Question 1

$8 + 88 + 888 + 8888 + 88888 + 888888 = x$

Accepted Answer

Concept & Strategy The problem asks for the sum of a series where the digit $8$ repeats and grows in length. The most efficient way to solve this is by factoring out the repeating digit or by vertically aligning the numbers and summing column by column from right to left. Step-by-Step Solution * Write the numbers vertically aligned by their unit digits: $$8$$ $$88$$ $$888$$ $$8888$$ $$88888$$ $$888888$$ * Sum the units column: There are six $8$s. $6 	imes 8 = 48$. Write down $8$, carry over $4$. * Sum the tens column: There are five $8$s. $5 	imes 8 = 40$. Add the carry ($40 + 4 = 44$). Write down $4$, carry over $4$. * Sum the hundreds column: There are four $8$s. $4 	imes 8 = 32$. Add the carry ($32 + 4 = 36$). Write down $6$, carry over $3$. * Sum the thousands column: There are three $8$s. $3 	imes 8 = 24$. Add the carry ($24 + 3 = 27$). Write down $7$, carry over $2$. * Sum the ten-thousands column: There are two $8$s. $2 	imes 8 = 16$. Add the carry ($16 + 2 = 18$). Write down $8$, carry over $1$. * Sum the lakh column: There is one $8$. Add the carry ($8 + 1 = 9$). Write down $9$. * The final sum is $987648$. Exam Strategy & Shortcut You can often solve this using just the last few digits. Calculate the units place: $8 	imes 6 = 48$ (ends in $8$). Calculate the tens place: $8 	imes 5 + 4 = 44$ (ends in $48$). Look at the options: only (a), (b), (c), and (d) end in $48$. Calculate the hundreds place: $8 	imes 4 + 4 = 36$ (ends in $648$). Only (a), (c), and (d) end in $648$. The thousands place: $8 	imes 3 + 3 = 27$ (ends in $7648$). Only (a) and (d) remain. Ten-thousands place: $8 	imes 2 + 2 = 18$ (ends in $87648$). This uniquely identifies option (d). Common Pitfall Attempting to add the numbers horizontally without proper alignment frequently leads to place value errors. Always stack them or use the digit counting method. Final Answer **Therefore, the correct answer is 987648.**

Question 2

From the sum of 17 and $-12$, subtract 48.

Accepted Answer

Concept & Logic The question tests basic arithmetic operations involving positive and negative integers. The keyword "sum" indicates addition, and "subtract" means to take away the specified value from that initial result. Step-by-Step Solution * Given: Sum of $17$ and $-12$. * Calculate the sum: $$17 + (-12) = 17 - 12 = 5$$ * Now, subtract $48$ from this sum. $$5 - 48 = -43$$ Exam Strategy & Shortcut Translate the English statement directly into a single algebraic equation: $(17 - 12) - 48$. Perform the operations left to right. Since you are subtracting a large positive number from a small positive number, the result must be negative. $48 - 5 = 43$, so the answer is $-43$. Common Pitfall A common mistake is subtracting the sum from $48$ instead of subtracting $48$ from the sum, leading to $48 - 5 = 43$, which isn't the requested operation. Final Answer **Therefore, the correct answer is -43.**

Question 3

$60840 \div 234 = x$

Accepted Answer

Concept & Strategy This is a direct division problem involving large numbers. The goal is to simplify the division by estimating or using the unit digit to verify the quotient quickly. Step-by-Step Solution * We need to find the value of $x$ in the equation: $$60840 \div 234 = x$$ * Let's perform long division. Divide the first three digits ($608$) by $234$. * $234 	imes 2 = 468$. * Subtract $468$ from $608$: $608 - 468 = 140$. * Bring down the next digit ($4$) to make $1404$. * Divide $1404$ by $234$. Since $4 	imes 6 = 24$, let's test $6$. * $234 	imes 6 = 1404$. * The remainder is $0$. Bring down the final $0$, making the quotient $260$. Exam Strategy & Shortcut Use option elimination. The dividend $60840$ ends in $0$. Since $234$ does not end in $5$ or $0$, the quotient must end in $0$ or $5$ to produce a $0$ in the dividend. $234 	imes 200 = 46800$ and $234 	imes 300 = 70200$. The answer must be between $200$ and $300$, but closer to $250$. Looking at the options ending in $0$ or $5$: $255$ and $260$. Check the unit digit of $234 	imes 26 = 6084$. Add the zero, giving $260$. Common Pitfall Getting bogged down in massive long division without using the options as a guide. Estimating first prevents calculation errors from going unnoticed. Final Answer **Therefore, the correct answer is 260.**

