Question 1

A natural number, when increased by 12, equals 160 times its reciprocal. What is the number?

Accepted Answer

Introduction / Context: This problem involves a natural number, its reciprocal and a simple algebraic equation. The statement says that when the number is increased by 12, the result is equal to 160 times the reciprocal of the number. Such questions are typical in algebra and aptitude exams, testing your ability to translate a word statement into an equation and solve a quadratic equation that results from the presence of a reciprocal. Given Data / Assumptions: The number is a natural number, say n. The reciprocal of n is 1 / n. n + 12 = 160 * (1 / n). We must find the value of n. Concept / Approach: We first convert the verbal statement into an algebraic equation: n + 12 = 160 / n. To eliminate the fraction, we multiply both sides by n, which leads to a quadratic equation in n. Solving this quadratic using the quadratic formula or factorisation gives two roots, but only the positive natural number root is acceptable for this question. Recognising that natural numbers are positive integers helps us discard invalid solutions. Step-by-Step Solution: Step 1: Let the natural number be n. Step 2: According to the problem, n + 12 = 160 * (1 / n) = 160 / n. Step 3: Multiply both sides by n to clear the denominator: n * (n + 12) = 160. Step 4: Expand the left side: n^2 + 12n = 160. Step 5: Bring all terms to one side to form a quadratic equation: n^2 + 12n − 160 = 0. Step 6: Compute the discriminant: D = 12^2 − 4 * 1 * (−160) = 144 + 640 = 784. Step 7: The square root of 784 is 28, so the solutions are n = (−12 ± 28) / 2. Step 8: Compute the two roots: n = (−12 + 28) / 2 = 16 / 2 = 8, and n = (−12 − 28) / 2 = −40 / 2 = −20. Verification / Alternative check: Since n is a natural number, it must be positive, so we discard n = −20. We take n = 8. Now verify it in the original equation. The reciprocal of 8 is 1 / 8. 160 times 1 / 8 is 160 / 8 = 20. The number 8 increased by 12 is 8 + 12 = 20. Both sides match, confirming that n = 8 is correct. This double check ensures that the quadratic equation was solved correctly and the correct root was chosen. Why Other Options Are Wrong: Option 20: If n = 20, then n + 12 = 32, while 160 / n = 8, so the two sides do not match. Option 12: For n = 12, n + 12 = 24 and 160 / 12 is not equal to 24. Option 4 or 16: Substituting either of these values will not satisfy the equation n + 12 = 160 / n when checked. Common Pitfalls: Students sometimes forget to multiply both sides by n or mishandle the expansion, leading to an incorrect quadratic equation. Another common issue is ignoring the requirement that the number is a natural number and accepting negative roots. Always re-read the question to check domain restrictions (natural, integer, non-zero, etc.) and verify the solution by substituting back into the original equation. Final Answer: The natural number that satisfies the condition is 8.

