Question 1

Out of six consecutive natural numbers if the sum of first three is $27$, what is the sum of the other three?

Accepted Answer

Concept & Strategy: For consecutive natural numbers, every number in the sequence increases by exactly 1. This predictable difference allows us to compare two subsets of a sequence mathematically without necessarily finding the individual numbers. Step-by-Step Solution: • Given: We have six consecutive natural numbers. Let's call them x, x+1, x+2, x+3, x+4, x+5. • Given: The sum of the first three is 27. • Formulate the equation for the first three: x + (x+1) + (x+2) = 27 3x + 3 = 27 • Solve for x: 3x = 24 x = 8 • The first three numbers are 8, 9, 10. • The next three numbers are 11, 12, 13. • Calculate the sum of the other three: 11 + 12 + 13 = 36 Exam Strategy & Shortcut: Use relational mapping for an instant answer. Compare the first three terms with the next three terms: Term 1 vs Term 4: Difference is +3 Term 2 vs Term 5: Difference is +3 Term 3 vs Term 6: Difference is +3 Since there are 3 terms, the second set is exactly 3 × 3 = 9 greater than the first set. New Sum = Old Sum + 9 New Sum = 27 + 9 = 36. This solves the problem mentally in 3 seconds. Common Pitfall: Students often spend unnecessary time writing out the full algebraic equation (3x + 3 = 27) and solving for the exact terms, which eats up the clock. Always look for relative differences between groups when dealing with consecutive series. Final Answer: Therefore, the correct answer is 36.

Question 2

The sum of seven consecutive numbers is $175$. What is the difference between twice the largest number and thrice the smallest number?

Accepted Answer

Concept & Strategy: The sum of an odd number of consecutive terms (which form an Arithmetic Progression) yields the middle term when divided by the number of terms. This is because the average of an AP is its perfectly centered value. Middle Term = \frac{Sum of Terms}{Number of Terms} Step-by-Step Solution: • Given: The sum of 7 consecutive numbers is 175. • Calculate the middle term directly by finding the average: Middle Term = (175/7) = 25 • Since 25 is the middle term of 7 consecutive integers, there are exactly 3 terms before it and 3 terms after it. • The sequence is: 22, 23, 24, 	extbf{25}, 26, 27, 28. • Identify the smallest and largest numbers: • Smallest Number = 22 • Largest Number = 28 • The question asks for the difference between twice the largest and thrice the smallest: • Twice the largest = 2 × 28 = 56 • Thrice the smallest = 3 × 22 = 66 • Calculate the absolute difference: |56 - 66| = 10 Exam Strategy & Shortcut: Never use the standard algebraic setup x + (x+1) + ... + (x+6) = 175, which simplifies to 7x + 21 = 175. While it works, it is mathematically heavier. Identifying the middle term via division (175 / 7 = 25) allows you to immediately step back by 3 to find the smallest (22) and step forward by 3 to find the largest (28). Common Pitfall: The phrasing "difference between A and B" in aptitude exams implies the absolute difference (larger value minus smaller value) unless a specific order of subtraction is stated. Calculating 56 - 66 = -10 and assuming the answer isn't present (leading to "None of these") is a frequent trap. Final Answer: Therefore, the correct answer is 10.

Question 3

The sum of five consecutive odd numbers is $575$. What is the sum of the next set of five consecutive odd numbers?

Accepted Answer

Concept & Strategy: When shifting from one set of consecutive numbers to the very next consecutive set, the total sum increases by a predictable fixed amount based on the number of items and the interval. Step-by-Step Solution: • Given: The sum of five consecutive odd numbers is 575. Let's find the numbers to visualize it. • The average (middle term) is (575/5) = 115. • The first set of odd numbers is: 111, 113, 	extbf{115}, 117, 119. • The "next set of five consecutive odd numbers" starts exactly after 119. • The second set is: 121, 123, 125, 127, 129. • Calculate the sum of this new set. The middle term is 125. Sum = 125 × 5 = 625 • Check the options: (a) 595, (b) 615, (c) 635. The value 625 is not listed. Exam Strategy & Shortcut: You don't need to find the actual numbers. Compare the two sets term by term: Set 1: A, B, C, D, E Set 2: F, G, H, I, J (where F is the odd number immediately following E) Since consecutive odd numbers have a gap of 2, there are 5 steps (a gap of 10) between any term in the first set and the corresponding term in the second set (e.g., F = A + 10, G = B + 10). Because there are 5 terms, the total sum increases by: 5 × 10 = 50. New Sum = 575 + 50 = 625 Since 625 is absent, immediately mark "None of these". Common Pitfall: A very common mistake is confusing "odd numbers" with regular "natural numbers." If they were natural numbers, the shift for 5 items would be +25, resulting in 600. Because they are odd numbers, the step size is 2, doubling the aggregate shift to +50. Final Answer: Therefore, the correct answer is None of these.

