Question 1

If a number is multiplied by two-thirds of itself the value so obtained is $864$. What is the number?

Accepted Answer

Concept & Formula: Unlike addition or subtraction problems that form linear equations, multiplying a number by a fraction of itself creates a quadratic equation (involving x^2). Step-by-step Solution: Given: • Let the unknown number be x. • The product of the number and (2/3) of itself is 864. Calculation: Formulate the algebraic equation: x × ((2/3)x) = 864 Multiply the variables: (2/3)x^2 = 864 Isolate x^2 by multiplying both sides by (3/2): x^2 = 864 × (3/2) x^2 = 432 × 3 x^2 = 1296 Find the square root of both sides to solve for x: x = \sqrt{1296} x = 36 Exam Strategy & Shortcut: Option Elimination via Unit Digits: You need a number x such that (2/3)x^2 = 864, meaning x^2 = 1296. Instead of calculating the square root of 1296, look at the options and their unit digits: (a) 34 \rightarrow 4^2 ends in 6 (b) 36 \rightarrow 6^2 ends in 6 (c) 38 \rightarrow 8^2 ends in 4 (Eliminate) (d) 44 \rightarrow 4^2 ends in 6 (e) 46 \rightarrow 6^2 ends in 6 Since 40^2 = 1600, the answer must be significantly less than 40. This immediately eliminates 44 and 46. We are left with 34 and 36. Since 35^2 = 1225, and our target (1296) is larger, the answer must be 36. Common Pitfall: The most frequent error is treating the multiplication like addition, resulting in (2/3)x = 864 instead of (2/3)x^2 = 864. Always remember that x × x = x^2. Final Answer: Therefore, the correct answer is 36.

Question 2

A number when multiplied by $13$ is increased by $180$. The number is

Accepted Answer

Concept & Logic: When a number is multiplied by a factor, the "increase" represents the difference between the new product and the original number. We can set up a basic linear equation to find the unknown value. Step-by-step Solution: Given: • Let the original number be x. • Multiplying it by 13 gives 13x. • This new value is 180 more than the original number x. Calculation: Set up the algebraic equation: 13x = x + 180 Subtract x from both sides to group the variables: 13x - x = 180 12x = 180 Divide by 12 to isolate x: x = (180/12) x = 15 Exam Strategy & Shortcut: Logical Deduction: If multiplying a number by 13 increases it by 180, it means you added 12 times the original number to itself (since 1x + 12x = 13x). Therefore, 12 times the number equals 180. Number = 180 \div 12 = 15. This skips writing the equation entirely. Common Pitfall: A frequent mistake is interpreting "increased by 180" as multiplying by 180 or setting the equation as 13x = 180. Always remember that "increased by" denotes addition relative to the original base value. Final Answer: Therefore, the correct answer is 15.

Question 3

If one-third of one-fourth of a number is $15$, then three-tenths of that number is

Accepted Answer

Concept & Logic: The word "of" in fractional word problems translates to multiplication. We can chain these fractions together to form a single coefficient for our unknown variable, solve for the variable, and then apply the final requested fraction. Step-by-step Solution: Step 1: Find the original number Let the unknown number be x. Set up the equation based on the given sequence: (1/3) × (1/4) × x = 15 Multiply the fractions: (1/12)x = 15 Multiply both sides by 12 to isolate x: x = 15 × 12 x = 180 Step 2: Calculate the required final value The question asks for three-tenths ((3/10)) of the original number. Result = (3/10) × 180 Result = 3 × 18 Result = 54 Exam Strategy & Shortcut: Direct Chain Multiplication: You can combine everything into a single line without finding the intermediate number x. You know x = 15 × 3 × 4. You need (3/10) × x. So, calculate: (3/10) × (15 × 3 × 4). Cancel out a 5 and 2 from the denominator (10 = 5 × 2): Cancel 5 from 15 (leaves 3). Cancel 2 from 4 (leaves 2). Result = 3 × (3 × 3 × 2) = 3 × 18 = 54. This minimizes large numbers and reduces calculation errors. Common Pitfall: A common mistake is forgetting the second part of the question. A student might calculate the original number (180) and panic if it's not in the options, or accidentally pick a distractor if one matches. Always highlight what the question is ultimately asking for. Final Answer: Therefore, the correct answer is 54.

Question 4

A number is doubled and $9$ is added. If the resultant is trebled, it becomes $75$. What is that number?