Question 2

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

Accepted Answer

Introduction / Context: This is an algebra problem involving fractions of an unknown number. You are told that if 50 is subtracted from two thirds of a number, the result is equal to the sum of 40 and one fourth of the same number. The task is to find that number. Such questions are common in quantitative aptitude tests and help you practice forming and solving linear equations involving fractional coefficients. Given Data / Assumptions: Let the number be N. Two-thirds of the number is (2 / 3) * N. One-fourth of the number is (1 / 4) * N. (2 / 3) * N − 50 = 40 + (1 / 4) * N. We must solve for N. Concept / Approach: We translate the word statement into an algebraic equation. Because the equation has fractions with different denominators (3 and 4), it is convenient to multiply through by the least common multiple of the denominators to clear the fractions. This gives a simpler linear equation with integer coefficients, which we can solve using basic algebra. After finding N, we should substitute it back into the original equation to verify that it satisfies the condition. Step-by-Step Solution: Step 1: Let the number be N. Step 2: According to the problem, two-thirds of the number minus 50 equals 40 plus one-fourth of the number: (2 / 3) * N − 50 = 40 + (1 / 4) * N. Step 3: To eliminate fractions, multiply the entire equation by 12, the least common multiple of 3 and 4. Step 4: Multiply term by term: 12 * (2 / 3) * N = 8N, 12 * (−50) = −600, 12 * 40 = 480, and 12 * (1 / 4) * N = 3N. Step 5: The new equation is 8N − 600 = 480 + 3N. Step 6: Subtract 3N from both sides to get 5N − 600 = 480. Step 7: Add 600 to both sides: 5N = 480 + 600 = 1,080. Step 8: Divide both sides by 5 to find N: N = 1,080 / 5 = 216. Verification / Alternative check: Substitute N = 216 into the original statement. Two-thirds of 216 is (2 / 3) * 216 = 144. Now compute (2 / 3) * 216 − 50 = 144 − 50 = 94. One-fourth of 216 is (1 / 4) * 216 = 54. The sum of 40 and one-fourth of the number is 40 + 54 = 94. Since both sides are equal, N = 216 satisfies the equation and is therefore correct. Why Other Options Are Wrong: Option 156, 179, 198 and 144: Substituting any of these values into the original equation results in different values on the two sides, so they do not satisfy the condition (2 / 3) * N − 50 = 40 + (1 / 4) * N. Common Pitfalls: A common mistake is to misinterpret the words and form the wrong equation, for example by subtracting 50 from the whole number instead of from two-thirds of the number. Another error is to handle the fractions incorrectly when clearing denominators, which leads to wrong coefficients. Carefully multiplying every term by the least common multiple and checking the final equation prevents such mistakes. Always verify your solution by substituting it back into the original statement. Final Answer: The number that satisfies the given condition is 216.

Question 3

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

Accepted Answer

Introduction / Context: This is a digit-based number puzzle involving a two-digit number and the number formed by reversing its digits. You are told that the difference between these two numbers is 63 and asked to find the smaller of the two numbers. Such questions test your understanding of place value and your ability to work with algebraic representations of digit-based numbers. Given Data / Assumptions: Let the original two-digit number have tens digit a and units digit b. The original number is 10a + b. The reversed number is 10b + a. The difference between the original number and the reversed number is 63. We must find the smaller of the two numbers. Concept / Approach: We translate the problem into an equation using the algebraic representations of the numbers. The difference between the original number and its reverse is (10a + b) − (10b + a). This simplifies to 9a − 9b, which equals 63 according to the question. Solving for a − b gives us the difference between the digits. Then we list all possible valid digit pairs (a, b) that satisfy this relation, ensuring that a and b are digits between 0 and 9 and that a is non-zero (because it is the tens digit). From these possibilities, we determine if the smaller number is uniquely determined or not. Step-by-Step Solution: Step 1: Let the original number be 10a + b, where a is the tens digit and b is the units digit. Step 2: The reversed number is 10b + a. Step 3: According to the question, (10a + b) − (10b + a) = 63. Step 4: Simplify the left-hand side: (10a + b) − (10b + a) = 10a + b − 10b − a = 9a − 9b = 9(a − b). Step 5: So 9(a − b) = 63, which implies a − b = 63 / 9 = 7. Step 6: Since a and b are digits, the pairs that satisfy a − b = 7 are (a, b) = (7, 0), (8, 1) and (9, 2). Step 7: These correspond to original numbers 70, 81 and 92. Their reversed numbers are 07 (that is 7), 18 and 29 respectively. Verification / Alternative check: Check the differences: 70 − 7 = 63, 81 − 18 = 63 and 92 − 29 = 63. All three pairs satisfy the condition in the question. The smaller numbers in these pairs are 7, 18 and 29 respectively. Since there are multiple possible smaller numbers that satisfy the condition, the problem does not have a unique answer for “the smaller of the two numbers”. Therefore, none of the specific numerical options like 12, 15 or 17 can be chosen as a unique correct answer. Why Other Options Are Wrong: Option 12, 15, 17 and 18: These are specific numbers, but each of them corresponds either to digit differences that do not give 63 when reversed or to only one of several valid solutions. The question expects a unique smaller number, which is not possible here, so no single one of these values can be the correct answer. Option “none of these answers can be determined”: This correctly reflects the fact that there are several valid pairs of numbers giving a difference of 63, and thus the smaller number is not uniquely determined from the information provided. Common Pitfalls: A frequent mistake is to assume that there is only one possible pair of digits and to stop after finding the first valid solution, for example (8, 1). This leads to incorrectly choosing 18 as the smaller number without checking for other digit pairs. Another error is to forget that b can be zero, which gives additional valid pairs. Always list all digit pairs satisfying the algebraic condition and then check whether the problem has a unique answer or multiple possibilities. Final Answer: Because more than one pair of numbers satisfies the condition, the smaller number cannot be uniquely determined. Hence the correct choice is none of these answers can be determined.