Question 4

$3578 + 5729 - x 	imes 581 = 5821$

Accepted Answer

Concept & Formula This question requires basic algebraic manipulation and the application of the BODMAS/PEMDAS rule. You need to isolate the unknown variable by moving known quantities to one side of the equation. Step-by-Step Solution * The given equation is: $$3578 + 5729 - x 	imes 581 = 5821$$ * First, calculate the sum on the left side: $$3578 + 5729 = 9307$$ * Substitute this back into the equation: $$9307 - 581x = 5821$$ * Rearrange the equation to isolate the term with $x$: $$9307 - 5821 = 581x$$ * Perform the subtraction: $$3486 = 581x$$ * Solve for $x$: $$x = \frac{3486}{581}$$ $$x = 6$$ Exam Strategy & Shortcut Use the unit digit method to skip the final division. The subtraction $9307 - 5821$ ends in $6$ ($7 - 1 = 6$). So, $581 	imes x$ must end in $6$. Since $581$ ends in $1$, the unit digit of $x$ must be $6$ because $1 	imes 6 = 6$. Looking at the options, $6$ is the only choice that fits perfectly. Common Pitfall A frequent error is misapplying the order of operations, such as trying to subtract $581$ from the sum before multiplying by $x$, which violates BODMAS rules. Final Answer **Therefore, the correct answer is 6.**

Question 5

$-95 \div 19 = x$

Accepted Answer

Concept & Formula This problem covers the division of a negative integer by a positive integer. When dividing numbers with opposite signs, the resulting quotient is always negative. Step-by-Step Solution * The given expression is: $$-95 \div 19$$ * First, ignore the sign and perform the standard division: $$95 \div 19 = 5$$ * Apply the sign rule for division. A negative divided by a positive yields a negative result: $$- \div + = -$$ * Attach the negative sign to the quotient: $$-5$$ Exam Strategy & Shortcut Recognize immediately that a negative divided by a positive is negative, which eliminates options (c) and (d). Since $19 	imes 5 = 95$ is a standard multiplication table fact, the answer instantly resolves to $-5$. Common Pitfall Overlooking the negative sign in the dividend is the most common error, leading students to incorrectly choose the positive counterpart (option d). Final Answer **Therefore, the correct answer is -5.**

Question 6

$12345679 	imes 72$ is equal to

Accepted Answer

Concept & Strategy This question relies on recognizing a special numerical pattern. The number $12345679$ (missing the digit 8) multiplied by multiples of $9$ yields a repeating digit sequence. Step-by-Step Solution * The expression is: $$12345679 	imes 72$$ * Recognize that $72$ is a multiple of $9$. We can split $72$ into $9 	imes 8$: $$12345679 	imes (9 	imes 8)$$ * Multiply $12345679$ by $9$ first. This is a known mathematical property: $$12345679 	imes 9 = 111111111$$ * Now multiply this intermediate result by the remaining factor, $8$: $$111111111 	imes 8 = 888888888$$ * Notice that the final result has nine 8s. Exam Strategy & Shortcut If you don't remember the pattern $12345679 	imes 9 = 111111111$, you can use the unit digit and digit sum. $9 	imes 2 = 18$, so it ends in $8$. Option (b) has nine 8s. Let's count digits: an 8-digit number times a 2-digit number will generally be a 9-digit or 10-digit number. $1.2 	imes 10^7 	imes 7.2 	imes 10^1 \approx 8.6 	imes 10^8$, which perfectly aligns with the 9-digit number $888888888$. Option (a) only has eight digits. Common Pitfall The most common mistake is choosing option (a) because students miscount the number of repeating digits. Always verify the order of magnitude or count the length of the expected answer. Final Answer **Therefore, the correct answer is 888888888.**

