Q: The product of two numbers is 9375 and the quotient, when the larger one is divided by the smaller, is 15. The sum of the numbers is

Concept & Logic: When you have the product and the quotient of two numbers, multiplying these two values together cleverly cancels out the smaller number, giving you the square of the larger number directly. (x × y) × ((x/y)) = x^2 Step-by-Step Solution: • Given: Let the larger number be x and the smaller number be y. Product: xy = 9375 Quotient: (x/y) = 15 • Calculation / Deduction: • From the quotient, we can express x in terms of y: x = 15y. • Substitute this into the product equation: • (15y) × y = 9375 • 15y^2 = 9375 • Divide by 15: • y^2 = (9375/15) • y^2 = 625 • Take the square root (ignoring the negative root as aptitude questions typically assume positive numbers unless stated otherwise): • y = 25 • Now find x: x = 15 × 25 = 375. • The question asks for their sum: x + y = 375 + 25 = 400. Exam Strategy & Shortcut: Direct Multiplication Method: Multiply the product by the quotient to get the square of the larger number: 9375 × 15. To avoid huge numbers, divide the product by the quotient to get the square of the smaller number: 9375 / 15 = 625. Immediately you know y = 25. Then x is just 15 times that (375). Sum is 400. Common Pitfall: Finding the two numbers (375 and 25) and then hastily looking for 375 in the options, forgetting that the final question asks for the sum of the two numbers. Always re-read the final sentence of the prompt before selecting an option. Final Answer: Therefore, the correct answer is 400.

Q: The difference between two numbers is 1365. When the larger number is divided by the smaller one, the quotient is 6 and the remainder is 15. The smaller number is

Concept & Formula: This problem utilizes the fundamental Division Algorithm rule, combined with a simple linear equation for the difference. Dividend = (Divisor × Quotient) + Remainder Step-by-Step Solution: • Given: Let the larger number be x (Dividend) and the smaller number be y (Divisor). Difference: x - y = 1365 Quotient = 6, Remainder = 15 • Calculation / Deduction: • Apply the division rule: x = (y × 6) + 15 • x = 6y + 15 • Substitute this expression for x into the difference equation: • (6y + 15) - y = 1365 • Combine like terms: • 5y + 15 = 1365 • Subtract 15 from both sides: • 5y = 1350 • Divide by 5: • y = 270 • The question asks for the smaller number, which is y. Exam Strategy & Shortcut: Back-Solving via Options: We know the larger number is x = y + 1365. We also know x = 6y + 15. This means y + 1365 = 6y + 15. Visually isolate y: 5y = 1350. Divide 1350 by 5 mentally (double it to 2700, drop a zero = 270). This avoids writing out the full formal steps. Common Pitfall: A common error is setting up the division equation backwards, writing y = 6x + 15, which will yield a negative or nonsensical fractional answer. Always clearly label your variables as "larger" and "smaller" to ensure the dividend and divisor are placed correctly. Final Answer: Therefore, the correct answer is 270.

Q: If the sum and difference of two numbers are 20 and 8 respectively, then the difference of their squares is

Concept & Formula: The difference of squares of two numbers can be directly calculated if you know their sum and their difference, by using the standard algebraic factorization identity. x^2 - y^2 = (x + y)(x - y) Step-by-Step Solution: • Given: Let the two numbers be x and y. Sum: x + y = 20 Difference: x - y = 8 • Calculation / Deduction: • We need to find the difference of their squares, which is x^2 - y^2. • According to the algebraic identity: • x^2 - y^2 = (x + y)(x - y) • Substitute the given values directly into the formula: • x^2 - y^2 = 20 × 8 • x^2 - y^2 = 160 Exam Strategy & Shortcut: Direct Multiplication: Recognize that "difference of squares" is simply the product of the sum and the difference. Instantly multiply 20 × 8 = 160. Do not waste time calculating the individual numbers (14 and 6) just to square them and subtract (196 - 36 = 160). Common Pitfall: A common inefficiency is solving for the individual numbers first. Adding the equations gives 2x = 28 \Rightarrow x = 14, and subtracting gives 2y = 12 \Rightarrow y = 6. While correct, squaring 14 and 6 takes more mental effort and time than simply multiplying the given sum and difference. Final Answer: Therefore, the correct answer is 160.

