Question 1

The difference between a two-digit number and the number obtained by interchanging the digits is $36$. What is the difference between the sum and the difference of the digits of the number if the ratio between the digits of the number is $1 : 2$?

Accepted Answer

Concept & Strategy: Break the problem into two sequential steps: first, find the actual digits using the reversal rule and the given ratio, then compute the required algebraic operation on those digits. Digit Difference = \frac{Number Difference}{9} Step-by-Step Solution: Given: • The difference between the number and its reverse is 36. • The ratio of the digits is 1 : 2. Calculation: • First, find the difference between the digits: 36 / 9 = 4. • Let the digits be a and 2a (since their ratio is 1:2). • The difference between these digits is |2a - a| = a. • Therefore, a = 4. • The digits are 4 and 2(4) = 8. (The number is either 48 or 84). • The question asks for the difference between the sum of the digits and the difference of the digits. • Sum of the digits = 8 + 4 = 12. • Difference of the digits = 8 - 4 = 4. • Difference between the sum and the difference = 12 - 4 = 8. Exam Strategy & Shortcut: Algebraic Shortcut: Let the digits be x and y (where x > y). The sum is (x + y) and the difference is (x - y). The difference between the sum and the difference is: (x + y) - (x - y) = 2y. This means the answer will always simply be twice the smaller digit. Since the difference between the digits is 4 and they are in a 1:2 ratio, the digits are 4 and 8. The smaller digit is 4. Twice the smaller digit is 2 × 4 = 8. Common Pitfall: Students often compute the digits (4 and 8) and then lose track of the final nested question ("difference between the sum and the difference"). They might incorrectly mark 4 (the difference of the digits) or 12 (the sum) by failing to read the final sentence carefully. Final Answer: Therefore, the correct answer is 8.

Question 2

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 3

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 4

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 5

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.

Question 6

A certain number of two digits is three times the sum of its digits and if 45 be added to it, the digits are reversed. The number is

Accepted Answer

Concept & Logic: Translate the numerical value of a number into an equation proportional to its digit sum to quickly isolate a single definitive digit ratio. 10x + y = 3(x + y) Step-by-Step Solution: Given: • The number is exactly 3 times the sum of its own digits. • Adding 45 to the number reverses its digits. Calculation / Deduction: • Let the original two-digit number be 10x + y. • Set up the first condition: 10x + y = 3(x + y). • Expand the terms: 10x + y = 3x + 3y. • Rearrange to find the ratio: 7x = 2y \Rightarrow x/y = 2/7. • Because x and y must be single, positive integers (from 1 to 9), the only numbers that can satisfy a ratio of 2:7 are x = 2 and y = 7. • The original number is 27. • Verify using the second condition: 27 + 45 = 72. 72 is indeed the reversed form of 27. Exam Strategy & Shortcut: Direct Ratio Mapping: The moment you simplify 10x + y = 3(x + y) into 7x = 2y, you can completely stop calculating. There are no other single-digit combinations that equal each other when multiplied by 7 and 2 respectively. The digits are locked to 2 and 7 instantly. Common Pitfall: Getting bogged down trying to create a complex system of equations by utilizing the +45 condition (9(y-x) = 45 \Rightarrow y-x = 5). While correct, solving the simultaneous equations 7x = 2y and y - x = 5 wastes valuable time when the initial ratio already provides the exclusive answer. Final Answer: Therefore, the correct answer is 27.

Question 7

In a number of three digits, the digits in the unit's place and in the hundred's place are equal and the sum of all the digits is 8. The number of such numbers is

