problem requires squaring a number that is very close to a round base (1400). We can simplify this by using the algebraic identity for the square of a difference: $$ (a - b)^2 = a^2 + b^2 - 2ab $$ ### Step-by-Step Solution * We can express $1398$ as $(1400 - 2)$. * Therefore, $1398 	imes 1398 = (1400 - 2)^2$. * Applying the formula where $a = 1400$ and $b = 2$: $$ (1400 - 2)^2 = (1400)^2 + (2)^2 - 2 	imes 1400 	imes 2 $$ * Calculate the individual terms: $$ (1400)^2 = 1960000 $$ $$ (2)^2 = 4 $$ $$ 2 	imes 1400 	imes 2 = 5600 $$ * Combine the terms: $$ 1960000 + 4 - 5600 = 1954404 $$ ### Exam Strategy & Shortcut Instead of performing a 4x4 digit multiplication, converting the problem to a base of 1400 reduces the math to simple subtraction. Also, check the unit digit: $8 	imes 8 = 64$, so the answer must end in 4. If only one option ends in 4, you can pick it immediately without calculating. ### Common Pitfall A common error is forgetting to subtract the $2ab$ term (5600) and instead adding it, which would yield the incorrect distractor 1965604. Pay close attention to the minus sign in the $(a - b)^2$ identity. ### Final Answer Therefore, the correct answer is **1954404**.

Question 17

Evaluate $475 	imes 475 + 125 	imes 125$

Accepted Answer

### Concept & Formula This expression represents the sum of two squares, $a^2 + b^2$. We can manipulate the standard algebraic identities for $(a+b)^2$ and $(a-b)^2$ to create a simpler calculation formula: $$ a^2 + b^2 = \frac{1}{2} [ (a + b)^2 + (a - b)^2 ] $$ ### Step-by-Step Solution * Let $a = 475$ and $b = 125$. * We need to find the value of $a^2 + b^2$. * Using the derived formula, substitute the values: $$ (475)^2 + (125)^2 = \frac{1}{2} [ (475 + 125)^2 + (475 - 125)^2 ] $$ * Calculate the sum and difference inside the brackets: $$ 475 + 125 = 600 $$ $$ 475 - 125 = 350 $$ * Substitute these back into the equation: $$ = \frac{1}{2} [ (600)^2 + (350)^2 ] $$ * Square the simpler numbers: $$ = \frac{1}{2} [ 360000 + 122500 ] $$ * Add the inner values and divide by 2: $$ = \frac{1}{2} 	imes 482500 = 241250 $$ ### Exam Strategy & Shortcut Whenever you see a pattern of $a^2 + b^2$ where $a$ and $b$ sum to a clean round number (like $475 + 125 = 600$), this specific identity is the fastest path. It replaces nasty 3-digit multiplications with much simpler mental math. ### Common Pitfall The most common mistake is correctly calculating the sum of the squares inside the bracket (482500) but forgetting to multiply by the $\frac{1}{2}$ at the very end. Test setters always include this halfway value as an option. ### Final Answer Therefore, the correct answer is **241250**.

Question 18

Evaluate $796 	imes 796 - 204 	imes 204$

Accepted Answer

### Concept & Formula The problem is formatted as the difference between two squares, $a^2 - b^2$. This is a classic algebraic structure that can be factored to eliminate the need for any actual squaring: $$ a^2 - b^2 = (a + b)(a - b) $$ ### Step-by-Step Solution * Let $a = 796$ and $b = 204$. * The expression given is $(796)^2 - (204)^2$. * Substitute the values into the difference of squares identity: $$ (796)^2 - (204)^2 = (796 + 204)(796 - 204) $$ * Calculate the sum in the first bracket: $$ 796 + 204 = 1000 $$ * Calculate the difference in the second bracket: $$ 796 - 204 = 592 $$ * Multiply the two results: $$ 1000 	imes 592 = 592000 $$ ### Exam Strategy & Shortcut Never manually square large numbers when a minus sign separates them. The difference of squares is one of the most important time-saving identities in quantitative aptitude. Notice how $(796 + 204)$ cleanly adds up to $1000$—this is a deliberate design by the examiner to reward students who spot the pattern. ### Common Pitfall A common pitfall is ignoring the pattern entirely and attempting the brute-force calculation. This wastes precious minutes on a timed exam and vastly increases the likelihood of a simple arithmetic mistake. ### Final Answer Therefore, the correct answer is **592000**.

Question 19

Simplify $387 	imes 387 + 113 	imes 113 + 2 	imes 387 	imes 113$

Accepted Answer

### Concept & Formula The expression follows the algebraic identity for the square of a binomial sum. We can avoid massive calculations by substituting the numbers into the pattern: $$ (a + b)^2 = a^2 + b^2 + 2ab $$ ### Step-by-Step Solution * Let $a = 387$ and $b = 113$. * The given expression is $(387)^2 + (113)^2 + 2 	imes 387 	imes 113$, which matches the exact form of $a^2 + b^2 + 2ab$. * According to the algebraic formula, this expression is equal to $(a + b)^2$. * Substitute the values of $a$ and $b$ back into the simplified formula: $$ (387 + 113)^2 $$ * Add the numbers inside the parenthesis: $$ (500)^2 $$ * Calculate the final square: $$ 250000 $$ ### Exam Strategy & Shortcut Never multiply out 3-digit numbers in this format. The presence of two squared terms and a $2ab$ term is a giant billboard indicating an algebraic identity. Furthermore, note that $387 + 113$ sums perfectly to $500$, a clean, round number. Examiners design these questions specifically to reward pattern recognition over brute-force arithmetic. ### Common Pitfall A common mistake is failing to recognize the pattern and attempting to calculate $387 	imes 387$ manually. This wastes valuable exam time and heavily increases the chance of a simple multiplication error costing you the mark. ### Final Answer Therefore, the correct answer is **250000**.

Question 20

Simplify $87 	imes 87 + 61 	imes 61 - 2 	imes 87 	imes 61$

Accepted Answer

### Concept & Formula The expression perfectly matches the algebraic identity for the square of a difference. By identifying this pattern, we reduce the complex arithmetic to a simple subtraction and square: $$ (a - b)^2 = a^2 + b^2 - 2ab $$ ### Step-by-Step Solution * Let $a = 87$ and $b = 61$. * The expression is structured as $(87)^2 + (61)^2 - 2 	imes 87 	imes 61$, corresponding directly to $a^2 + b^2 - 2ab$. * Substitute this expanded form with the compact identity $(a - b)^2$. * Calculate the difference inside the parenthesis: $$ (87 - 61)^2 = (26)^2 $$ * Expand the square of 26 (using $(20 + 6)^2 = 400 + 36 + 240$ if you do not have it memorized): $$ 676 $$ ### Exam Strategy & Shortcut Spotting the minus sign before the $2ab$ term tells you immediately the formula is $(a - b)^2$. A quick shortcut after finding $(87 - 61) = 26$ is to look at the unit digit. The square of a number ending in 6 must also end in 6 because $6 	imes 6 = 36$. You can often eliminate multiple multiple-choice options just by checking this final unit digit. ### Common Pitfall Students sometimes confuse the identity formulas and add the numbers $(87 + 61)$ instead of subtracting them, missing the crucial minus sign in front of the $2 	imes 87 	imes 61$ term. Always double-check the signs in the formula before doing the arithmetic. ### Final Answer Therefore, the correct answer is **676**.

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