Question 4

The sum of three consecutive odd numbers and three consecutive even numbers together is $231$. Also, the smallest odd number is $11$ less than the smallest even number. What is the sum of the largest odd number and the largest even number?

Accepted Answer

Concept & Formula: This problem requires forming a system of linear equations to manage two independent sequences (one odd, one even) that are linked by a specific condition regarding their starting values. Odd Sequence: o, o+2, o+4 Even Sequence: e, e+2, e+4 Step-by-Step Solution: • Given: The total sum of the three odds and three evens is 231. (o + o+2 + o+4) + (e + e+2 + e+4) = 231 3o + 6 + 3e + 6 = 231 3o + 3e + 12 = 231 3(o + e) = 219 • Divide by 3 to simplify: o + e = 73 • Given: The smallest odd number (o) is 11 less than the smallest even number (e). o = e - 11 • Substitute this condition into our simplified sum equation: (e - 11) + e = 73 2e = 84 e = 42 • Find the smallest odd number (o): o = 42 - 11 = 31 • Find the largest odd number and largest even number: • Largest odd = o + 4 = 31 + 4 = 35 • Largest even = e + 4 = 42 + 4 = 46 • Calculate their sum: 35 + 46 = 81 • Comparing 81 to the options, it is not present. Exam Strategy & Shortcut: Instead of expanding o, o+2, o+4 separately, note that the sum of the largest terms is directly related to the sum of the smallest terms. We know o + e = 73. The largest odd is o + 4. The largest even is e + 4. Their sum is (o + 4) + (e + 4) = o + e + 8. Substitute o + e = 73: Sum = 73 + 8 = 81. This bypasses solving for the individual values of e and o entirely, drastically cutting down computation time. Common Pitfall: Students often get bogged down solving for the exact value of every single integer in both sequences. As shown in the shortcut, you can often find the sum of the target numbers by manipulating the combined variable (o+e) without ever breaking it apart. Final Answer: Therefore, the correct answer is None of these.

Question 5

The sum of three consecutive multiples of $3$ is $72$. What is the largest number?

Accepted Answer

Concept & Strategy: Any set of consecutive multiples (like multiples of 3) forms an Arithmetic Progression (AP). The most powerful property of an AP with an odd number of terms is that its average strictly equals its middle term. Middle Term = \frac{Sum}{Number of Terms} Step-by-Step Solution: • Given: The sum of 3 consecutive multiples of 3 is 72. • Calculate the average to find the middle term directly: Middle Term = (72/3) = 24 • Since the numbers are consecutive multiples of 3, they are spaced by 3. • The middle number is 24. • The sequence is: (24 - 3), 	extbf{24}, (24 + 3). • The sequence is: 21, 24, 27. • The question asks for the largest number, which is 27. Exam Strategy & Shortcut: Do not use the traditional algebra method of setting up x + (x+3) + (x+6) = 72. While correct, it forces you to solve 3x + 9 = 72 \Rightarrow 3x = 63 \Rightarrow x = 21, and then calculate 21 + 6 = 27. Using the average property (72 / 3 = 24, then 24 + 3 = 27) requires zero scratch paper and solves the problem in under 5 seconds mentally. Common Pitfall: A standard trap is identifying the first number (21) or the middle number (24) and accidentally choosing it from the options. Always box or underline the specific condition requested ("largest number") before glancing at the choices. Final Answer: Therefore, the correct answer is 27.