Question 1

The sum of three numbers is 264. If the first number be twice the second and third number be one-third of the first, then the second number is:

Accepted Answer

Concept & Logic: When multiple variables are related to each other, express all of them in terms of a single common variable to create a simple linear equation based on their sum. Step-by-Step Solution: • Given: Let the three numbers be A, B, and C. Sum = A + B + C = 264. • Calculation / Deduction: • First condition: The first number is twice the second. • A = 2B \Rightarrow B = (A/2) • Second condition: The third number is one-third of the first. • C = (A/3) • Substitute B and C in terms of A into the sum equation: • A + (A/2) + (A/3) = 264 • Find a common denominator (6): • (6A + 3A + 2A/6) = 264 • (11A/6) = 264 • 11A = 264 × 6 = 1584 • A = (1584/11) = 144 • The question asks for the second number (B): • B = (A/2) = (144/2) = 72 Exam Strategy & Shortcut: Ratio Method: Establish a direct ratio. A:B = 2:1 (which is 6:3) A:C = 3:1 (which is 6:2) Combined ratio A:B:C = 6:3:2. Total parts = 6 + 3 + 2 = 11 parts. 11 parts = 264 \Rightarrow 1 part = 24. Second number is 3 parts = 3 × 24 = 72. This is much faster than fractions. Common Pitfall: Students often get bogged down by fractions when using algebraic substitution. Switching to LCM-based ratios as shown in the shortcut completely bypasses the risk of fraction arithmetic errors. Final Answer: Therefore, the correct answer is 72.

Question 2

The sum of two numbers is 22. Five times one number is equal to 6 times the other. The bigger of the two numbers is

Accepted Answer

Concept & Logic: When a multiple of one quantity equals a multiple of another, their ratio is the inverse of those multipliers. This ratio can then be distributed over their total sum. a \cdot x = b \cdot y \Rightarrow (x/y) = (b/a) Step-by-Step Solution: • Given: Let the numbers be x and y. Sum x + y = 22. Also, 5x = 6y. • Calculation / Deduction: • From 5x = 6y, we find the ratio of x to y: • (x/y) = (6/5) • This means the numbers are in the ratio of 6:5. Let them be 6k and 5k. • Substitute into the sum equation: • 6k + 5k = 22 • 11k = 22 • k = 2 • The numbers are 6(2) = 12 and 5(2) = 10. • The bigger number is 12. Exam Strategy & Shortcut: Mental Math via Proportions: 5 times one is 6 times the other, meaning the numbers are in a 6:5 ratio. Total parts = 11. Sum is 22, which is exactly double the total parts. Therefore, the numbers are just double the ratio figures: 12 and 10. The larger is instantly 12. Common Pitfall: A frequent mistake is writing the ratio as 5:6 instead of 6:5 when translating 5x = 6y. Always double-check inverse proportionality: the number with the smaller coefficient must be the larger value to balance the equation. Final Answer: Therefore, the correct answer is 12.

Question 3

The sum of two numbers is 40 and their difference is 4. The ratio of the numbers is

Accepted Answer

Concept & Formula: When required to find the ratio of two numbers given their sum (S) and difference (D), you can use the direct formula to bypass calculating the actual numbers. Ratio = (S + D/S - D) Step-by-Step Solution: • Given: The sum of two numbers (x + y) is 40. Their difference (x - y) is 4. • Calculation / Deduction: • Add the equations to find the larger number (x): • 2x = 44 \Rightarrow x = 22 • Subtract the equations to find the smaller number (y): • 2y = 36 \Rightarrow y = 18 • The ratio of the numbers is x : y = 22 : 18 • Simplify the ratio by dividing by 2: • 11 : 9 Exam Strategy & Shortcut: Direct Ratio Formula: Use the formula derived in the concept section for immediate results. Ratio = (40 + 4/40 - 4) = (44/36). Divide the numerator and denominator by 4 to get 11/9, or 11:9. This eliminates the intermediate variables entirely. Common Pitfall: Failing to simplify the final ratio. An unsimplified ratio like 22:18 will not appear in standard exam options, potentially causing panic. Always reduce fractions to their simplest terms. Final Answer: Therefore, the correct answer is 11 : 9.

Question 4

What is the greater of the two numbers whose product is 1092 and the sum of the two numbers exceeds their difference by 42?