Question 7

$8899 - 6644 - 3322 = x - 1122$

Accepted Answer

Concept & Logic This is a linear algebraic equation involving subtraction. The goal is to isolate the unknown variable $x$ on one side by simplifying the constants on the other side. Step-by-Step Solution * The given equation is: $$8899 - 6644 - 3322 = x - 1122$$ * First, simplify the left-hand side (LHS) by subtracting $6644$ from $8899$: $$8899 - 6644 = 2255$$ * Substitute this back into the LHS and perform the next subtraction: $$2255 - 3322 = -1067$$ * Now substitute the simplified LHS back into the full equation: $$-1067 = x - 1122$$ * Isolate $x$ by adding $1122$ to both sides: $$x = 1122 - 1067$$ $$x = 55$$ Exam Strategy & Shortcut You can solve this purely by looking at the last two digits of each number since the options are all two-digit numbers ending in 5. LHS last two digits: $99 - 44 - 22 = 33$. Equation becomes: $33 = x - 22$ (looking only at the tens and units places). So, $x = 33 + 22 = 55$. This matches option (a) instantly. Common Pitfall A common error is mishandling the negative signs, particularly calculating $2255 - 3322$ as positive $1067$ instead of negative, which would lead to an incorrect larger number when adding $1122$. Final Answer **Therefore, the correct answer is 55.**

Question 8

$74844 \div x = 54 	imes 63$

Accepted Answer

Concept & Formula This problem tests simplification and the relationship between multiplication and division. You need to simplify the right side of the equation and then isolate the divisor $x$. Step-by-Step Solution * The equation is: $$74844 \div x = 54 	imes 63$$ * First, compute the product on the right-hand side (RHS): $$54 	imes 63 = 3402$$ * Substitute this back into the equation: $$\frac{74844}{x} = 3402$$ * Rearrange the formula to solve for $x$: $$x = \frac{74844}{3402}$$ * Perform the final division. Notice that $3402 	imes 20 = 68040$, so the answer is slightly more than $20$. * Let's test $22$: $$3402 	imes 22 = 3402 	imes (20 + 2) = 68040 + 6804 = 74844$$ * Thus, $x = 22$. Exam Strategy & Shortcut Use the unit digit concept. Let's look at the equation $\frac{74844}{x} = 54 	imes 63$. The unit digit of the RHS ($54 	imes 63$) is $4 	imes 3 = 12$, so it ends in $2$. This means $74844 = x 	imes (	ext{a number ending in } 2)$. The unit digit of the LHS is $4$. For $x 	imes 2$ to end in $4$, the unit digit of $x$ must be either $2$ or $7$. Looking at the options, only $22$ and $42$ end in $2$. By quickly estimating $\frac{74000}{3400} \approx 22$, we can confirm the answer is $22$ without doing full multiplication. Common Pitfall Students often waste time trying to divide $74844$ by $54$ and then by $63$ sequentially, which can be prone to calculation errors, rather than evaluating the RHS as a single block first. Final Answer **Therefore, the correct answer is 22.**

Question 9

$1256 	imes 3892 = x$

Accepted Answer

Concept & Strategy This is a straight arithmetic question requiring the multiplication of two four-digit numbers. Breaking down one of the numbers into simpler components (distributive property) is the most reliable manual method. Step-by-Step Solution * We need to calculate: $$1256 	imes 3892$$ * Let's break down $3892$ into $(4000 - 100 - 8)$ for easier multiplication. $$1256 	imes (4000 - 100 - 8)$$ * Multiply $1256$ by each component: $$1256 	imes 4000 = 5024000$$ $$1256 	imes 100 = 125600$$ $$1256 	imes 8 = 10048$$ * Now combine them according to the signs: $$5024000 - 125600 - 10048$$ * Subtract $125600$ from $5024000$: $$5024000 - 125600 = 4898400$$ * Finally, subtract $10048$: $$4898400 - 10048 = 4888352$$ Exam Strategy & Shortcut Let's use the last two digits to eliminate options. Multiply the last two digits of both numbers: $56 	imes 92$. $56 	imes (100 - 8) = 5600 - 448 = 5152$. The product must end in $52$. Options (a), (c), and (e) are left. Now use the digital root (sum of digits until a single digit is reached): Digital root of $1256 = 1+2+5+6 = 14 ightarrow 1+4 = 5$. Digital root of $3892 = 3+8+9+2 = 22 ightarrow 2+2 = 4$. Product of digital roots = $5 	imes 4 = 20 ightarrow 2+0 = 2$. Check digital roots of remaining options: (a) $4883852 ightarrow 4+8+8+3+8+5+2 = 38 ightarrow 11 ightarrow 2$. (Possible) (c) $4888352 ightarrow 4+8+8+8+3+5+2 = 38 ightarrow 11 ightarrow 2$. (Possible) We need to evaluate deeper. If we approximate: $1250 	imes 3900 = 4875000$. Our exact number should be slightly higher. $4888352$ is a better fit than $4883852$. Common Pitfall Doing a standard 4-by-4 long multiplication is extremely prone to addition errors in the middle rows. Breaking it into manageable chunks reduces this risk. Final Answer **Therefore, the correct answer is 4888352.**