Question 6

What is the sum of two consecutive even numbers, the difference of whose squares is $84$?

Accepted Answer

Concept & Formula: This problem elegantly unravels if you apply the fundamental algebraic identity for the difference of two squares. It directly links the sum, difference, and the squared difference of two variables. a^2 - b^2 = (a - b)(a + b) Step-by-Step Solution: • Given: We have two consecutive even numbers. Let the larger be a and the smaller be b. • Because they are consecutive even numbers, their difference is always 2. a - b = 2 • Given: The difference of their squares is 84. a^2 - b^2 = 84 • Substitute these values into the difference of squares identity: (a - b)(a + b) = 84 (2)(a + b) = 84 • Divide both sides by 2 to isolate (a + b): a + b = (84/2) a + b = 42 • The value (a + b) represents the sum of the two numbers, which is exactly what the question asks for. The sum is 42. Exam Strategy & Shortcut: Recognize the pattern instantly: Sum = \frac{Difference of Squares}{Difference of Numbers}. Since consecutive even numbers always differ by 2, you simply divide the given difference of squares by 2. 84 / 2 = 42. Mark the answer and move on in 3 seconds. Common Pitfall: Many students unnecessarily calculate the individual numbers by creating a system of equations: a - b = 2 and a + b = 42, finding 2a = 44 \Rightarrow a = 22, and b = 20. Then they add 20 + 22 to get 42. This redundant step wastes valuable time since the formula already isolated the sum (a+b). Final Answer: Therefore, the correct answer is 42.

Question 7

The sum of the squares of three consecutive natural numbers is $2030$. What is the middle number?

Accepted Answer

Concept & Logic: When solving for consecutive numbers involving squares, defining the sequence symmetrically around a middle term x (e.g., x-1, x, x+1) eliminates the middle linear terms during expansion, making the quadratic equation vastly simpler to solve. Step-by-Step Solution: • Given: Three consecutive natural numbers. Let them be (x-1), x, and (x+1). Here, x represents the middle number. • Condition: The sum of their squares is 2030. (x-1)^2 + x^2 + (x+1)^2 = 2030 • Expand the squared binomials: (x^2 - 2x + 1) + x^2 + (x^2 + 2x + 1) = 2030 • Combine like terms. Notice how the -2x and +2x neatly cancel out: 3x^2 + 2 = 2030 • Subtract 2 from both sides: 3x^2 = 2028 • Divide by 3: x^2 = 676 • Take the square root to find x: x = \sqrt{676} = 26 • The middle number x is exactly 26. Exam Strategy & Shortcut: Use Approximation. The sum of the three squares is 2030. Since the numbers are consecutive, they are very close to each other in value. Therefore, 3 × (Middle Number)^2 \approx 2030. (Middle Number)^2 \approx (2030/3) \approx 676.6. Looking at the options, we need the square root of a number very close to 676. If you memorize squares up to 30, you will instantly know that 26^2 = 676. Option (b) is the clear winner without doing any algebraic expansion. Common Pitfall: Setting the numbers up as x, x+1, and x+2 creates a much harder quadratic: x^2 + (x^2 + 2x + 1) + (x^2 + 4x + 4) = 2030, leading to 3x^2 + 6x - 2025 = 0 \Rightarrow x^2 + 2x - 675 = 0. Factoring large quadratics under pressure often leads to errors. Always use symmetrical variables like (x-1, x, x+1). Final Answer: Therefore, the correct answer is 26.

Question 8

If a number of two digits is $k$ times the sum of its digits, then the number formed by interchanging the digits is the sum of the digits multiplied by