Accepted Answer

Concept & Logic: This problem requires forming an equation with nested operations. We must use parentheses carefully to ensure the order of operations described in the text translates correctly into algebra. Step-by-step Solution: Given: • Let the original number be x. • "Doubled and 9 is added": 2x + 9 • "Resultant is trebled": Multiply the entire previous expression by 3. Calculation: Set up the equation: 3(2x + 9) = 75 Divide both sides by 3 to simplify the equation immediately: 2x + 9 = (75/3) 2x + 9 = 25 Subtract 9 from both sides: 2x = 25 - 9 2x = 16 Divide by 2: x = 8 Exam Strategy & Shortcut: Reverse Operations (Backtracking): Instead of using algebra, work backward from the final answer by doing the opposite operations. 1. Final result is 75. 2. Before that, it was trebled (multiplied by 3). So, divide by 3 \rightarrow 75 \div 3 = 25. 3. Before that, 9 was added. So, subtract 9 \rightarrow 25 - 9 = 16. 4. Before that, it was doubled (multiplied by 2). So, divide by 2 \rightarrow 16 \div 2 = 8. The number is 8. This takes less than 10 seconds mentally. Common Pitfall: The most common algebraic error is failing to use parentheses for the "resultant". Students often write 2x + 9 × 3 = 75 or 3 × 2x + 9 = 75, which only trebles a portion of the expression. "The resultant" always implies evaluating everything before that point as a single block. Final Answer: Therefore, the correct answer is 8.

Question 5

When 24 is subtracted from a number, it reduces to its four-seventh.  What is the sum of the digits of that number?

Accepted Answer

Concept & Logic: When a number is reduced by a certain value, the relationship can be expressed algebraically using a single variable to represent the original number. x - 24 = (4/7)x Step-by-Step Solution: • Given: A number decreases to its four-sevenths when 24 is subtracted from it. • Let the original number be x. • According to the problem statement: x - 24 = (4/7)x • Bring variables to one side: x - (4/7)x = 24 • Simplify the left side: (3/7)x = 24 • Solve for x: x = (24 × 7/3) = 56 • The question asks for the sum of the digits of this number: 5 + 6 = 11 Exam Strategy & Shortcut: Notice that subtracting 24 reduces the number to (4/7) of itself. This means the reduction of 24 is equivalent to (3/7) of the number. If (3/7) is 24, then (1/7) is 8. The whole number (which is (7/7)) must be 8 × 7 = 56. The sum of digits is 5 + 6 = 11. This mental math approach is much faster. Common Pitfall: Students often solve for the unknown number (56) and immediately look for it in the options, forgetting that the question specifically asks for the sum of the digits of that number. Always re-read the final question line! Final Answer: Therefore, the correct answer is 11.

Question 6

If a number, when divided by 4, is reduced by 21, the number is

Accepted Answer

Concept & Logic: Translating verbal relationships into a mathematical equation allows us to find the unknown value. Here, dividing by 4 equates to subtracting 21. (x/4) = x - 21 Step-by-Step Solution: • Let the unknown number be x. • When the number is divided by 4, the result is (x/4). • The problem states this new value is equal to the original number reduced by 21: (x/4) = x - 21 • Multiply the entire equation by 4 to eliminate the fraction: x = 4(x - 21) x = 4x - 84 • Rearrange the equation to solve for x: 84 = 4x - x 3x = 84 • Divide by 3: x = (84/3) = 28 Exam Strategy & Shortcut: If dividing a number by 4 reduces it by 21, it means the removed portion represents (3/4) of the original number. If (3/4) of the number is 21, then (1/4) of the number is 21 \div 3 = 7. The whole number is 7 × 4 = 28. This logical deduction is much faster than writing out the algebraic steps. Common Pitfall: Students might incorrectly translate "reduced by 21" as 21 - x instead of x - 21. Remember that reducing a value means subtracting from the original amount. Final Answer: Therefore, the correct answer is 28.

