Question 1

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 2

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 3

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 4

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 5

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 6

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 7

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 8

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 9

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 10

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 11

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 12

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 13

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 14

Three numbers are in the ratio of $3 : 4 : 6$ and their product is $1944$. The largest of these numbers is

Accepted Answer

Concept & Formula: When multiplying numbers based on ratios, remember that the common multiplier variable (e.g., x) will be raised to a power corresponding to the number of terms. Let the common multiplier be x. Step-by-Step Solution: • Given: The ratio of the three numbers is 3 : 4 : 6. Let the three numbers be 3x, 4x, and 6x. Their product is 1944. • Calculation: Set up the product equation: 3x × 4x × 6x = 1944 Multiply the coefficients and the variables: (3 × 4 × 6) × (x \cdot x \cdot x) = 1944 72x^3 = 1944 Isolate x^3 by dividing by 72: x^3 = (1944/72) To simplify, you can divide step-by-step. Let's divide by 12 first: 1944 \div 12 = 162 72 \div 12 = 6 x^3 = (162/6) x^3 = 27 Take the cube root: x = \sqrt[3]{27} = 3 The question asks for the largest number. The largest ratio part is 6x. Largest number = 6 × 3 = 18. Exam Strategy & Shortcut: Digital Sum / Divisibility Rules can speed up finding x^3. 1944 sum of digits is 1+9+4+4 = 18, meaning it's divisible by 9. 72 is also divisible by 9. If you suspect x=3, test the final answer using the options. If the largest number is 18 (Option C), then 6x = 18 \Rightarrow x=3. Check the product: 3(3) × 4(3) × 6(3) = 9 × 12 × 18. 9 × 12 = 108. 108 × 18 = 1944. Matches perfectly. Common Pitfall: The most dangerous pitfall is forgetting the x^3 and writing 72x = 1944 \Rightarrow x = 27. This happens when students treat a product of ratios like a sum of ratios. Always mind your exponents when multiplying variables. Final Answer: Therefore, the correct answer is 18.

Question 15

Two numbers are such that the square of one is 224 less than 8 times the square of the other. If the numbers be in the ratio of 3 : 4, the numbers are

Accepted Answer

Concept & Logic: When two numbers are given as a ratio, they can be expressed using a common variable multiplier (e.g., 3x and 4x). The problem translates to an algebraic equation relating their squares. (4x)^2 = 8(3x)^2 - 224 Step-by-Step Solution: • Given: The ratio of the numbers is 3:4. Let the numbers be 3x and 4x. • Calculation / Deduction: • Square of the first number = (3x)^2 = 9x^2 • Square of the second number = (4x)^2 = 16x^2 • According to the condition, the square of one is 224 less than 8 times the square of the other. Let's test the relation: 16x^2 = 8(9x^2) - 224 • 16x^2 = 72x^2 - 224 • 56x^2 = 224 • x^2 = 4 • x = 2 • The numbers are 3(2) = 6 and 4(2) = 8. Exam Strategy & Shortcut: Option Elimination: Instead of solving the quadratic equation, directly test the options. Option (a) 6 and 8: Square of 8 is 64. 8 times the square of 6 is 8 × 36 = 288. The difference is 288 - 64 = 224. This matches the condition perfectly, saving you valuable time. Common Pitfall: Students often misinterpret "square of one is 224 less than 8 times the square of the other" and set up the equation backwards, subtracting from the wrong variable. Always use common sense or option verification to confirm the logical relationship. Final Answer: Therefore, the correct answer is 6, 8.

Question 16

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 17

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 18

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 19

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 20

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.

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