Accepted Answer

Concept & Logic: The fundamental property of two-digit numbers and their reversals. The sum of any two-digit number and the number formed by reversing its digits is always a perfect multiple of 11, specifically: Original + Reversed = 11 × (Sum of Digits) Step-by-Step Solution: Let the original two-digit number be 10x + y, where x and y are its digits. The problem states the number is k times the sum of its digits: 10x + y = k(x + y) --- (Equation 1) The number formed by interchanging the digits is 10y + x. Let this reversed number be m times the sum of its digits: 10y + x = m(x + y) --- (Equation 2) Add Equation 1 and Equation 2 together: (10x + y) + (10y + x) = k(x + y) + m(x + y) Group the terms on the left side: 11x + 11y = (k + m)(x + y) Factor out 11 on the left: 11(x + y) = (k + m)(x + y) Since (x + y) is not zero, we can divide both sides by (x + y): 11 = k + m We need to find the multiplier for the interchanged number, which is m: m = 11 - k Exam Strategy & Shortcut: Use the Assumption Method to bypass the abstract algebra entirely. Pick an easy, real two-digit number, say 24. • Sum of digits = 6. • The number 24 is 4 × 6, so k = 4. • The interchanged number is 42. • 42 = 7 × 6, so the new multiplier is 7. Now check the options using k = 4 to see which one yields 7: (a) 4 - 1 = 3 (b) 11 - 4 = 7 (Matches!) (c) 9 + 4 = 13 (d) 10 - 4 = 6 Option (b) is the only correct choice. Common Pitfall: Getting hopelessly bogged down trying to isolate individual variables x and y algebraically. This leads to a dead end. The trick relies on recognizing the structural relationship of adding the original and reversed numbers. Final Answer: Therefore, the correct answer is 11 - k.

Question 9

A two-digit number exceeds the sum of the digits of that number by 18. If the digit at the unit's place is double the digit in the ten's place, what is the number?

Accepted Answer

Concept & Formula: Algebraic representation of two-digit numbers. A two-digit number is written as 10t + u, where t is the ten's digit and u is the unit's digit. Step-by-Step Solution: Let the ten's digit be x and the unit's digit be y. The number is 10x + y. Condition 1: The number exceeds the sum of its digits by 18. 10x + y - (x + y) = 18 9x = 18 x = 2 So, the ten's digit is 2. Condition 2: The digit at the unit's place is double the digit in the ten's place. y = 2x y = 2(2) = 4 The number is 10(2) + 4 = 24. Exam Strategy & Shortcut: Option Verification is the fastest route. Check (a) 24: Unit digit (4) is double ten's digit (2). Number (24) minus sum of digits (6) = 18. Matches perfectly. Common Pitfall: Students might overcomplicate the algebra instead of recognizing that 10x + y - (x + y) simply simplifies to 9x, immediately giving the ten's digit without needing the second condition. Final Answer: Therefore, the correct answer is 24.

Question 10

The sum of the digits of a two-digit number is 15 and the difference between the digits is 3. What is the two-digit number?

Accepted Answer

Concept & Logic: Linear equations with two variables. The difference between digits implies two absolute scenarios depending on which digit is larger. Step-by-Step Solution: Let the two digits be x and y. Condition 1: Sum of digits is 15. x + y = 15 Condition 2: Difference between digits is 3. This gives two possibilities: Case A: x - y = 3 Case B: y - x = 3 Solving Case A: Add x + y = 15 and x - y = 3. 2x = 18 \Rightarrow x = 9 y = 15 - 9 = 6 If x is the ten's digit, the number is 96. Solving Case B: Add x + y = 15 and y - x = 3. 2y = 18 \Rightarrow y = 9 x = 15 - 9 = 6 If x is the ten's digit, the number is 69. Since the question does not specify which digit is larger (ten's or unit's), both 69 and 96 are valid possibilities. Thus, a unique number cannot be found. Exam Strategy & Shortcut: Look at the options immediately. Options (a) 69 and (c) 96 both satisfy the conditions (6+9=15 and |6-9|=3). Since both are present and valid, the unique answer must be "Cannot be determined". Common Pitfall: Assuming that the ten's digit must be larger than the unit's digit (or vice versa) and prematurely selecting either 69 or 96 without checking both cases. Final Answer: Therefore, the correct answer is Cannot be determined.

Question 11

A two-digit number is 7 times the sum of its two digits. The number that is formed by reversing its digits is 18 less than the original number. What is the number?