Question 7

A number whose fifth part increased by 4 is equal to its fourth part diminished by 10, is

Accepted Answer

Concept & Formula: This problem requires constructing an equation where two different modifications to fractions of the same unknown number are set equal to each other. (x/5) + 4 = (x/4) - 10 Step-by-Step Solution: • Let the unknown number be x. • Its fifth part increased by 4 can be written as: (x/5) + 4 • Its fourth part diminished by 10 can be written as: (x/4) - 10 • Set these two expressions equal to each other: (x/5) + 4 = (x/4) - 10 • Bring all the numerical constants to one side and the variable terms to the other: 4 + 10 = (x/4) - (x/5) 14 = (x/4) - (x/5) • Find a common denominator (20) for the fractions on the right side: 14 = (5x - 4x/20) 14 = (x/20) • Multiply by 20 to find x: x = 14 × 20 = 280 Exam Strategy & Shortcut: Notice the fractions involved: (1/4) and (1/5). The difference between them is (1/20). The problem tells us that a smaller fraction ((1/5)) plus 4 equals a larger fraction ((1/4)) minus 10. The total gap between the fractional parts is 4 + 10 = 14. So, (1/20) of the number is 14. The number must be 14 × 20 = 280. Common Pitfall: A common mistake is mixing up the operations, for example, writing "diminished by 10" as 10 - (x/4) instead of (x/4) - 10. Always preserve the order of operations described in the text. Final Answer: Therefore, the correct answer is 280.

Question 8

If one-seventh of a number exceeds its eleventh part by 100, then the number is

Accepted Answer

Concept & Formula: The word "exceeds" indicates a difference between two quantities. We can establish an equation representing the subtraction of the smaller fractional part from the larger one. (x/7) - (x/11) = 100 Step-by-Step Solution: • Let the original number be x. • Its one-seventh is (x/7) and its eleventh part is (x/11). • The problem states the one-seventh part is greater than the eleventh part by 100: (x/7) - (x/11) = 100 • Find a common denominator (77) for the left side: (11x - 7x/77) = 100 • Simplify the numerator: (4x/77) = 100 • Solve for x: 4x = 7700 x = (7700/4) = 1925 Exam Strategy & Shortcut: Work with the fractions directly. The difference between (1/7) and (1/11) is (11 - 7/77) = (4/77). The problem states that this (4/77) gap is equal to 100. Therefore, (1/77) of the number is 100 \div 4 = 25. The total number is 25 × 77. Calculate 25 × 77 quickly: (100 \div 4) × 77 = 7700 \div 4 = 1925. Common Pitfall: A common calculation mistake occurs during cross-multiplication or finding the common denominator. Always verify that (11x/77) and (7x/77) correctly map back to the original fractions before subtracting. Final Answer: Therefore, the correct answer is 1925.

Question 9

If the sum of one-half and one-fifth of a number exceeds one-third of that number by $7 \frac{1}{3}$, the number is

Accepted Answer

Concept & Formula: This problem tests your ability to add and subtract fractional parts of a single variable and handle mixed fractions appropriately. ((1/2)x + (1/5)x) - (1/3)x = 7 (1/3) Step-by-Step Solution: • Let the unknown number be x. • Convert the mixed fraction 7 (1/3) into an improper fraction: 7 (1/3) = (22/3) • Set up the equation based on the given conditions: ((x/2) + (x/5)) - (x/3) = (22/3) • Simplify the sum in the parenthesis first (common denominator is 10): (5x + 2x/10) - (x/3) = (22/3) (7x/10) - (x/3) = (22/3) • Find a common denominator (30) for the remaining terms on the left: (21x - 10x/30) = (22/3) (11x/30) = (22/3) • Solve for x: 11x = (22/3) × 30 11x = 22 × 10 11x = 220 x = 20 Exam Strategy & Shortcut: To avoid dealing with large fractions mid-way, find the Least Common Multiple (LCM) of all denominators involved (2, 5, and 3), which is 30. Assume the number is 30k. One-half = 15k. One-fifth = 6k. One-third = 10k. The equation becomes: (15k + 6k) - 10k = 7 (1/3) 21k - 10k = (22/3) 11k = (22/3) k = (2/3) Since the number is 30k, the result is 30 × (2/3) = 20. This method is highly resistant to arithmetic errors. Common Pitfall: Forgetting to convert the mixed fraction 7 (1/3) into an improper fraction before attempting algebraic manipulation will almost certainly lead to an incorrect result. Final Answer: Therefore, the correct answer is 20.