Accepted Answer

Concept & Logic: To find the possible numbers, we can represent the three-digit number algebraically and use the given constraints to find integer solutions. A three-digit number can be written based on its place values, but here we only need to focus on the individual digits. Let the three-digit number be represented by the digits x y x, where x is the digit in the hundred's and unit's place, and y is the digit in the ten's place. Step-by-Step Solution: • Given: The sum of the digits is 8. • Equation: x + y + x = 8 • Simplification: 2x + y = 8 • Deduction: Since x is the leading digit of a three-digit number, it cannot be 0. Thus, x \ge 1. Both x and y must be single-digit integers (0 to 9). • Let's test possible values for x: • If x = 1: 2(1) + y = 8 \implies y = 6. The number is 161. • If x = 2: 2(2) + y = 8 \implies y = 4. The number is 242. • If x = 3: 2(3) + y = 8 \implies y = 2. The number is 323. • If x = 4: 2(4) + y = 8 \implies y = 0. The number is 404. • If x = 5: 2(5) + y = 8 \implies 10 + y = 8 \implies y = -2. (Not possible as digits cannot be negative). There are exactly 4 valid combinations. Exam Strategy & Shortcut: For digit-sum problems with symmetrical conditions, systematically plug in values for the repeated digit starting from 1. You can quickly write out 2x + y = 8 and list the pairs (1,6), (2,4), (3,2), (4,0) in your head within seconds. Stop as soon as 2x \ge 8. Common Pitfall: Students often forget that the hundred's digit cannot be zero. If x=0, y=8, giving the number 080, which is technically a two-digit number (80), not a three-digit number. Always remember the leading digit constraint x 
eq 0. Final Answer: Therefore, the correct answer is 4.

Question 8

The sum of the numerator and denominator of a fraction is 11. If 1 is added to the numerator and 2 is subtracted from the denominator, it becomes $\frac{2}{3}$. The fraction is

Accepted Answer

Concept & Strategy: This problem translates simple arithmetic operations on the components of a fraction into algebraic equations. However, because we are looking for a specific starting fraction, checking the provided multiple-choice options against the conditions is the absolute fastest and most reliable method. Let the fraction be (n/d). Step-by-Step Solution: • Given: 1. n + d = 11 2. (n + 1/d - 2) = (2/3) • Calculation (Option Verification): Let's check the options to see which satisfies both conditions. • Check Option A ((5/6)): • Sum: 5 + 6 = 11 (Condition 1 satisfied). • New fraction: (5 + 1/6 - 2) = (6/4) = (3/2). This is not (2/3). • Check Option C ((3/8)): (Note: Checking C before B because (3/8) looks more standard as a proper fraction, though any order works). • Sum: 3 + 8 = 11 (Condition 1 satisfied). • New fraction: (3 + 1/8 - 2) = (4/6) = (2/3). • This perfectly satisfies both conditions. *(For completion, if we solved algebraically: From n+d=11, we get d = 11-n. Substitute into the second equation: (n+1/11-n-2) = (2/3) \implies (n+1/9-n) = (2/3) \implies 3(n+1) = 2(9-n) \implies 3n+3 = 18-2n \implies 5n = 15 \implies n=3. If n=3, then d=11-3=8. The fraction is (3/8).)* Exam Strategy & Shortcut: Use the "Sum" condition to eliminate options instantly. Option B ((6/5)) has sum 11. Option D ((8/3)) has sum 11. Wait, all options sum to 11! So the first condition doesn't help eliminate. Proceed directly to the second condition mentally: "Add 1 to top, subtract 2 from bottom." Apply to A: \rightarrow (6/4) (No) Apply to C: \rightarrow (4/6) (Yes, simplifies to (2/3)). Done in 10 seconds. Common Pitfall: When doing algebra, students might write the second equation incorrectly, such as (n+1/d) - 2 = (2/3), misinterpreting "subtracted from the denominator" as subtracted from the whole fraction. Always ensure operations on numerators/denominators are properly grouped in the fraction. Final Answer: Therefore, the correct answer is (3/8).

Question 9

If 1 is added to the denominator of a fraction, the fraction becomes $\frac{1}{2}$.  If 1 is added to the numerator of the fraction, the fraction becomes 1.  The fraction is