Accepted Answer

Concept & Formula: Algebraic relationships between a number, the sum of its digits, and its reversed form. When a number is reversed, the difference is always a multiple of 9: 9(x - y). Step-by-Step Solution: Let the number be 10x + y. Condition 2: The reversed number (10y + x) is 18 less than the original. (10x + y) - (10y + x) = 18 9x - 9y = 18 9(x - y) = 18 x - y = 2 Condition 1: The number is 7 times the sum of its digits. 10x + y = 7(x + y) 10x + y = 7x + 7y 3x = 6y x = 2y Substitute x = 2y into x - y = 2: 2y - y = 2 y = 2 Now find x: x = 2(2) = 4 The number is 10(4) + 2 = 42. Exam Strategy & Shortcut: Option elimination is much faster here. Test condition 2 (reversing digits is 18 less) first: (a) 42 - 24 = 18. (Matches! Check condition 1: 4+2=6, and 7 × 6 = 42. Perfect.) (b) 52 - 25 = 27 
eq 18. (c) 62 - 26 = 36 
eq 18. (d) 72 - 27 = 45 
eq 18. Common Pitfall: Spending too much time setting up simultaneous equations when quickly checking the reversal difference for each option gives the answer in under 10 seconds. Final Answer: Therefore, the correct answer is 42.

Question 12

If the digit in the unit's place of a two-digit number is halved and the digit in the ten's place is doubled, the number thus obtained is equal to the number obtained by interchanging the digits. Which of the following is definitely true?

Accepted Answer

Concept & Logic: Manipulating place values algebraically. We must carefully construct the expression for the "new" number based on the fractional and multiplicative changes to its individual digits. Step-by-Step Solution: Let the ten's digit be t and the unit's digit be u. The original number is 10t + u. Condition 1: Unit's digit is halved (u/2) and ten's digit is doubled (2t). The new number formed is: 10(2t) + (u/2) = 20t + 0.5u. Condition 2: This new number is equal to the number obtained by interchanging the original digits (10u + t). 20t + 0.5u = 10u + t Rearrange to find the relationship between t and u: 20t - t = 10u - 0.5u 19t = 9.5u Divide both sides by 9.5: 2t = u This equation directly translates to: The unit's digit (u) is twice the ten's digit (t). Exam Strategy & Shortcut: Plug in options to create a valid number. For (d), if unit is twice ten's, let's pick t=2, u=4. Original number = 24. Halve unit (2), double ten's (4). New digits: tens=4, units=2. New number = 42. Interchanged original number (24) is also 42. 42 = 42. The condition holds perfectly. Common Pitfall: Constructing the new number incorrectly. A common mistake is writing the new number as 2t + u/2 instead of properly multiplying the new ten's digit (2t) by its place value of 10. Final Answer: Therefore, the correct answer is Digit in the unit's place is twice the digit in the ten's place..

Question 13

In a two-digit number, if it is known that its unit's digit exceeds its ten's digit by 2 and that the product of the given number and the sum of its digits is equal to 144, then the number is

Accepted Answer

Concept & Logic: Solving non-linear equations derived from digit place values and their products. Utilizing multiple constraints simultaneously. Step-by-Step Solution: Let the ten's digit be x and the unit's digit be y. The original number is 10x + y. Condition 1: Unit's digit exceeds ten's digit by 2. y = x + 2 Condition 2: The product of the number and the sum of its digits is 144. (10x + y) × (x + y) = 144 Substitute y = x + 2 into the product equation: (10x + x + 2) × (x + x + 2) = 144 (11x + 2) × (2x + 2) = 144 Divide the equation by 2 to simplify: (11x + 2)(x + 1) = 72 11x^2 + 11x + 2x + 2 = 72 11x^2 + 13x - 70 = 0 Solve the quadratic (or just test integers): x = 2 is a clear solution because 11(4) + 13(2) - 70 = 44 + 26 - 70 = 0. If x = 2, then y = 2 + 2 = 4. The number is 24. Exam Strategy & Shortcut: The quadratic equation takes too long. Immediately test the options using the given conditions! (a) 24: 4 - 2 = 2 (Pass). Product: 24 × (2+4) = 24 × 6 = 144. (Pass). We have our answer instantly. (b) 26: 6 - 2 
eq 2. (Fail) (c) 42: 2 - 4 
eq 2. (Fail) Common Pitfall: Attempting to solve the full quadratic equation 11x^2 + 13x - 70 = 0 under exam pressure instead of realizing that digit problems are almost always faster to solve by testing the four given options. Final Answer: Therefore, the correct answer is 24.