Accepted Answer

Concept & Logic: Translating a word problem into algebraic expressions is key here. The phrase "sum exceeds difference by" can be translated into a direct equation. Interestingly, the variables representing the "greater" number will cancel out in this translation, instantly revealing the smaller number. (x + y) - (x - y) = Excess Amount Step-by-Step Solution: • Given: Let the two numbers be x and y, where x > y. The product x × y = 1092. • Calculation / Deduction: • The sum is (x + y). The difference is (x - y). • Set up the equation based on the condition "sum exceeds difference by 42": • (x + y) - (x - y) = 42 • Simplify the left side: • x + y - x + y = 42 • 2y = 42 • y = 21 • We now know the smaller number is 21. Use the product to find the greater number (x): • x × 21 = 1092 • x = (1092/21) • x = 52 Exam Strategy & Shortcut: Logical Deduction: Realize that the difference between the sum (A+B) and the difference (A-B) of two numbers is ALWAYS simply twice the smaller number (2B). Therefore, if the difference is 42, the smaller number is instantly 21. Divide 1092 by 21 to get 52. To quickly divide 1092 by 21, look at the unit digit: 1 × what = 2? It must end in 2. Look at the options; 52 is the only logical choice that works (20 × 50 = 1000). Common Pitfall: A major pitfall is trying to set up complex quadratic equations using substitution before simplifying the "sum exceeds difference" clause. Always simplify your linear conditions completely before plugging them into a product condition. Final Answer: Therefore, the correct answer is 52.

Question 5

The difference between two integers is 5. Their product is 500. Find the numbers.

Accepted Answer

Concept & Logic: This problem can be represented algebraically as x - y = 5 and xy = 500. This leads to a quadratic equation. However, since the problem asks to find both numbers and they are provided as pairs in the options, this is a prime candidate for back-solving. Step-by-Step Solution: • Given: Two integers x and y. Difference x - y = 5. Product xy = 500. • Calculation / Deduction: • Let the smaller integer be y. The larger integer is y + 5. • Their product is y(y + 5) = 500. • Expand to a quadratic equation: y^2 + 5y - 500 = 0. • We need two numbers that multiply to -500 and add to 5. Those numbers are 25 and -20. • Factor the equation: (y + 25)(y - 20) = 0. • So, y = -25 or y = 20. • Assuming positive integers based on the options, if y = 20, then the other number is 20 + 5 = 25. • The numbers are 20 and 25. Exam Strategy & Shortcut: Option Elimination: Do not write any equations. Just test the options to see which pair has a difference of 5 and a product of 500. • (a) 15, 20: Difference is 5. Product is 15 × 20 = 300 (Incorrect). • (b) 20, 25: Difference is 5. Product is 20 × 25 = 500 (Correct!). This takes literally 3 seconds. Common Pitfall: Getting tunnel vision and automatically grinding through the quadratic equation (y^2 + 5y - 500 = 0) consumes unnecessary time. In objective exams, if the variables you are solving for are in the options, plug them in. Final Answer: Therefore, the correct answer is 20, 25.

Question 6

Two numbers differ by 5. If their product is 336, then the sum of the two numbers is

Accepted Answer

Concept & Formula: When you are given the difference and the product of two numbers and need to find their sum, you should use the core algebraic identity relating sum, difference, and product. This avoids finding the individual numbers entirely. (x + y)^2 = (x - y)^2 + 4xy Step-by-Step Solution: • Given: The difference (x - y) = 5. The product xy = 336. • Calculation / Deduction: • We need to find the sum (x + y). • Substitute the known values into the algebraic identity: • (x + y)^2 = (5)^2 + 4(336) • Calculate the components: • (x + y)^2 = 25 + 1344 • (x + y)^2 = 1369 • Take the square root of both sides to find the sum: • x + y = \sqrt{1369} • x + y = 37 Exam Strategy & Shortcut: Unit Digit Method: You need the square root of 1369. Look at the last digit, which is 9. A perfect square ends in 9 only if its square root ends in 3 or 7 (since 3 × 3 = 9 and 7 × 7 = 49). Look at the options. The only options ending in 3 or 7 is 37. You don't even need to fully calculate \sqrt{1369}! Common Pitfall: Students often attempt to set up a quadratic equation x(x - 5) = 336 \Rightarrow x^2 - 5x - 336 = 0. Factoring large numbers like 336 under pressure is highly prone to errors and takes too much time. Use the algebraic identity instead. Final Answer: Therefore, the correct answer is 37.