Question 10

What is $786$ times $964$?

Accepted Answer

Concept & Strategy This requires calculating the product of two 3-digit numbers. The distributive property helps simplify the math by rounding one number to the nearest thousand. Step-by-Step Solution * The problem asks for the product: $$786 	imes 964$$ * Express $964$ in terms of a base of $1000$ to simplify the multiplication: $$964 = 1000 - 36$$ * Substitute this into the expression: $$786 	imes (1000 - 36)$$ * Distribute $786$: $$(786 	imes 1000) - (786 	imes 36)$$ $$786000 - (786 	imes 36)$$ * Now, calculate $786 	imes 36$. You can break this down further as $786 	imes (30 + 6)$: $$786 	imes 30 = 23580$$ $$786 	imes 6 = 4716$$ Sum = $23580 + 4716 = 28296$ * Finally, subtract this from $786000$: $$786000 - 28296 = 757704$$ Exam Strategy & Shortcut Use the last two digits technique. We need to multiply $86 	imes 64$. $86 	imes 64 = 86 	imes (60 + 4) = 5160 + 344 = 5504$. The last two digits of the answer must be $04$. Looking at the given choices: (a) ends in $04$ (b) ends in $64$ (c) ends in $76$ (d) ends in $44$ Option (a) is the only specific numerical choice that matches. Common Pitfall Attempting rote column multiplication without checking the unit and tens digits first costs valuable time. Always scan the options to see if a partial calculation can reveal the answer. Final Answer **Therefore, the correct answer is 757704.**

Question 11

What is $348$ times $265$?

Accepted Answer

Concept & Strategy This is a direct three-digit multiplication problem. To solve it efficiently without a calculator, you can split one of the numbers into hundreds, tens, and units (distributive property) to make the calculations easier to manage. Step-by-Step Solution * We need to find the product: $$348 	imes 265$$ * Expand $265$ into $(200 + 60 + 5)$: $$348 	imes (200 + 60 + 5)$$ * Multiply $348$ by each part: $$348 	imes 200 = 69600$$ $$348 	imes 60 = 20880$$ $$348 	imes 5 = 1740$$ * Sum the partial products: $$69600 + 20880 + 1740 = 92220$$ Exam Strategy & Shortcut Use the unit digit and tens digit to eliminate options rapidly. First, the unit digit is $8 	imes 5 = 40$, which ends in $0$. All options end in $0$. Next, find the last two digits by multiplying the last two digits of the numbers: $48 	imes 65$. $48 	imes 65 = 48 	imes (60 + 5) = 2880 + 240 = 3120$. The product must end in $20$. Looking at the given options, only $92220$ ends in $20$. Common Pitfall Attempting standard column multiplication for large numbers can easily lead to carrying errors. Splitting into partial products or checking the last two digits is much safer and faster. Final Answer **Therefore, the correct answer is 92220.**