Question 4

In a divisibility test, the 7-digit number 78K928L is known to be divisible by 6. For which of the following pairs of digits (K, L) will this number be divisible by 6?

Accepted Answer

Introduction / Context: This question tests the basic divisibility rules for the number 6 and how they apply to a multi-digit number containing unknown digits. Such problems are common in quantitative aptitude exams because they combine number properties with simple reasoning about digits. Understanding divisibility by 2 and 3 is very helpful for quickly checking whether large numbers are divisible by 6 without doing full division. Here, we must decide which choice of digits K and L makes the 7-digit number 78K928L divisible by 6. Given Data / Assumptions: The number is 78K928L, where K and L are single digits from 0 to 9. The number is divisible by 6. We are given four possible pairs of values for (K, L) and must select the correct one. Standard divisibility rules for 2 and 3 apply. Concept / Approach: A number is divisible by 6 if and only if it is divisible by both 2 and 3. Divisibility by 2 requires that the last digit of the number is even. Divisibility by 3 requires that the sum of all digits of the number is a multiple of 3. We will apply these two checks for each pair of K and L values given in the options. Any pair that fails either of the two conditions cannot be correct. The pair that satisfies both conditions will be the required answer. Step-by-Step Solution: Step 1: Write the number as 7, 8, K, 9, 2, 8, L. Step 2: First enforce divisibility by 2. The last digit L must be even (0, 2, 4, 6, or 8). Step 3: Compute the sum of the fixed digits: 7 + 8 + 9 + 2 + 8 = 34. Step 4: For each option, add K and L to 34 and check if the total is a multiple of 3, while also making sure L is even. Step 5: Option (a) K=2, L=3: last digit 3 is odd, so the number is not divisible by 2, hence not divisible by 6. Step 6: Option (b) K=1, L=4: sum of digits is 34 + 1 + 4 = 39, which is a multiple of 3, and L=4 is even, so both divisibility rules are satisfied. Step 7: Option (c) K=1, L=2: sum is 34 + 1 + 2 = 37, not a multiple of 3, so it fails the divisibility by 3 test. Step 8: Option (d) K=3, L=3: again the last digit 3 is odd, so divisibility by 2 fails. Step 9: Therefore only option (b) gives a number divisible by 6. Verification / Alternative check: We can verify the result quickly by checking divisibility directly for the candidate in option (b). With K=1 and L=4, the number becomes 7819284. The last digit is 4, so the number is divisible by 2. The digit sum 7+8+1+9+2+8+4 equals 39, which is a multiple of 3, making the number divisible by 3. Since both conditions are met, the number is divisible by 6. This confirms that our earlier reasoning is correct and that no other option satisfies both rules simultaneously. Why Other Options Are Wrong: Option (a): L=3 makes the number odd, so it fails divisibility by 2. Option (c): The digit sum becomes 37, which is not a multiple of 3, so it fails divisibility by 3. Option (d): L=3 again makes the number odd, so it fails divisibility by 2. Common Pitfalls: A common mistake is to apply only the divisibility by 3 rule and forget that divisibility by 6 also requires divisibility by 2. Another error is computing the digit sum incorrectly or misreading the digits in the number. Some students also mistakenly check K and L independently instead of examining their combined effect on the digit sum. Careful and systematic checking of both conditions avoids these errors. Final Answer: The only pair of digits that makes 78K928L divisible by 6 is K = 1 and L = 4.

Question 5

The sum of one-half, one-third and one-fourth of a positive number is greater than the number itself by 22. What is the value of this number?