Question 14

The sum of the digits of a two-digit number is 9 less than the number. Which of the following digits is at unit's place of the number?

Accepted Answer

Concept & Logic: Evaluating algebraic constraints to see if they yield a unique solution. When variables cancel out or leave multiple possibilities open, the data is insufficient. Step-by-Step Solution: Let the ten's digit be x and the unit's digit be y. The numerical value of the number is 10x + y. The sum of its digits is x + y. According to the given condition: Sum of digits = Number - 9 (x + y) = (10x + y) - 9 Simplify the equation by subtracting y from both sides: x = 10x - 9 9x = 9 x = 1 The algebra tells us definitively that the ten's digit (x) must be 1. The number is in the 10s (e.g., 10, 11, 12, ... 19). However, the unit's digit (y) completely canceled out of the equation. This means y can be ANY digit from 0 to 9, and the original condition will still hold true. Since we cannot determine a single, unique value for the unit's place, the information provided is not enough. Exam Strategy & Shortcut: Test numbers in the 10s to see if they fit. Try 12: Sum = 3. Number - 9 = 12 - 9 = 3. (Matches) Try 15: Sum = 6. Number - 9 = 15 - 9 = 6. (Matches) Since both 12 and 15 satisfy the condition, the unit digit could be 2, 5, or any other digit. Therefore, the answer is immediately "Data inadequate". Common Pitfall: Solving for x = 1 and mistakenly thinking this 1 represents the unit's digit, thereby choosing option (a). It is crucial to remember which variable represents which place value. Final Answer: Therefore, the correct answer is Data inadequate.

Question 15

The difference between a two-digit number and the number obtained by interchanging the two digits is $63$. Which is the smaller of the two numbers?

Accepted Answer

Concept & Logic: The difference between a two-digit number and its reversed counterpart is directly proportional to the difference between its individual digits. Difference = 9(x - y) Step-by-Step Solution: Given: • The difference between the two-digit number and its reversed form is 63. Calculation: • Let the original number be 10x + y and the reversed number be 10y + x. • The difference is 9(x - y) = 63. • Solving for the difference of the digits: (x - y) = 63 / 9 = 7. • We need to find pairs of digits (x, y) that have a difference of 7. • The possible pairs of digits are (8, 1) and (9, 2). • This means the original number and its reverse could be 81 and 18, or 92 and 29. • The smaller number could be 18 or 29. Because there are multiple valid possibilities, a unique smaller number cannot be isolated. Exam Strategy & Shortcut: Divide the difference by 9 to get the difference between the digits (63 / 9 = 7). Immediately recognize that multiple pairs of digits satisfy a difference of 7 (9 and 2, 8 and 1). Since the exact digits are not fixed, the exact numbers aren't either. Mark "Cannot be determined" and move on. Common Pitfall: A common mistake is finding one pair that works (like 9 and 2), generating the numbers 92 and 29, and prematurely selecting 29 from the options without checking if other pairs (like 81 and 18) also satisfy the condition. Final Answer: Therefore, the correct answer is Cannot be determined.

Question 16

The sum of the digits of a two-digit number is $1/5$ of the difference between the number and the number obtained by interchanging the positions of the digits. What is definitely the difference between the digits of that number?

Accepted Answer

Concept & Formula: This problem bridges the sum of the digits with the algebraic expression for the difference between a number and its reversed form. Sum of digits = (x + y) Difference of reversed numbers = 9(x - y) Step-by-Step Solution: Given: • (x + y) = (1/5) × Difference of the numbers Calculation: • We know the difference between a two-digit number (10x + y) and its reverse (10y + x) is 9(x - y). • Substitute this into the given condition: (x + y) = (1/5) × 9(x - y). • Multiply both sides by 5 to clear the fraction: 5(x + y) = 9(x - y). • Expand the terms: 5x + 5y = 9x - 9y. • Rearrange to group variables: 5y + 9y = 9x - 5x. • Simplify: 14y = 4x, which reduces to 7y = 2x. • This gives us a ratio: x / y = 7 / 2. • Since x and y must be single-digit integers (from 0 to 9) and x cannot be 0 for a two-digit number, the only valid integer solution is x = 7 and y = 2. • The original number is 72 and its reverse is 27. • The question asks for the difference between the digits: x - y = 7 - 2 = 5. Exam Strategy & Shortcut: Translate the word problem directly into the simplified ratio: 5(x+y) = 9(x-y). Rearranging gives 7y = 2x. The moment you see a ratio between single digits like x:y = 7:2, you know the digits must be exactly 7 and 2. The difference is instantly 5. Common Pitfall: Students might arrive at the equation 7y = 2x and incorrectly conclude that the data is inadequate because it's a single equation with two variables. However, the hidden constraint that x and y must be single-digit integers (1 through 9) forces a unique solution. Final Answer: Therefore, the correct answer is 5.