Question 12

$(71 	imes 29 + 27 	imes 15 + 8 	imes 4)$ equals

Accepted Answer

Concept & Formula This problem requires the application of the BODMAS rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction). The multiplications inside the bracket must be resolved first before summing the final values. Step-by-Step Solution * The given expression is: $$(71 	imes 29) + (27 	imes 15) + (8 	imes 4)$$ * Calculate the first product. Use $29 = 30 - 1$: $$71 	imes (30 - 1) = 2130 - 71 = 2059$$ * Calculate the second product. Use $15 = 10 + 5$: $$27 	imes (10 + 5) = 270 + 135 = 405$$ * Calculate the third product: $$8 	imes 4 = 32$$ * Sum all the computed values: $$2059 + 405 + 32 = 2496$$ Exam Strategy & Shortcut The unit digit concept is the fastest way to crack this. Calculate only the last digit of each multiplication component: $1 	imes 9 = 9$ $7 	imes 5 = 35 ightarrow 5$ $8 	imes 4 = 32 ightarrow 2$ Sum of unit digits = $9 + 5 + 2 = 16$. The final answer must end with a $6$. Looking at the options, only $2496$ ends in $6$. Common Pitfall A common error is getting bogged down in traditional multiplication for $71 	imes 29$ and making arithmetic mistakes. Using the nearest multiple of ten (like $30-1$) simplifies the math drastically. Final Answer **Therefore, the correct answer is 2496.**

Question 13

$x 	imes (|a| 	imes |b|) = -ab$

Accepted Answer

Concept & Logic This algebraic question deals with the properties of absolute values. The absolute value of a product is equal to the product of their absolute values: $|a| 	imes |b| = |ab|$. Step-by-Step Solution * Simplify the given equation: $$x 	imes (|a| 	imes |b|) = -ab$$ $$x 	imes |ab| = -ab$$ * Isolate $x$: $$x = \frac{-ab}{|ab|}$$ * The value of $x$ depends entirely on the sign of the product $ab$: * Case 1: If $a$ and $b$ have the SAME sign (both positive or both negative), then $ab > 0$. Therefore, $|ab| = ab$. $$x = \frac{-ab}{ab} = -1$$ * Case 2: If $a$ and $b$ have OPPOSITE signs (one positive, one negative), then $ab < 0$. Therefore, $|ab| = -ab$. $$x = \frac{-ab}{-ab} = 1$$ * Since the value of $x$ is not a single constant and changes based on the signs of $a$ and $b$, neither $1$ nor $-1$ is universally correct. Exam Strategy & Shortcut Whenever dealing with absolute values equal to variables without specified domains (e.g., $a,b > 0$), expect multiple cases. Since $x$ can evaluate to either $1$ or $-1$ depending on the integers chosen, a single definitive numeric option is impossible. Instantly choose "None of these". Common Pitfall Assuming that variables $a$ and $b$ are strictly positive integers. If you make this assumption, you will erroneously conclude that $x = -1$ and select option (b). Final Answer **Therefore, the correct answer is None of these.**

Question 14

$(46)^2 - (x)^2 = 4398 - 3066$

Accepted Answer

Concept & Formula This problem involves solving a quadratic equation for an unknown variable $x$. You must evaluate the squares and simple arithmetic on both sides to isolate $x^2$. Step-by-Step Solution * The equation is: $$(46)^2 - x^2 = 4398 - 3066$$ * First, simplify the right-hand side (RHS) of the equation: $$4398 - 3066 = 1332$$ * Next, calculate the square of $46$. You can use the formula $(50 - 4)^2 = 2500 - 400 + 16 = 2116$: $$2116 - x^2 = 1332$$ * Rearrange the equation to isolate $x^2$: $$x^2 = 2116 - 1332$$ $$x^2 = 784$$ * Finally, take the square root of both sides to find $x$: $$x = \sqrt{784}$$ $$x = 28$$ Exam Strategy & Shortcut Use the unit digit trick to save time finding the square root. The RHS subtraction $(4398 - 3066)$ ends in $2$. The square of $46$ ends in $6$ (since $6 	imes 6 = 36$). So, the equation in terms of unit digits is: $6 - (	ext{unit digit of } x^2) = 2$. This implies the unit digit of $x^2$ must be $4$. A square ending in $4$ means the original number $x$ must end in $2$ or $8$. Looking at the options, we have $28$ and $42$. Since $40^2 = 1600$, which is already greater than $784$, $42$ is too large. Thus, the answer must be $28$. Common Pitfall A common error is manually multiplying $46 	imes 46$ and making an addition mistake, then getting stuck on an imperfect square subtraction. Final Answer **Therefore, the correct answer is 28.**