Accepted Answer

Introduction / Context: This problem involves forming and solving a linear equation using fractions of an unknown number. Such questions are standard in aptitude tests to check whether a learner can translate a verbal description involving fractions into an algebraic equation, and then solve it correctly. Here, three fractional parts of a number are added, and the result is said to exceed the original number by a fixed amount. Our goal is to find that number using basic algebra and fraction operations. Given Data / Assumptions: Let the unknown positive number be represented by N. One-half of the number is N/2. One-third of the number is N/3. One-fourth of the number is N/4. The sum of these three fractional parts exceeds the original number N by 22. Concept / Approach: We convert the English statement into an equation. The phrase "sum of one-half, one-third and one-fourth of a number exceeds the number by 22" can be written as N/2 + N/3 + N/4 = N + 22. Then we simplify by finding a common denominator for the fractions and reducing the equation to a simple linear equation in N. Finally, we solve for N and verify that the result matches one of the provided options. Step-by-Step Solution: Step 1: Let the number be N. Step 2: Translate the condition into an equation: N/2 + N/3 + N/4 = N + 22. Step 3: Find the least common denominator of 2, 3 and 4, which is 12. Step 4: Express each term with denominator 12: N/2 = 6N/12, N/3 = 4N/12, and N/4 = 3N/12. Step 5: Add the fractions: N/2 + N/3 + N/4 = (6N + 4N + 3N) / 12 = 13N/12. Step 6: The equation becomes 13N/12 = N + 22. Step 7: Multiply both sides by 12 to clear the denominator: 13N = 12N + 264. Step 8: Subtract 12N from both sides: N = 264. Step 9: Therefore, the required number is 264. Verification / Alternative check: To verify, substitute N = 264 back into the original statement. One-half of 264 is 132, one-third is 88, and one-fourth is 66. Their sum is 132 + 88 + 66 = 286. The original number is 264, and 286 minus 264 equals 22. This matches the given condition that the sum exceeds the number by 22, so N = 264 is correct. Why Other Options Are Wrong: Option (b) 284: Fractions would sum to a value that does not exceed 284 by exactly 22. Option (c) 215: The difference between the sum of fractions and 215 is not 22. Option (d) 302: Again, the resulting difference is not exactly 22, so it does not satisfy the condition. Common Pitfalls: Learners sometimes misinterpret the phrase "exceeds the number by 22" and set up the equation with 22 on the wrong side or with the wrong sign. Others add the fractions incorrectly by not using a common denominator or by making arithmetic errors in the numerator. Writing and simplifying the equation carefully helps avoid these mistakes. Final Answer: The value of the number is 264.

Question 6

Find the smallest positive integer by which 16200 must be multiplied so that the resulting product becomes a perfect cube.

Accepted Answer

Introduction / Context: This question checks understanding of prime factorization and the conditions required for a number to be a perfect cube. In aptitude and number theory problems, converting numbers into their prime factors is a powerful technique, especially when determining whether they are perfect squares, cubes, or higher powers. Here, we are given 16200 and asked for the least integer that, when multiplied with it, turns the product into a perfect cube. That means we want all prime exponents in the factorization to be multiples of 3. Given Data / Assumptions: Given number: 16200. We need to multiply 16200 by a positive integer k. The product 16200 * k must be a perfect cube. A perfect cube has all prime factor exponents as multiples of 3. Concept / Approach: We first factorize 16200 into its prime factors. After obtaining the exponents of each prime, we check which exponents are not multiples of 3. For each such prime, we find the smallest additional power needed to make the exponent a multiple of 3. The required multiplier k is then formed by multiplying these extra prime factors together. This ensures that the combined exponent for each prime in 16200 * k is divisible by 3, making the product a perfect cube. Step-by-Step Solution: Step 1: Factorize 16200. Step 2: Note that 16200 = 162 * 100. Step 3: Factorize 162 as 2 * 81 = 2 * 3^4. Step 4: Factorize 100 as 2^2 * 5^2. Step 5: Combine the factors: 16200 = 2^3 * 3^4 * 5^2. Step 6: For a perfect cube, the exponents of all primes must be multiples of 3. Step 7: The exponent of 2 is 3, already a multiple of 3. Step 8: The exponent of 3 is 4. To reach the next multiple of 3 (which is 6), we need 2 more factors of 3, that is 3^2. Step 9: The exponent of 5 is 2. To reach the next multiple of 3 (which is 3), we need one more factor of 5, that is 5^1. Step 10: Therefore, the required multiplier k is 3^2 * 5 = 9 * 5 = 45. Verification / Alternative check: Multiply 16200 by 45: the combined factorization is 2^3 * 3^4 * 5^2 * 3^2 * 5^1 = 2^3 * 3^6 * 5^3. Now each exponent is a multiple of 3: 3 for base 2, 6 for base 3, and 3 for base 5. Therefore, the product is indeed a perfect cube of the form (2 * 3^2 * 5)^3. This confirms that 45 is the correct smallest multiplier. Why Other Options Are Wrong: Option (b) 48: Its factors do not fix all exponents to multiples of 3 with the smallest possible product, and it is larger than necessary. Option (c) 360: This is much larger and not the least multiplier, even though it may also yield a perfect cube. Option (d) 36: This changes the exponents but does not make all of them multiples of 3, so the product is not a perfect cube. Common Pitfalls: Common errors include incorrect prime factorization or forgetting that exponents must be multiples of 3 for a cube. Some learners also choose a multiplier that works but is not the smallest possible one. Focusing on each prime factor exponent and adjusting it to the nearest higher multiple of 3 helps avoid mistakes and ensures minimality. Final Answer: The least integer that must be multiplied with 16200 is 45.