Question 10

If doubling a number and adding 20 to the result gives the same answer as multiplying the number by 8 and taking away 4 from the product, the number is

Accepted Answer

Concept & Logic: This word problem straightforwardly describes an equality between two distinct operations performed on the same initial unknown variable. 2x + 20 = 8x - 4 Step-by-Step Solution: • Let the unknown number be x. • Operation 1: Doubling the number and adding 20 yields: 2x + 20 • Operation 2: Multiplying the number by 8 and taking away 4 yields: 8x - 4 • The problem states these two operations give the same answer: 2x + 20 = 8x - 4 • Group the variable terms on one side by subtracting 2x from both sides: 20 = 6x - 4 • Add 4 to both sides: 24 = 6x • Divide by 6 to isolate x: x = (24/6) = 4 Exam Strategy & Shortcut: This question can be answered extremely quickly using the Option Elimination Method. • Test (a) 2: (2 × 2) + 20 = 24. (2 × 8) - 4 = 12. Not equal. • Test (c) 4: (4 × 2) + 20 = 28. (4 × 8) - 4 = 28. They match! For simple linear equations with small integer options, substituting the answers directly is often faster than writing out the algebraic steps. Common Pitfall: When reading "taking away 4 from the product", some students mistakenly write 4 - 8x instead of 8x - 4. Order matters in subtraction; always subtract the specified amount *from* the base value. Final Answer: Therefore, the correct answer is 4.

Question 11

If 50 is subtracted from two-third of a number, the result is equal to sum of 40 and one-fourth of that number. What is the number?

Accepted Answer

Concept & Formula: This problem translates a word sentence directly into a linear equation. We set the expression for subtracting 50 from a fraction of the number equal to adding 40 to another fraction of the number. (2/3)x - 50 = (1/4)x + 40 Step-by-Step Solution: • Let the unknown number be x. • "50 is subtracted from two-third of a number" translates to: (2/3)x - 50 • "sum of 40 and one-fourth of that number" translates to: (1/4)x + 40 • Equate both expressions as stated in the problem: (2/3)x - 50 = (1/4)x + 40 • Group the variable terms (x) on one side and constants on the other by adding 50 and subtracting (1/4)x from both sides: (2/3)x - (1/4)x = 40 + 50 (2/3)x - (1/4)x = 90 • Find a common denominator (12) for the fractions: (8x - 3x/12) = 90 (5x/12) = 90 • Isolate x: 5x = 90 × 12 x = (1080/5) x = 216 Exam Strategy & Shortcut: To avoid dealing with fractions, find the Least Common Multiple (LCM) of the denominators 3 and 4, which is 12. Assume the number is 12k. Two-thirds of 12k = 8k. One-fourth of 12k = 3k. The equation becomes: 8k - 50 = 3k + 40. 5k = 90, which means k = 18. Since the number is 12k, calculate 12 × 18 = 216. This LCM method heavily reduces calculation time and errors. Common Pitfall: A standard mistake is subtracting the fraction from 50 (i.e., 50 - (2/3)x) instead of subtracting 50 from the fraction ((2/3)x - 50). Always respect the "subtracted from" wording. Final Answer: Therefore, the correct answer is 216.

Question 12

One-third of a two-digit number exceeds one-fourth of its successive number by 1. The number is

Accepted Answer

Concept & Logic: This problem defines a relationship between a number and its successive (next consecutive) integer using fractional parts. "Exceeds" means the difference between the larger and smaller value is 1. (x/3) - (x + 1/4) = 1 Step-by-Step Solution: • Let the required two-digit number be x. • Its successive number is x + 1. • One-third of the number is (x/3). • One-fourth of its successive number is (x + 1/4). • According to the problem, the first quantity exceeds the second by 1: (x/3) - (x + 1/4) = 1 • Find a common denominator (12) for the left side: (4x - 3(x + 1)/12) = 1 • Distribute the negative sign carefully: (4x - 3x - 3/12) = 1 (x - 3/12) = 1 • Solve for x: x - 3 = 12 x = 15 Exam Strategy & Shortcut: Use Option Elimination, which is incredibly fast for small integers. You need a number that is easily divisible by 3, and whose successor is divisible by 4. • Check (a) 12: Successor is 13 (not cleanly divisible by 4). Skip. • Check (b) 15: One-third is 15 \div 3 = 5. Successor is 16. One-fourth is 16 \div 4 = 4. Difference is 5 - 4 = 1. Matches perfectly! You can arrive at the answer in under 10 seconds without algebra. Common Pitfall: A frequent algebraic error is forgetting to distribute the negative sign when finding the common denominator. Writing 4x - 3x + 3 instead of 4x - 3x - 3 will completely derail the final answer. Final Answer: Therefore, the correct answer is 15.