Question 15

$(800 \div 64) 	imes (1296 \div 36) = x$

Accepted Answer

Concept & Strategy This question tests the simplification of bracketed division operations followed by multiplication (BODMAS). Recognizing perfect squares makes the second bracket much faster to solve. Step-by-Step Solution * The given expression is: $$(800 \div 64) 	imes (1296 \div 36) = x$$ * Simplify the first bracket by writing it as a fraction and reducing: $$\frac{800}{64} = \frac{100 	imes 8}{8 	imes 8} = \frac{100}{8} = 12.5$$ * Simplify the second bracket. Notice that $1296$ is the square of $36$ ($36 	imes 36 = 1296$): $$\frac{1296}{36} = 36$$ * Multiply the simplified terms together: $$x = 12.5 	imes 36$$ * To calculate this easily, treat $12.5$ as $\frac{100}{8}$: $$x = \frac{100}{8} 	imes 36$$ $$x = 100 	imes \frac{36}{8}$$ $$x = 100 	imes 4.5$$ $$x = 450$$ * Review the given options: (a) $420$, (b) $460$, (c) $500$, (d) $540$. The calculated answer $450$ is not present. Exam Strategy & Shortcut Memorizing squares up to $40$ is crucial for banking exams; knowing $36^2 = 1296$ instantly turns the second bracket into $36$. For the first bracket, $800 \div 64 ightarrow \frac{100}{8}$. Multiplying $\frac{100}{8}$ by $36$ gives $\frac{3600}{8}$, which easily divides into $450$. Checking the options immediately confirms that "None of these" is required. Common Pitfall Converting $\frac{100}{8}$ into the decimal $12.5$ and then performing long decimal multiplication by $36$ is slower and increases the chance of placing the decimal point in the wrong position. Always keep it as a fraction when multiplying by another whole number. Final Answer **Therefore, the correct answer is None of these.**

Question 16

$5358 	imes 51 = x$

Accepted Answer

### Concept & Strategy To multiply a large number by a number close to a multiple of 10 (like 51), we can use the distributive property of multiplication over addition. $$a 	imes (b + c) = (a 	imes b) + (a 	imes c)$$ ### Step-by-Step Solution * Break down 51 into simpler additive components: $$51 = 50 + 1$$ * Apply the distributive property: $$5358 	imes 51 = 5358 	imes (50 + 1)$$ * Multiply 5358 by 50 and by 1: $$5358 	imes 50 = 5358 	imes \frac{100}{2} = 267900$$ $$5358 	imes 1 = 5358$$ * Add the results together: $$267900 + 5358 = 273258$$ ### Exam Strategy & Shortcut Check the last two digits using the unit digit method for faster elimination. The last two digits of the product will be determined by the last two digits of the multiplicands. $$58 	imes 51 ightarrow 58 	imes (50 + 1) = 2900 + 58 = 2958$$ The last two digits must be 58. We can eliminate options (b) and (c) immediately. Since $53 	imes 5 = 265$, the answer must be around 270,000, confirming 273258. ### Common Pitfall Students often attempt direct vertical multiplication, which consumes excessive time and increases the risk of carrying errors during addition. Always look for base-10 approximations. ### Final Answer **Therefore, the correct answer is 273258.**

Question 17

$587 	imes 999 = x$

Accepted Answer

### Concept & Formula When multiplying by 99, 999, 9999, etc., use the distributive property over subtraction. Express the multiplier as a power of 10 minus 1. $$a 	imes (10^n - 1) = a 	imes 10^n - a$$ ### Step-by-Step Solution * Express 999 as $1000 - 1$: $$587 	imes 999 = 587 	imes (1000 - 1)$$ * Distribute the 587: $$(587 	imes 1000) - (587 	imes 1)$$ $$587000 - 587$$ * Perform the subtraction: $$587000 - 587 = 586413$$ ### Exam Strategy & Shortcut When multiplying any number by a sequence of 9s (where the number of digits matches or is less), subtract 1 from the number to get the first part of the answer, and then find the 9's complement for each digit of that result to get the second part. 1. $587 - 1 = 586$ (First part) 2. $9 - 5 = 4$, $9 - 8 = 1$, $9 - 6 = 3$ (Second part: 413) Combine them: 586413. ### Common Pitfall A common mistake is misaligning the place values during subtraction (e.g., subtracting 587 from 58700 instead of 587000), leading to off-by-one place value errors. ### Final Answer **Therefore, the correct answer is 586413.**