Accepted Answer

Concept & Logic: This problem requires setting up a system of two linear equations based on independent conditions applied to a single fraction. By assigning variables to the numerator and denominator, each sentence in the problem translates into a distinct algebraic equation that can be solved using substitution. Step-by-Step Solution: • Given: • Condition 1: Adding 1 to the denominator results in (1/2). • Condition 2: Adding 1 to the numerator results in 1. • Calculation: 1. Let the fraction be (x/y). 2. Translate Condition 1 into an equation: (x/y + 1) = (1/2) Cross-multiply: 2x = y + 1 \Rightarrow y = 2x - 1 3. Translate Condition 2 into an equation: (x + 1/y) = 1 Cross-multiply: x + 1 = y 4. Substitute the expression for y from the second condition into the first: x + 1 = 2x - 1 5. Solve for x: 1 + 1 = 2x - x x = 2 6. Substitute the value of x back into the second equation to find y: y = 2 + 1 = 3 7. The original fraction is (x/y) = (2/3). Exam Strategy & Shortcut: Option Elimination (Fastest Method): Instead of solving a system of equations, plug the given options into the second condition (adding 1 to the numerator makes the fraction equal to 1). A fraction equals 1 when its numerator and denominator are exactly equal. • (a) (1+1/3) = (2/3) 
eq 1 • (b) (2+1/3) = (3/3) = 1 (Valid) • (c) (3+1/4) = (4/4) = 1 (Valid) • (d) (3+1/2) = (4/2) = 2 
eq 1 Now test the remaining valid options (b and c) against the first condition (adding 1 to the denominator makes it (1/2)): • Try (b) (2/3) ightarrow (2/3+1) = (2/4) = (1/2). This is a perfect match! You have your answer instantly. Common Pitfall: Students often get confused and try to combine both conditions into a single operation, such as adding 1 to both the numerator and denominator simultaneously. It is crucial to recognize that the problem presents two separate, independent scenarios to form two distinct equations. Final Answer: Therefore, the correct answer is (2/3).

Question 10

If the numerator of a fraction is increased by $\frac{1}{4}$ and the denominator is decreased by $\frac{1}{3}$, the new fraction obtained is $\frac{33}{64}$.  What was the original fraction?

Accepted Answer

Concept & Logic: This problem tests the understanding of linear equations with two variables. When absolute values (like (1/4) and (1/3)) are added to or subtracted from the numerator and denominator of a fraction respectively, it generates a single linear equation with two independent variables. A unique original fraction cannot be determined without a second condition or equation linking the numerator and denominator. Step-by-Step Solution: • Given: • An original fraction exists. Let's call its numerator x and denominator y. • The numerator is increased by the absolute value (1/4). • The denominator is decreased by the absolute value (1/3). • The resulting fraction is (33/64). • Calculation: 1. Let the original fraction be (x/y). 2. Set up the equation based on the given conditions: (x + \frac{1/4)}{y - (1/3)} = (33/64) 3. Simplify the complex fraction by finding common denominators in the numerator and denominator expressions: (\frac{4x + 1/4)}{(3y - 1/3)} = (33/64) 4. Multiply by the reciprocal: (3(4x + 1)/4(3y - 1)) = (33/64) 5. Cross-multiply to linearize the equation: 64 × 3(4x + 1) = 33 × 4(3y - 1) 192(4x + 1) = 132(3y - 1) 6. Divide both sides by 12 to simplify: 16(4x + 1) = 11(3y - 1) 64x + 16 = 33y - 11 64x - 33y = -27 7. We are left with one equation and two variables (x and y). Without a second equation, it is impossible to find a unique value for the fraction (x/y). Exam Strategy & Shortcut: Logical Deduction: Whenever a question states that an absolute number (or fixed fraction) is added to or subtracted from the unknown numerator and denominator, immediately look for a relationship given between the original numerator and denominator. If no such relationship (like a ratio or a sum) is provided, the answer is always "Cannot be determined". You do not need to write down any equations. Common Pitfall: Students often misread "increased by (1/4)" as "increased by (1/4) of itself" (which would mean multiplying by (5/4)). If it were a proportional increase, the variables could be isolated as a ratio (x/y), and an answer could be found. Always interpret phrasing literally unless specified otherwise. Final Answer: Therefore, the correct answer is Cannot be determined.