Question 17

In a two-digit positive number, the digit in the unit's place is equal to the square of the digit in ten's place, and the difference between the number and the number obtained by interchanging the digits is $54$. What is $40\%$ of the original numbe…

Accepted Answer

Concept & Logic: This problem combines non-linear digit relationships (squares) with standard linear reversal properties. Units digit y = x^2 Step-by-Step Solution: Given: • Unit's digit (y) is the square of the ten's digit (x): y = x^2. • The absolute difference between the number and its reverse is 54. Calculation: • From the reversal rule, the difference between the digits is 54 / 9 = 6. So, |x - y| = 6. • Substitute y = x^2 into the equation. We have two cases for the absolute difference: Case 1: x - x^2 = 6 \Rightarrow x^2 - x + 6 = 0. This quadratic has no real roots. Case 2: x^2 - x = 6 \Rightarrow x^2 - x - 6 = 0. • Factor the quadratic equation: (x - 3)(x + 2) = 0. • Since x must be a positive single digit, x = 3. • Find the unit's digit: y = 3^2 = 9. • The original number is 39. • The question asks for 40% of the original number. • 40% of 39 = 0.4 × 39 = 15.6. Exam Strategy & Shortcut: Logical Deduction: The unit's digit is a perfect square of the ten's digit. The possible digits (x, y) where y = x^2 are (1, 1), (2, 4), and (3, 9). Check the differences between these digits: • For (1, 1), difference is 0. • For (2, 4), difference is 2. • For (3, 9), difference is 6. We need a difference of digits equal to 54 / 9 = 6. Therefore, the digits must be 3 and 9. The number is 39. Calculate 40% of 39 mentally: 10% is 3.9, so 40% is 3.9 × 4 = 15.6. Common Pitfall: A major pitfall is ignoring the final part of the question. After correctly identifying the original number as 39, students often immediately look for 39 in the options and select option (d). The question specifically asks for 40% of that original number. Final Answer: Therefore, the correct answer is 15.6.

Question 18

A two-digit number becomes five-sixth of itself when its digits are reversed. The two digits differ by one. The number is

Accepted Answer

Concept & Strategy: By setting up the algebraic ratio between a two-digit number and its reversed form, we can simplify the equation to find the exact ratio between the two digits. 10y + x = (5/6)(10x + y) Step-by-Step Solution: Given: • The reversed number is (5/6) of the original number. • The digits differ by 1. Calculation / Deduction: • Let the original number be 10x + y. The reversed number is 10y + x. • Set up the equation: 10y + x = (5/6)(10x + y). • Multiply both sides by 6 to clear the fraction: 60y + 6x = 50x + 5y. • Rearrange to group the x and y terms: 60y - 5y = 50x - 6x. • Simplify: 55y = 44x. • Divide by 11: 5y = 4x, which gives the ratio x/y = 5/4. • We are given that the digits differ by 1 (x - y = 1). Since the digits are in a 5:4 ratio, the only single-digit values that fit are x = 5 and y = 4. • The original number is 54. Exam Strategy & Shortcut: Option Elimination via Fractions: The reversed number becomes (5/6) of itself, meaning the reversed number must be smaller. (a) 45 reverses to 54 (Bigger, eliminate). (b) 54 reverses to 45 (Smaller, check it: is 45 exactly (5/6) of 54? Yes, 54 × (5/6) = 9 × 5 = 45). (c) 56 reverses to 65 (Bigger, eliminate). (d) 65 reverses to 56 (Smaller, check it: 65 × (5/6) is a fraction, not 56). Common Pitfall: Confusing the order of the fraction multiplication, which leads to setting up (5/6)(10y + x) = 10x + y. This will yield negative or non-integer digit values, causing confusion. Final Answer: Therefore, the correct answer is 54.