Question 7

Two different natural numbers are such that their product is less than their sum. One of the numbers must be

Accepted Answer

Concept & Logic: This is a conceptual number theory question. For natural numbers (positive integers: 1, 2, 3...), the product of two numbers grows much faster than their sum. We must analyze the boundary conditions to see when the sum can exceed the product. Step-by-Step Solution: • Given: Two distinct natural numbers x and y. Condition: x × y < x + y. • Calculation / Deduction: • Let's test the smallest possible natural numbers. • Assume neither number is 1. The smallest distinct natural numbers would then be 2 and 3. • Test condition: Product = 2 × 3 = 6. Sum = 2 + 3 = 5. • Is 6 < 5? No. • As the numbers get larger (e.g., 3 and 4), the product (12) outpaces the sum (7) even more. • Therefore, the only way the sum can be greater than the product is if we use the absolute smallest natural number, which is 1. • Let's verify: Let one number be 1, and the other be any distinct natural number, say 5. • Product = 1 × 5 = 5. Sum = 1 + 5 = 6. • Is 5 < 6? Yes. The condition holds true. Exam Strategy & Shortcut: Logical Substitution: Instantly test the base cases. Try 2 and 3: 2 × 3 = 6, 2 + 3 = 5. 6 is not less than 5. The only number smaller than 2 in the natural number set is 1. Therefore, one of the numbers absolutely must be 1 to drag the product down below the sum. Common Pitfall: A common mistake is confusing "natural numbers" (starts at 1) with "whole numbers" (starts at 0) or "integers" (includes negatives). If 0 or negative numbers were allowed, there would be infinite solutions, but the strict constraint of "natural numbers" makes 1 the only correct logical anchor. Final Answer: Therefore, the correct answer is 1.

Question 8

The product of two numbers is 9375 and the quotient, when the larger one is divided by the smaller, is 15. The sum of the numbers is

Accepted Answer

Concept & Logic: When you have the product and the quotient of two numbers, multiplying these two values together cleverly cancels out the smaller number, giving you the square of the larger number directly. (x × y) × ((x/y)) = x^2 Step-by-Step Solution: • Given: Let the larger number be x and the smaller number be y. Product: xy = 9375 Quotient: (x/y) = 15 • Calculation / Deduction: • From the quotient, we can express x in terms of y: x = 15y. • Substitute this into the product equation: • (15y) × y = 9375 • 15y^2 = 9375 • Divide by 15: • y^2 = (9375/15) • y^2 = 625 • Take the square root (ignoring the negative root as aptitude questions typically assume positive numbers unless stated otherwise): • y = 25 • Now find x: x = 15 × 25 = 375. • The question asks for their sum: x + y = 375 + 25 = 400. Exam Strategy & Shortcut: Direct Multiplication Method: Multiply the product by the quotient to get the square of the larger number: 9375 × 15. To avoid huge numbers, divide the product by the quotient to get the square of the smaller number: 9375 / 15 = 625. Immediately you know y = 25. Then x is just 15 times that (375). Sum is 400. Common Pitfall: Finding the two numbers (375 and 25) and then hastily looking for 375 in the options, forgetting that the final question asks for the sum of the two numbers. Always re-read the final sentence of the prompt before selecting an option. Final Answer: Therefore, the correct answer is 400.

Question 9

The difference between two numbers is 1365. When the larger number is divided by the smaller one, the quotient is 6 and the remainder is 15. The smaller number is

Accepted Answer

Concept & Formula: This problem utilizes the fundamental Division Algorithm rule, combined with a simple linear equation for the difference. Dividend = (Divisor × Quotient) + Remainder Step-by-Step Solution: • Given: Let the larger number be x (Dividend) and the smaller number be y (Divisor). Difference: x - y = 1365 Quotient = 6, Remainder = 15 • Calculation / Deduction: • Apply the division rule: x = (y × 6) + 15 • x = 6y + 15 • Substitute this expression for x into the difference equation: • (6y + 15) - y = 1365 • Combine like terms: • 5y + 15 = 1365 • Subtract 15 from both sides: • 5y = 1350 • Divide by 5: • y = 270 • The question asks for the smaller number, which is y. Exam Strategy & Shortcut: Back-Solving via Options: We know the larger number is x = y + 1365. We also know x = 6y + 15. This means y + 1365 = 6y + 15. Visually isolate y: 5y = 1350. Divide 1350 by 5 mentally (double it to 2700, drop a zero = 270). This avoids writing out the full formal steps. Common Pitfall: A common error is setting up the division equation backwards, writing y = 6x + 15, which will yield a negative or nonsensical fractional answer. Always clearly label your variables as "larger" and "smaller" to ensure the dividend and divisor are placed correctly. Final Answer: Therefore, the correct answer is 270.