Question 11

When the numerator of a fraction increases by 4, the fraction increases by $\frac{2}{3}$.  The denominator of the fraction is

Accepted Answer

Concept & Formula: This problem tests the algebraic relationship between modifying a fraction's numerator and the corresponding change in the fraction's overall value. When you add a value k to the numerator of a fraction (x/y), the new fraction can be split into the original fraction plus a new term: (x + k/y) = (x/y) + (k/y). The increase in the fraction's value is exactly represented by this new term, (k/y). Step-by-Step Solution: • Given: • Increase in numerator = 4 • Overall increase in the fraction's value = (2/3) • Calculation: 1. Let the original fraction be (x/y). 2. The new numerator is x + 4. 3. The new fraction is (x + 4/y). 4. According to the problem, the new fraction equals the original fraction plus (2/3): (x + 4/y) = (x/y) + (2/3) 5. Separate the terms on the left side: (x/y) + (4/y) = (x/y) + (2/3) 6. Subtract (x/y) from both sides to isolate y: (4/y) = (2/3) 7. Cross-multiply to solve for y: 2y = 12 y = 6 8. The denominator y is 6. Exam Strategy & Shortcut: Direct Concept Application: Recognize immediately that adding to a numerator splits the fraction. An addition of 4 to the numerator creates an extra value of (4/Denominator). The problem states this extra value is (2/3). Mentally set up: (4/D) = (2/3). Scale the numerator of (2/3) to match 4 by multiplying top and bottom by 2. (2 × 2/3 × 2) = (4/6). By direct comparison, the denominator D must be 6. Common Pitfall: A common mistake is trying to assign arbitrary values to the original numerator x to test options, which wastes time. The original numerator x is completely irrelevant to finding the denominator in this specific scenario because it cancels out algebraically. Final Answer: Therefore, the correct answer is 6.

Question 12

A fraction is such that if the double of the numerator and the triple of the denominator is changed by $+10$ percent and $-30$ percent respectively, then we get 11 percent of $\frac{16}{21}$.  Find the fraction.

Accepted Answer

Concept & Strategy: This problem combines fraction scaling with percentage modifications. To solve it systematically, apply the initial multipliers to the variables, apply the respective percentage changes as decimal multipliers, set the result equal to the target value, and then isolate the original fraction (x/y). Step-by-Step Solution: • Given: • Numerator is doubled (2x) and then changed by +10%. • Denominator is tripled (3y) and then changed by -30%. • Result equals 11% of (16/21). • Calculation: 1. Let the original fraction be (x/y). 2. Double the numerator: 2x. Change by +10%: New Numerator = 2x × 1.10 = 2.2x 3. Triple the denominator: 3y. Change by -30%: New Denominator = 3y × (1 - 0.30) = 3y × 0.70 = 2.1y 4. The new fraction expression is: (2.2x/2.1y) = (22x/21y) 5. Calculate the target value on the right side (11% of (16/21)): (11/100) × (16/21) 6. Set up the full equation: (22/21) × (x/y) = (11/100) × (16/21) 7. Isolate (x/y) by multiplying both sides by (21/22): (x/y) = ( (11 × 16/100 × 21) ) × (21/22) 8. Cancel out common terms (cancel 21 with 21, and simplify (11/22) to (1/2)): (x/y) = (1 × 16/100 × 2) (x/y) = (16/200) 9. Reduce the fraction by dividing the numerator and denominator by 8: (x/y) = (2/25) Exam Strategy & Shortcut: Fractional Ratios: Work entirely in fractions instead of decimals for cleaner cancellations. +10% is an increase of (1/10), so the multiplier is (11/10). -30% is a decrease of (3/10), so the multiplier is (7/10). Setup: (2x × \frac{11/10)}{3y × (7/10)} = (11/100) × (16/21) The denominators of 10 on the left cancel out immediately: (22x/21y) = (11 × 16/100 × 21) Cancel the 21s on the bottom across the equals sign. Cancel 11 on the right into 22 on the left (leaves 2). 2x / y = (16/100) Divide by 2: (x/y) = (8/100) = (2/25) Common Pitfall: Students often apply the percentage change to the original variable rather than the doubled/tripled variable. For example, calculating 1.1x instead of 1.1(2x). Carefully follow the chronological sequence of operations dictated by the wording. Final Answer: Therefore, the correct answer is (2/25).