Question 13

If the sum of a number and its square is 182, what is the number?

Accepted Answer

Concept & Formula: This problem requires establishing a quadratic equation since it involves the addition of a variable and its squared self. x + x^2 = 182 Step-by-Step Solution: • Let the unknown number be x. • The sum of the number and its square is given as 182: x + x^2 = 182 • Rearrange into standard quadratic form (ax^2 + bx + c = 0): x^2 + x - 182 = 0 • We need to find two numbers that multiply to -182 and add to 1. Testing near squares: 13^2 = 169 and 14^2 = 196. 14 × (-13) = -182 and 14 + (-13) = 1. • Factor the quadratic equation: x^2 + 14x - 13x - 182 = 0 x(x + 14) - 13(x + 14) = 0 (x - 13)(x + 14) = 0 • The possible solutions are x = 13 or x = -14. • Now look at the options provided: (a) 15, (b) 26, (c) 28, (d) 91. Neither 13 nor -14 is listed. Exam Strategy & Shortcut: Instead of factoring a large quadratic, use Option Elimination and mental math based on squares. • 15^2 = 225, which is already larger than 182. (Eliminated) • 26, 28, and 91 are much larger. Their squares will be massive. (Eliminated) Since all given numerical options are trivially too large, you can instantly conclude the answer must be "None of these" without doing any deep calculation. Common Pitfall: Students sometimes overcomplicate their factoring or waste time trying to force one of the given options (like 15) to work. If simple estimations show the numbers are too big, trust your intuition and select the "None of these" trapdoor. Final Answer: Therefore, the correct answer is None of these.

Question 14

If $(73)^2$ is subtracted from the square of a number, the answer so obtained is 5075. What is the number?

Accepted Answer

Concept & Formula: This problem translates directly into a straightforward algebraic equation involving squares. We isolate the unknown variable's square to solve. x^2 - 73^2 = 5075 Step-by-Step Solution: • Let the unknown number be x. • According to the problem statement: x^2 - 73^2 = 5075 • Calculate the square of 73: 73 × 73 = 5329 • Substitute this back into the equation: x^2 - 5329 = 5075 • Isolate x^2 by adding 5329 to both sides: x^2 = 5075 + 5329 x^2 = 10404 • Find the square root to get x: x = \sqrt{10404} • Knowing that 100^2 = 10000, the answer must be slightly larger. Checking 102: 102 × 102 = 10404. Thus, x = 102. Exam Strategy & Shortcut: Use the Unit Digit Strategy to solve this incredibly fast without calculating big squares. • The unit digit of 73^2 is the unit digit of 3 × 3, which is 9. • The equation is x^2 - (...9) = (...5). • This means the unit digit of x^2 must be 4 (because a number ending in 4, like 14, minus 9 equals 5). • Look at the options: (a) 96^2 ends in 6 (6 × 6 = 36). (b) 98^2 ends in 4 (8 × 8 = 64). (c) 102^2 ends in 4 (2 × 2 = 4). (d) 106^2 ends in 6 (6 × 6 = 36). • It must be 98 or 102. Since 73^2 \approx 5000, and 5000 + 5075 \approx 10000, the root must be slightly above 100. Hence, 102 is the logical choice. Common Pitfall: A common error is making arithmetic mistakes when manually calculating 73 × 73. Relying on unit digits and approximations provides a much safer cross-check mechanism. Final Answer: Therefore, the correct answer is 102.

Question 15

Twenty times a positive integer is less than its square by 96. What is the integer?