Question 10

If the sum and difference of two numbers are 20 and 8 respectively, then the difference of their squares is

Accepted Answer

Concept & Formula: The difference of squares of two numbers can be directly calculated if you know their sum and their difference, by using the standard algebraic factorization identity. x^2 - y^2 = (x + y)(x - y) Step-by-Step Solution: • Given: Let the two numbers be x and y. Sum: x + y = 20 Difference: x - y = 8 • Calculation / Deduction: • We need to find the difference of their squares, which is x^2 - y^2. • According to the algebraic identity: • x^2 - y^2 = (x + y)(x - y) • Substitute the given values directly into the formula: • x^2 - y^2 = 20 × 8 • x^2 - y^2 = 160 Exam Strategy & Shortcut: Direct Multiplication: Recognize that "difference of squares" is simply the product of the sum and the difference. Instantly multiply 20 × 8 = 160. Do not waste time calculating the individual numbers (14 and 6) just to square them and subtract (196 - 36 = 160). Common Pitfall: A common inefficiency is solving for the individual numbers first. Adding the equations gives 2x = 28 \Rightarrow x = 14, and subtracting gives 2y = 12 \Rightarrow y = 6. While correct, squaring 14 and 6 takes more mental effort and time than simply multiplying the given sum and difference. Final Answer: Therefore, the correct answer is 160.

Question 11

The product of two numbers is $45$ and the sum of their squares is $106$. The numbers are

Accepted Answer

Concept & Strategy: When given a system of non-linear equations involving products and sums of squares, substituting the options back into the conditions is often faster than solving the algebra algebraically. (a × b) = Product a^2 + b^2 = Sum of Squares Step-by-Step Solution: • Given: Let the two numbers be a and b. • We know that a × b = 45. • We know that a^2 + b^2 = 106. • Let's test the given options against these two conditions. • Test Option (a) 3 and 5: • Product: 3 × 5 = 15 
eq 45. (Incorrect) • Test Option (b) 5 and 9: • Product: 5 × 9 = 45. (Matches) • Sum of Squares: 5^2 + 9^2 = 25 + 81 = 106. (Matches) • Since both conditions are perfectly satisfied, 5 and 9 are the correct numbers. Exam Strategy & Shortcut: Option Elimination: For questions asking for "the numbers" when equations are provided, never solve the quadratic equations manually. Quickly scan the product condition first because factoring is easier mental math. Since 3 × 5 = 15 and 5 × 19 = 95, options (a) and (c) are instantly eliminated. Only (b) and (d) yield 45. A quick glance at (d) shows 45^2 is massive, leaving (b) as the only logical answer within 5 seconds. Common Pitfall: The most common mistake is attempting to solve (a+b)^2 = a^2 + b^2 + 2ab to find (a+b) and (a-b), leading to a lengthy derivation. While algebraically correct, doing this under timed exam conditions eats up valuable minutes that could be saved by simply plugging in the choices. Final Answer: Therefore, the correct answer is 5 and 9.

Question 12

The sum of the squares of two numbers is $3341$ and the difference of their squares is $891$. The numbers are