Question 13

54 is to be divided into two parts such that the sum of 10 times the first and 22 times the second is 780.  The bigger part is

Accepted Answer

Concept & Logic: This is a classic linear equation problem based on dividing a known total into two unknown parts. By expressing one part as x, the other part is naturally forced to be (Total - x). This allows us to translate the given conditions into a single variable linear equation. Step-by-Step Solution: • Given: • Total sum = 54 • Condition: 10 × (First Part) + 22 × (Second Part) = 780 • Calculation: 1. Let the first part be x. 2. Since the two parts add up to 54, the second part must be (54 - x). 3. Translate the condition into an algebraic equation: 10x + 22(54 - x) = 780 4. Distribute the 22 into the parenthesis: 10x + (22 × 54) - 22x = 780 5. Calculate 22 × 54. (Hint: 22 × 50 = 1100, and 22 × 4 = 88. So, 1188). 10x + 1188 - 22x = 780 6. Combine the x terms: -12x + 1188 = 780 7. Isolate the x term by moving constants to one side: 1188 - 780 = 12x 408 = 12x 8. Solve for x: x = (408/12) x = 34 9. The first part is 34. The second part is 54 - 34 = 20. 10. The question asks for the bigger part, which is 34. Exam Strategy & Shortcut: Mixture & Alligation Method (Fastest Method): Treat this as finding the weighted average that blends two groups. Imagine if the whole 54 was just the first part. The sum would be 54 × 10 = 540. Imagine if the whole 54 was just the second part. The sum would be 54 × 22 = 1188. The actual sum is 780. Using Alligation layout: Part 1 (10x) : 540 Part 2 (22x) : 1188 Middle (Target) : 780 Diagonal Differences: Ratio of Part 1 to Part 2 = (1188 - 780) : (780 - 540) = 408 : 240 Divide by 24 to simplify: = 17 : 10 The parts are split in the ratio 17 : 10. Total parts = 17 + 10 = 27. Value of 1 ratio unit = (54/27) = 2. Bigger part (17 units) = 17 × 2 = 34. Common Pitfall: A common error is solving for x and assuming it is automatically the final answer. If you set x as the second part in your initial assumption, you would find x = 20, and you might incorrectly select an option if 20 was listed. Always verify which part is the "bigger part" before answering. Final Answer: Therefore, the correct answer is 34.

Question 14

The sum of the squares of three numbers is 138, while the sum of their products taken two at a time is 131. Their sum is

Accepted Answer

Concept & Formula: This problem can be instantly solved by recognizing the standard algebraic identity for the square of a trinomial. (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca) Step-by-Step Solution: • Given: The sum of the squares: a^2 + b^2 + c^2 = 138 The sum of the products two at a time: ab + bc + ca = 131 • Calculation: Substitute the given values into the algebraic identity. (a + b + c)^2 = 138 + 2(131) (a + b + c)^2 = 138 + 262 (a + b + c)^2 = 400 • Deduction: Taking the square root of both sides yields the sum of the numbers. a + b + c = \sqrt{400} = 20 Exam Strategy & Shortcut: Memorize the structure Sum^2 = SquaresSum + 2 × ProductsSum. In an exam, you should mentally calculate 138 + 2(131) = 400, and immediately recognize that its square root is 20. No writing is necessary. Common Pitfall: A frequent mistake is forgetting to multiply the product sum by 2 in the formula, leading to an incorrect sum of 138 + 131 = 269, which doesn't yield a clean integer square root. Final Answer: Therefore, the correct answer is 20.

Question 15

Of the three numbers, the sum of the first two is 73; the sum of the second and the third is 77 and the sum of the third and thrice the first is 104. The third number is

Accepted Answer

Concept & Strategy: This requires solving a system of three linear equations. The fastest approach is to eliminate variables by subtracting equations that share a common term. Step-by-Step Solution: • Given: Let the numbers be x, y, and z. Eq 1: x + y = 73 Eq 2: y + z = 77 Eq 3: z + 3x = 104 • Calculation: Subtract Eq 1 from Eq 2 to eliminate y. (y + z) - (x + y) = 77 - 73 z - x = 4 \implies z = x + 4 • Substitute this expression for z into Eq 3. (x + 4) + 3x = 104 4x + 4 = 104 4x = 100 \implies x = 25 • Deduction: Now, substitute x = 25 back into our expression for z. z = 25 + 4 = 29 The first number is 25, the second is 48, and the third is 29. Since 29 is not listed among the first three options, the correct choice is "None of these". Exam Strategy & Shortcut: Instead of finding all three variables, strictly focus on the target. You need z. From x + y = 73 and y + z = 77, the difference is clearly z - x = 4. Rearranging gives x = z - 4. Substitute into z + 3x = 104: z + 3(z - 4) = 104 \implies 4z - 12 = 104 \implies 4z = 116 \implies z = 29. This prevents calculating unneeded variables. Common Pitfall: A common trap is solving for the first variable (x = 25) and accidentally selecting option (a) because it matches, completely forgetting that the question specifically asks for the *third* number. Always reread the final prompt. Final Answer: Therefore, the correct answer is None of these.