Accepted Answer

Concept & Formula: This word problem constructs a quadratic equation by relating a number to its square and a constant difference. 20x = x^2 - 96 Step-by-Step Solution: • Let the positive integer be x. • Twenty times the integer is 20x. • The problem states this value is 96 less than its square (x^2): 20x = x^2 - 96 • Rearrange the equation into a standard quadratic form (ax^2 + bx + c = 0): x^2 - 20x - 96 = 0 • We need two numbers that multiply to -96 and add up to -20. Factors of 96 include: 1 × 96, 2 × 48, 3 × 32, 4 × 24, 6 × 16, 8 × 12. The pair -24 and +4 satisfies both conditions: (-24) × 4 = -96 and -24 + 4 = -20. • Factor the quadratic: x^2 - 24x + 4x - 96 = 0 x(x - 24) + 4(x - 24) = 0 (x - 24)(x + 4) = 0 • The possible solutions are x = 24 and x = -4. • Since the problem specifies a *positive* integer, we disregard -4. Exam Strategy & Shortcut: Use the Option Elimination Strategy to bypass factoring entirely. • Test (a) 20: 20 × 20 = 400. Is 400 less than 20^2 (400) by 96? No, they are equal. • Test (b) 24: 24 × 20 = 480. Square of 24 is 576. Difference: 576 - 480 = 96. This perfectly satisfies the condition. Plugging in the answers takes only seconds compared to solving a quadratic equation. Common Pitfall: When writing the initial equation, a common mistake is translating "less than its square by 96" as 96 - x^2 instead of x^2 - 96. Always subtract the specific amount *from* the base value. Final Answer: Therefore, the correct answer is 24.

Question 16

Two-third of a positive number and $\frac{25}{216}$ of its reciprocal are equal.  The number is

Accepted Answer

Concept & Formula: Equating fractional parts of a variable and its reciprocal allows us to isolate the square of the variable. Let the positive number be x. Step-by-Step Solution: • Given: (2/3) of x equals (25/216) of (1/x). • Calculation: Write down the equation: (2/3)x = (25/216) × (1/x) Multiply both sides by x to group the variables: (2/3)x^2 = (25/216) Isolate x^2 by multiplying both sides by (3/2): x^2 = (25/216) × (3/2) Simplify the fraction: x^2 = (25 × 1/72 × 2) x^2 = (25/144) Take the square root of both sides. Since x is a positive number, we take the principal root: x = \sqrt{(25/144)} = (5/12) Exam Strategy & Shortcut: When equating A \cdot x = B \cdot (1/x), always remember it immediately simplifies to x^2 = (B/A). Jump straight to: x^2 = (25/216) × (3/2). Simplifying gives (25/144). Taking the square root gives (5/12) instantly without writing down intermediate steps. Common Pitfall: A major pitfall is forgetting to take the square root at the very end. Notice that (25/144) is explicitly offered as Option (c) to trap students who stop calculating too early. Final Answer: Therefore, the correct answer is 5/12.

Question 17

Find the whole number which when increased by $20$ is equal to $69$ times the reciprocal of the number.

Accepted Answer

Concept & Formula: This requires formulating a simple quadratic equation based on the text. Let the whole number be x. Step-by-Step Solution: • Given: The number increased by 20 is x + 20. This is equal to 69 times the reciprocal of the number, which is (69/x). • Calculation: Equate the two expressions: x + 20 = (69/x) Multiply the entire equation by x to clear the denominator: x^2 + 20x = 69 Rearrange into standard quadratic form: x^2 + 20x - 69 = 0 We need two numbers that multiply to -69 and add to 20. These are 23 and -3. x^2 + 23x - 3x - 69 = 0 x(x + 23) - 3(x + 23) = 0 (x - 3)(x + 23) = 0 Therefore, x = 3 or x = -23. Since the problem specifically asks for a "whole number", we discard the negative value. x = 3. Exam Strategy & Shortcut: Option Elimination is perfect here. The question asks for a "whole number", so we can immediately ignore Option (a) 2.5. Test Option (b) which is 3: Increased by 20 \rightarrow 3 + 20 = 23 Does this equal 69 times its reciprocal? 69 × (1/3) = 23. Since 23 = 23, we immediately know Option (b) is correct without solving the quadratic. Common Pitfall: Students might waste time trying to factorize x^2 + 20x - 69 = 0 if they aren't comfortable with their multiplication tables (recognizing that 23 × 3 = 69). In competitive exams, verifying options is often much safer. Final Answer: Therefore, the correct answer is 3.

Question 18

The sum and product of two numbers are $12$ and $35$ respectively. The sum of their reciprocals will be

Accepted Answer

Concept & Formula: The sum of the reciprocals of two numbers can be found directly if their sum and product are known, without needing to calculate the individual numbers. For any two numbers x and y, the sum of their reciprocals is: (1/x) + (1/y) = (x + y/xy) Step-by-Step Solution: • Given: The sum of the two numbers (x + y) = 12 The product of the two numbers (xy) = 35 • Calculation: We need to find the sum of their reciprocals: (1/x) + (1/y). Using the algebraic identity established in the concept: (1/x) + (1/y) = (x + y/xy) Substitute the given values directly into the formula: Sum of reciprocals = (12/35) Exam Strategy & Shortcut: Direct Substitution is the ultimate shortcut here. Whenever you see "sum and product are S and P, find the sum of reciprocals", instantly write down (S/P). You do not need to figure out that the numbers are 7 and 5. The answer is just (12/35) immediately. Common Pitfall: Students often waste time trying to solve the quadratic equation to find the individual numbers (finding 7 and 5), and then adding their reciprocals (1/7) + (1/5) = (5+7/35) = (12/35). While this gets the right answer, it consumes precious time unnecessarily. Final Answer: Therefore, the correct answer is 12/35.