Accepted Answer

Concept & Formula: This problem simplifies to a basic system of linear equations if you treat the squares of the numbers as single variables (e.g., let A = x^2 and B = y^2). x^2 + y^2 = Sum x^2 - y^2 = Difference Step-by-Step Solution: • Given: Let the two numbers be x and y. • Equation 1: x^2 + y^2 = 3341 • Equation 2: x^2 - y^2 = 891 • To find x^2, add Equation 1 and Equation 2: (x^2 + y^2) + (x^2 - y^2) = 3341 + 891 2x^2 = 4232 • Divide by 2: x^2 = 2116 • Take the square root to find x: x = \sqrt{2116} = 46 • To find y^2, subtract Equation 2 from Equation 1: (x^2 + y^2) - (x^2 - y^2) = 3341 - 891 2y^2 = 2450 • Divide by 2: y^2 = 1225 • Take the square root to find y: y = \sqrt{1225} = 35 • The numbers are 46 and 35. Exam Strategy & Shortcut: Instead of full square root extraction, use Unit Digits. We know 2x^2 = 4232, so x^2 ends in 6. The square roots of numbers ending in 6 must end in 4 or 6. Option (c) has 46, which fits. Similarly, 2y^2 = 2450 \Rightarrow y^2 = 1225. A square ending in 25 guarantees the root ends in 5. Option (c) contains 35 and 46, seamlessly matching the unit digit logic without doing the heavy lifting of calculating \sqrt{2116}. Common Pitfall: Students often get intimidated by large numbers like 3341 and 2116 and assume they made an arithmetic error. Do not panic; trust the linear elimination method. Keep track of which variable is the larger one to match the options correctly (though options here are pairs). Final Answer: Therefore, the correct answer is 35, 46.

Question 13

The difference between two positive integers is $3$. If the sum of their squares is $369$, then the sum of the numbers is

Accepted Answer

Concept & Formula: You can find the sum of two variables if you know their difference and the sum of their squares by using standard algebraic identities linking (a+b)^2, (a-b)^2, and (a^2+b^2). (a+b)^2 + (a-b)^2 = 2(a^2 + b^2) Step-by-Step Solution: • Given: Let the two positive integers be x and y. • Difference: x - y = 3 • Sum of squares: x^2 + y^2 = 369 • We need to find the sum of the numbers, which is (x + y). • Substitute the given values into the algebraic identity: (x+y)^2 + (3)^2 = 2(369) • Square the 3 and multiply the right side: (x+y)^2 + 9 = 738 • Isolate (x+y)^2: (x+y)^2 = 738 - 9 (x+y)^2 = 729 • Take the square root to find the sum: x + y = \sqrt{729} x + y = 27 Exam Strategy & Shortcut: An alternative approach is to use the formula (a-b)^2 = a^2 + b^2 - 2ab to find 2ab, and then plug that into (a+b)^2 = a^2 + b^2 + 2ab. 3^2 = 369 - 2ab \Rightarrow 2ab = 360. Then (a+b)^2 = 369 + 360 = 729 \Rightarrow a+b = 27. Both algebraic paths are similarly fast, but knowing the direct relationship (a+b)^2 + (a-b)^2 = 2(a^2 + b^2) saves one substitution step and is highly recommended for speed. Common Pitfall: A common error is to try and guess the numbers by factoring 369, or guessing numbers that are 3 units apart (like 12 and 15) and manually squaring them (144 + 225 = 369). While guessing can sometimes be fast if you are good at mental math, it is unreliable and can waste time if the numbers are not immediately obvious. Stick to the algebraic identity. Final Answer: Therefore, the correct answer is 27.

Question 14

A, B, C, D and E are five consecutive odd numbers. The sum of A and C is $146$. What is the value of E?

Accepted Answer

Concept & Logic: Consecutive odd numbers always have a common difference of 2. By defining the first term, the rest follow predictably. A = x, B = x+2, C = x+4, D = x+6, E = x+8 Step-by-Step Solution: • Given: We have five consecutive odd numbers: A, B, C, D, and E. • Given: The sum of A and C is 146. • Substitute the algebraic expressions for A and C into the equation: x + (x + 4) = 146 • Combine like terms: 2x + 4 = 146 • Isolate the variable term: 2x = 142 • Divide to find the value of the first odd number (A): x = 71 • We need to find the value of E, which is x + 8: E = 71 + 8 = 79 Exam Strategy & Shortcut: Because consecutive odd numbers form an Arithmetic Progression (AP), the middle term between any two symmetric terms is their average. The number exactly between A and C is B. Therefore, B is the average of A and C: B = (146/2) = 73. Since the sequence is A, B, C, D, E with a difference of 2, you can just count up from B: B = 73 \Rightarrow C = 75 \Rightarrow D = 77 \Rightarrow E = 79. This skips the formal algebra entirely. Common Pitfall: A frequent mistake is solving for x (getting 71) and immediately selecting option (a) without re-reading the final question. The problem asks for the value of E, not A. Always underline the final target variable in the question. Final Answer: Therefore, the correct answer is 79.

Question 15

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 16

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 17

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 18

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 19

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 20

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.

Problems on Numbers Questions

Questions

Practice smarter