Question 16

The sum of the squares of two positive integers is 100 and the difference of their squares is 28. The sum of the numbers is

Accepted Answer

Concept & Formula Use basic algebraic operations to solve simultaneous linear equations formed by the squares of variables. x^2 + y^2 = a x^2 - y^2 = b Step-by-Step Solution • Given: Let the two positive integers be x and y. The sum of their squares is 100, and the difference is 28. • Calculation: x^2 + y^2 = 100 (Equation 1) x^2 - y^2 = 28 (Equation 2) • Add Equation 1 and Equation 2 to eliminate y^2: 2x^2 = 128 x^2 = 64 • Since x is a positive integer, take the positive square root: x = 8 • Substitute x^2 = 64 into Equation 1 to find y: 64 + y^2 = 100 y^2 = 36 y = 6 • Calculate the required sum of the numbers: x + y = 8 + 6 = 14 Exam Strategy & Shortcut Simultaneous Addition/Subtraction: Instead of full substitution, immediately add the two constants and halve the result to get x^2, then subtract them and halve the result to get y^2. x^2 = (100 + 28) / 2 = 64 \Rightarrow x = 8 y^2 = (100 - 28) / 2 = 36 \Rightarrow y = 6 Sum is 8 + 6 = 14. This skips writing out the formal equations. Common Pitfall A frequent mistake is finding the values of x^2 and y^2 (64 and 36) but forgetting to take the square root before adding them, leading to an incorrect sum. Always re-read what the question is asking for at the final step. Final Answer Therefore, the correct answer is 14.

Question 17

The sum of two number is 37 and the difference of their squares is 185, then the difference between the two numbers is:

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Concept & Formula Utilize the standard algebraic identity for the difference of squares to relate the sum, difference, and squares of two numbers. x^2 - y^2 = (x - y)(x + y) Step-by-Step Solution • Given: Let the two numbers be x and y. Sum x + y = 37. Difference of squares x^2 - y^2 = 185. • Calculation: Expand the difference of squares identity: x^2 - y^2 = (x - y)(x + y) • Substitute the known values into the equation: 185 = (x - y)(37) • Solve for the difference (x - y): x - y = 185 / 37 • Divide 185 by 37: x - y = 5 Exam Strategy & Shortcut Direct Formula Application: Recognize instantly that (x^2 - y^2) / (x + y) = (x - y). Simply divide the difference of the squares by the sum of the numbers to get the difference of the numbers. 185 / 37 = 5. This takes 5 seconds and requires no paper. Common Pitfall Students sometimes try to solve for x and y individually by creating a quadratic equation or guessing numbers. This wastes immense time when the identity directly yields the required difference. Final Answer Therefore, the correct answer is 5.

Question 18

In a Mathematics examination the number scored by 5 candidates are 5 successive odd integers. If their total marks are 185, the highest score is

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Concept & Strategy For an odd number of consecutive integers (even or odd), the average of the set is exactly equal to the middle integer. Middle Term = \frac{Sum}{Number of Terms} Step-by-Step Solution • Given: 5 candidates scored successive odd integers. Total sum = 185. • Calculation: Let the 5 consecutive odd integers be (x - 4), (x - 2), x, (x + 2), (x + 4). • Sum these terms: (x - 4) + (x - 2) + x + (x + 2) + (x + 4) = 185 • Notice how the constants cancel out perfectly: 5x = 185 • Solve for x (the middle score): x = 185 / 5 = 37 • The terms are 33, 35, 37, 39, 41. The question asks for the highest score, which is (x + 4). Highest Score = 37 + 4 = 41 Exam Strategy & Shortcut Average equals Middle Term: Immediately divide the sum by the number of terms to find the middle number. 185 / 5 = 37. Because there are 5 terms, there are two terms above the middle. The consecutive odd numbers above 37 are 39 and 41. Thus, 41 is the highest. This can be done entirely mentally. Common Pitfall Setting the integers as x, x+1, x+2... instead of x, x+2, x+4.... Successive odd (or even) integers are separated by 2, not 1. Final Answer Therefore, the correct answer is 41.