Question 19

If $2\frac{1}{2}$ is added to a number and the sum multiplied by $4\frac{1}{2}$ and $3$ is added to the product and the sum is divided by $1\frac{1}{5}$, the quotient becomes $25$. What is the number?

Accepted Answer

Concept & Strategy: This is a classic "working backwards" or direct algebraic translation problem. Converting mixed fractions to improper fractions or decimals is crucial for clean calculation. Let the unknown number be x. Step-by-Step Solution: • Preparation: Convert all mixed fractions to decimals for easier manipulation: 2(1/2) = 2.5 4(1/2) = 4.5 1(1/5) = 1.2 • Calculation (Working Forward algebraically): Step 1: Add 2.5 to x \rightarrow (x + 2.5) Step 2: Multiply by 4.5 \rightarrow 4.5(x + 2.5) Step 3: Add 3 \rightarrow 4.5(x + 2.5) + 3 Step 4: Divide by 1.2, and the result is 25. (4.5(x + 2.5) + 3/1.2) = 25 Now, solve for x: Multiply by 1.2: 4.5(x + 2.5) + 3 = 25 × 1.2 4.5(x + 2.5) + 3 = 30 Subtract 3: 4.5(x + 2.5) = 30 - 3 4.5(x + 2.5) = 27 Divide by 4.5: x + 2.5 = (27/4.5) x + 2.5 = 6 Subtract 2.5: x = 6 - 2.5 x = 3.5 Convert back to a mixed fraction: x = 3(1/2). Exam Strategy & Shortcut: Working Backwards is highly efficient here and skips setting up a massive equation. Start at the end: 25. It was divided by 1.2 \rightarrow multiply by 1.2 = 30. 3 was added to it \rightarrow subtract 3 = 27. It was multiplied by 4.5 \rightarrow divide by 4.5 = 6. 2.5 was added to it \rightarrow subtract 2.5 = 3.5. The answer is 3.5 or 3(1/2). Common Pitfall: Attempting to keep everything as improper fractions (like (9/2) and (6/5)) can lead to messy, error-prone stacked fractions. Switching to decimals (4.5, 1.2) makes the arithmetic visually simpler and faster. Final Answer: Therefore, the correct answer is 3(1/2).

Question 20

Three numbers are in the ratio $4 : 5 : 6$ and their average is $25$. The largest number is

Accepted Answer

Concept & Formula: When numbers are given in a ratio, we can represent them as multiples of a common variable (e.g., 4x, 5x, 6x). The average is the sum of the items divided by the number of items. Average = \frac{Sum of Terms}{Number of Terms} Step-by-Step Solution: • Given: The ratio of the three numbers is 4:5:6. Let the numbers be 4x, 5x, and 6x. The average of these three numbers is 25. • Calculation: Set up the average formula: (4x + 5x + 6x/3) = 25 Sum the terms in the numerator: (15x/3) = 25 Simplify the fraction: 5x = 25 Solve for x: x = (25/5) = 5 The problem asks for the largest number. Based on the ratio (4x, 5x, 6x), the largest number is 6x. Largest number = 6 × 5 = 30. Exam Strategy & Shortcut: Notice that the numbers are in an arithmetic progression (ratio 4:5:6 implies the terms are evenly spaced). For an odd number of evenly spaced terms, the average is always the middle term. Therefore, the middle term is exactly 25. Since the ratio is 4:5:6, and the "5" part represents 25, the multiplier is 5. The largest part is "6". So, 6 × 5 = 30. You can solve this entirely in your head in seconds. Common Pitfall: A frequent trap is calculating x = 5 and then mistakenly selecting an option if 5 were present, forgetting to multiply by 6 to find the *largest* number. Always reread what the question specifically demands. Final Answer: Therefore, the correct answer is 30.

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