Question 7

The number 96 is divided into two positive parts such that one-seventh of the first part is equal to one-ninth of the second part. What is the smaller of the two parts?

Accepted Answer

Introduction / Context: This question is a classic example of a linear equation word problem involving the division of a total quantity into two parts with a relation between them. Such problems commonly appear in aptitude exams and help test whether candidates can convert a verbal condition into algebraic equations and solve them systematically. Here, the total is 96, and we are told that certain fractions of each part are equal. We must determine the smaller part using basic algebra. Given Data / Assumptions: Total sum of two parts is 96. Let the two parts be x and y, both positive. x + y = 96. One-seventh of the first part equals one-ninth of the second part: x/7 = y/9. We are asked to find the smaller part. Concept / Approach: We will model the problem using two equations: one from the total (x + y = 96) and another from the fractional relationship (x/7 = y/9). The second equation allows us to express one variable in terms of the other. Substituting this expression into the first equation results in a single-variable linear equation that we can solve. Once x and y are found, we simply pick the smaller value. Step-by-Step Solution: Step 1: Let the two parts be x and y. Step 2: The total is 96, so x + y = 96. Step 3: From the given relation x/7 = y/9, cross multiply to obtain 9x = 7y. Step 4: Express y in terms of x: y = 9x / 7. Step 5: Substitute y = 9x / 7 into x + y = 96. Step 6: This gives x + 9x / 7 = 96. Step 7: Combine like terms: (7x + 9x) / 7 = 96, so 16x / 7 = 96. Step 8: Multiply both sides by 7: 16x = 96 * 7. Step 9: Compute 96 * 7 = 672, hence 16x = 672. Step 10: Divide both sides by 16 to get x = 672 / 16 = 42. Step 11: Find y from x + y = 96: y = 96 - 42 = 54. Step 12: The smaller part is x = 42. Verification / Alternative check: Check the fractional condition: one-seventh of the first part is 42 / 7 = 6, and one-ninth of the second part is 54 / 9 = 6. Since both are equal, the relationship is satisfied. Also, their sum is 42 + 54 = 96, matching the given total. This confirms that the values are correct and the smaller part is indeed 42. Why Other Options Are Wrong: Option (b) 54: This is the larger part, not the smaller one, although it still fits the equations when paired with 42. Option (c) 46: Using 46 as one part does not give integer values that satisfy both conditions. Option (d) 58: Similarly, 58 cannot produce consistent values for the fractional relationship and the total of 96. Common Pitfalls: Learners sometimes confuse which part is smaller or make errors while cross multiplying the equation x/7 = y/9. Another frequent mistake is mishandling the substitution into the total equation and performing arithmetic incorrectly. Carefully writing each step and rechecking the operations helps to avoid such issues. Final Answer: The smaller of the two parts is 42.