Question 19

If the square of a two-digit number is reduced by the square of the number formed by reversing the digits of the number, the final result is

Accepted Answer

Concept & Formula: This problem leverages the algebraic identity for the difference of two squares applied to the standard properties of reversing two-digit numbers. a^2 - b^2 = (a - b)(a + b) Step-by-Step Solution: Given: • An expression for the difference between the square of a two-digit number and the square of its reverse. Calculation / Deduction: • Let the original number be N = 10x + y and the reversed number be M = 10y + x. • The question asks for the properties of N^2 - M^2. • Expand using the difference of squares: N^2 - M^2 = (N - M)(N + M). • We know that the difference between a number and its reverse is always a multiple of 9: N - M = 9(x - y). • We know that the sum of a number and its reverse is always a multiple of 11: N + M = 11(x + y). • Substitute these back: N^2 - M^2 = 9(x - y) × 11(x + y) = 99(x - y)(x + y). • Since the final expression contains the factor 99, the result must be divisible by 99, meaning it is simultaneously divisible by both 9 and 11. Exam Strategy & Shortcut: Test with Simple Numbers: Pick a simple two-digit number like 21. Its reverse is 12. Calculate 21^2 - 12^2 = 441 - 144 = 297. Check divisibility by 11: 297 / 11 = 27 (Divisible). Check divisibility by 9: 297 / 9 = 33 (Divisible). Since it is divisible by both, the answer is immediately "Both (a) and (b)". Common Pitfall: Trying to brute-force square the algebraic term (10x + y)^2 = 100x^2 + 20xy + y^2 and subtract the reversed squared term. This leads to a massive, easily-bungled polynomial instead of the clean factored form. Final Answer: Therefore, the correct answer is Both (a) and (b).

Question 20

The digit in the unit's place of a number is equal to the digit in the ten's place of half of that number and the digit in the ten's place of that number is less than the digit in unit's place of half of the number by 1. If the sum of the digits of …

Accepted Answer

Concept & Logic: This problem introduces complex, interdependent positional relationships between the digits of a number N and the digits of its quotient N/2. Step-by-Step Solution: Given: • Let the original number be N. • Condition 1: Unit digit of N = Ten's digit of N/2. • Condition 2: Ten's digit of N = (Unit digit of N/2) - 1. • Condition 3: Sum of the digits of N = 7. Calculation / Deduction: • Creating algebraic formulations for extracting digits of N/2 creates complicated parity trees. We will deduce the answer by systematically testing the given valid options against the conditions. • Test Option (a) 34: N = 34, so N/2 = 17. The unit digit of N is 4. The ten's digit of N/2 is 1. Since 4 
eq 1, Condition 1 fails immediately. • Test Option (b) 52: N = 52, so N/2 = 26. Check Condition 1: Unit digit of N is 2. Ten's digit of N/2 is 2. (Matches). Check Condition 2: Ten's digit of N is 5. Unit digit of N/2 is 6. Is 5 = 6 - 1? Yes, 5 = 5. (Matches). Check Condition 3: Sum of digits of N is 5 + 2 = 7. (Matches). • Since option (b) flawlessly satisfies every distinct condition, it is the correct number. Exam Strategy & Shortcut: Pure Option Elimination: Problems that mandate extracting specific digits from "half" of a variable number are traps designed to make you write messy algebraic bounds. Skip the algebra entirely. Grab the options, divide them by 2, and read the digits. You can verify the conditions for 52 (N=52, N/2=26) in roughly 15 seconds mentally. Common Pitfall: Attempting to mathematically define the digits of N/2 as (10x+y/2). If y is an odd digit, this formula creates decimals, meaning you have to borrow from the tens place which completely breaks standard digit-extraction algebra. Final Answer: Therefore, the correct answer is 52.

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