Question 18

$3897 	imes 999 = x$

Accepted Answer

### Concept & Formula Similar to previous problems, calculating products involving 999 is highly efficient when utilizing the distributive property over subtraction. $$a 	imes (10^n - 1) = a 	imes 10^n - a$$ ### Step-by-Step Solution * Express 999 as a base of 10: $$999 = 1000 - 1$$ * Substitute into the expression: $$3897 	imes 999 = 3897 	imes (1000 - 1)$$ * Multiply and subtract: $$3897000 - 3897$$ * Complete the subtraction: $$3897000 - 3897 = 3893103$$ ### Exam Strategy & Shortcut Use the subtraction shortcut quickly: $3897000 - 3897$. To do this mentally, subtract 4000 (which is easier) and then add back 103 (since $4000 - 3897 = 103$). $$3897000 - 4000 = 3893000$$ $$3893000 + 103 = 3893103$$ ### Common Pitfall Trying to use the "subtract 1 and find the 9's complement" trick without adjusting for the fact that 3897 has 4 digits while 999 has only 3 digits. That specific trick only works when the number of 9s is greater than or equal to the digits in the other number. The zero-appending subtraction method is foolproof. ### Final Answer **Therefore, the correct answer is 3893103.**

Question 19

$72519 	imes 9999 = x$

Accepted Answer

### Concept & Formula This involves multiplying a 5-digit number by 9999. We continue applying the algebraic identity for distributive subtraction to minimize calculation time. $$x 	imes (y - z) = x 	imes y - x 	imes z$$ ### Step-by-Step Solution * Rewrite 9999 to its closest base-10 equivalent: $$9999 = 10000 - 1$$ * Apply to the problem: $$72519 	imes (10000 - 1)$$ * Distribute 72519: $$725190000 - 72519$$ * Perform the final subtraction: $$725190000 - 72519 = 725117481$$ ### Exam Strategy & Shortcut Since 72519 has 5 digits and 9999 has 4 digits, we append four zeros and subtract the original number. To subtract quickly, look at the last four digits of 725190000, which is 0000. Subtracting 2519 from 10000 yields 7481. The options end in 7481, 6481, etc. Only options (a) and (b) end in 17481. Check the leading digits: $72519 - 7 = 72512$, but with the borrow it becomes $72511$. Thus, the answer starts with 72511. ### Common Pitfall Forgetting to account for the "borrow" across multiple zeros when subtracting, causing an error in the hundreds or thousands place. ### Final Answer **Therefore, the correct answer is 725117481.**

Question 20

$2056 	imes 987 = x$

Accepted Answer

### Concept & Formula When a multiplier is close to a thousand (like 987), we can express it as 1000 minus a small integer to use the distributive property, significantly simplifying the math. $$a 	imes (b - c) = ab - ac$$ ### Step-by-Step Solution * Express 987 as $1000 - 13$: $$2056 	imes 987 = 2056 	imes (1000 - 13)$$ * Distribute the multiplication: $$(2056 	imes 1000) - (2056 	imes 13)$$ * Calculate the two parts: $$2056000 - 26728$$ * Perform the final subtraction: $$2056000 - 26728 = 2029272$$ ### Exam Strategy & Shortcut Use the last two digits to eliminate options! The last two digits of the answer will be the last two digits of $56 	imes 87$. $56 	imes 87 ightarrow 56 	imes (100 - 13) = 5600 - 728 = 4872$. The last two digits are 72. All options have 72! Expand to the last three digits: $056 	imes 987$. $056 	imes (1000 - 13) = 56000 - 728 = 55272$. The last three digits must be 272. Only options (b) and (d) end in 272. Estimate: $2000 	imes 1000 = 2000000$. The exact answer will be close to this, making 2029272 the logical choice over 1923472. ### Common Pitfall Attempting a three-by-four digit standard multiplication line by line. This takes at least triple the time and introduces a high probability of simple addition errors in the final sum. ### Final Answer **Therefore, the correct answer is 2029272.**

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