Question 19

Directions : Each of the questions given below consists of a statement and/or a question and two statements numbered I and II given below it. You have to decide whether the data provided in the statement(s) is/are sufficient to answer the question. …

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Concept & Logic The core logic revolves around the properties of digits in a two-digit number. Digits can only range from 0 to 9, and the leading digit (tens place) cannot be 0. Step-by-Step Solution • Evaluating Statement I: The difference between the two digits is 9. Since the maximum possible digit is 9 and the minimum is 0, the only pair of digits with a difference of 9 is (9, 0). For a two-digit number, the tens digit cannot be 0. Therefore, the tens digit must be 9 and the unit digit must be 0. The number is definitively 90. Thus, Statement I alone is sufficient. • Evaluating Statement II: The sum of the digits is equal to the difference between the two digits. Let the digits be x and y. x + y = x - y 2y = 0 \implies y = 0 This tells us the unit digit is 0. However, the tens digit (x) could be any number from 1 to 9 (e.g., 10, 20, 30 \dots 90). Since we cannot determine a unique number, Statement II alone is not sufficient. Exam Strategy & Shortcut (MANDATORY) For data sufficiency, always test statements completely independently before combining them. When dealing with digit differences, remember that extreme differences (like 9) severely restrict the possible digit pairs. Recognizing that 9 and 0 are the *only* single digits separated by 9 solves Statement I instantly without calculation. Common Pitfall (MANDATORY) Students often mistakenly combine both statements immediately without analyzing them individually first. Another pitfall is forgetting that a two-digit number cannot start with 0 (i.e., 09 is a one-digit number, not a two-digit number). Final Answer Therefore, the correct answer is that the data in Statement I alone are sufficient to answer the question, while the data in Statement II alone are not sufficient to answer the question.

Question 20

Directions : Each of the questions given below consists of a statement and/or a question and two statements numbered I and II given below it. You have to decide whether the data provided in the statement(s) is/are sufficient to answer the question. …

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Concept & Logic In a two-digit number ab, knowing the sum and absolute difference of the digits allows you to find the values of the individual digits. However, unless the order or relative size of the digits is specified, you cannot determine which digit belongs in the tens place and which belongs in the units place. Step-by-Step Solution • Evaluating Statement I: The difference between the digits is 2. This means |a - b| = 2. It could be a - b = 2 or b - a = 2. Possible pairs include (3,1), (1,3), (4,2), (2,4) \dots This yields multiple possible numbers (31, 13, 42, 24 \dots). Statement I is not sufficient. • Evaluating Statement II: The sum of the digits is 4. a + b = 4 Possible pairs include (4,0), (3,1), (1,3), (2,2). This yields multiple numbers (40, 31, 13, 22). Statement II is not sufficient. • Evaluating Both Statements Together: We know a + b = 4 and |a - b| = 2. If a > b, we solve a + b = 4 and a - b = 2. Adding them gives 2a = 6 \implies a = 3 and b = 1. The number would be 31. If b > a, we solve a + b = 4 and b - a = 2. Adding them gives 2b = 6 \implies b = 3 and a = 1. The number would be 13. Because the problem does not specify whether a or b is larger, the number could be 31 or 13. Exam Strategy & Shortcut (MANDATORY) When a problem asks for an exact two-digit number but only provides symmetrical equations (like sum and absolute difference), you will always end up with two interchangeable digit values. Without an asymmetrical condition (e.g., "the units digit is larger" or "the number is even"), the data will never be sufficient. Spotting this symmetry saves you from doing the algebra. Common Pitfall (MANDATORY) The most common mistake is assuming that a - b = 2 is the only interpretation of "the difference between its digits". Students often forget the b - a = 2 case, wrongly concluding that 31 is the single unique answer and incorrectly marking (e) instead of (d). Final Answer Therefore, the correct answer is that the data even in both Statements I and II together are not sufficient to answer the question.

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