Question 8

A certain number gives a quotient of 271 and a remainder of 0 when divided by 96. If the same number is divided by 95, what remainder will be obtained?

Accepted Answer

Introduction / Context: Remainder and quotient problems like this one are fundamental in number system questions. They test understanding of the division algorithm and how to manipulate a number when the divisor changes. Instead of working with the full division every time, we can use algebraic representations of the number in terms of divisor, quotient, and remainder. Here, we first reconstruct the number from one division and then redivide it by a different divisor to find the new remainder. Given Data / Assumptions: When the number N is divided by 96, quotient is 271 and remainder is 0. This means N is exactly divisible by 96. We then divide the same N by 95. We must find the remainder when N is divided by 95. Concept / Approach: The division algorithm states that for a divisor d, quotient q and remainder r, the number N can be written as N = d * q + r. Since the remainder is 0 in the first division, N is exactly equal to 96 * 271. Once we compute N, we apply the division algorithm again with divisor 95 to determine the remainder. Instead of performing long division manually, we can use simple multiplication and modular reasoning to find the remainder quickly. Step-by-Step Solution: Step 1: Let the number be N. Step 2: From the given information, N = 96 * 271 + 0, so N = 96 * 271. Step 3: Compute N: 96 * 271 = 26016. Step 4: Now divide N by 95 and find the remainder. Step 5: Use the division idea: N = 95 * q + r, where r is the remainder and 0 ≤ r < 95. Step 6: Compute 26016 / 95. We find the integer part of the quotient first. Step 7: 95 * 273 = 25935 and 95 * 274 = 26030. Step 8: Since 26030 is greater than 26016, the quotient is 273. Step 9: Compute the remainder: r = 26016 − 95 * 273 = 26016 − 25935 = 81. Step 10: Hence, the remainder when N is divided by 95 is 81. Verification / Alternative check: As an alternative, we can note that 96 is 1 more than 95. Therefore N = 96 * 271 can be written as (95 + 1) * 271 = 95 * 271 + 271. When divided by 95, 95 * 271 gives remainder 0, and 271 gives remainder 271 mod 95. Since 95 * 2 = 190 and 95 * 3 = 285, we have 271 − 190 = 81. This again shows that the remainder is 81, confirming our previous calculation. Why Other Options Are Wrong: Option (a) 87: Does not match the remainder derived from either direct division or modular reasoning. Option (b) 59: Also inconsistent with the remainder calculation of 81. Option (c) 92: Greater than the correct remainder and does not satisfy the division relationship for N. Common Pitfalls: Some learners mistakenly try to divide 271 by 95 instead of reconstructing N. Others forget that the number is exactly divisible by 96 and might mis-handle the remainder. Arithmetic mistakes in multiplication or subtraction steps can also lead to wrong remainders. Working systematically and, where possible, using the representation N = (95 + 1) * 271 helps reduce errors. Final Answer: The remainder when the number is divided by 95 is 81.

Question 9

The ratio of two positive numbers is 3 : 5 and their arithmetic mean (average) is 48. What is the product of these two numbers?

Accepted Answer

Introduction / Context: This is a standard ratio and average problem from quantitative aptitude. The question combines the concept of expressing numbers in terms of a common ratio multiplier with the concept of arithmetic mean. By using the given ratio and the average, we can find the actual values of the two numbers and then compute their product. Such questions test algebraic representation skills and comfort with ratios. Given Data / Assumptions: The ratio of the two positive numbers is 3 : 5. The average (arithmetic mean) of the two numbers is 48. We are asked to find the product of these two numbers. Both numbers are positive real numbers, and in this context they will turn out to be integers. Concept / Approach: If the ratio of two numbers is 3 : 5, we can represent them as 3k and 5k, where k is a common positive multiplier. The arithmetic mean of these two numbers is given as 48. The mean of 3k and 5k is (3k + 5k) / 2 = 4k. Equating this to 48 allows us to find k. Once k is known, we compute the actual numbers and then multiply them to obtain the product. This approach uses simple algebra and avoids guesswork. Step-by-Step Solution: Step 1: Let the two numbers be 3k and 5k, since their ratio is 3 : 5. Step 2: The arithmetic mean of 3k and 5k is (3k + 5k) / 2. Step 3: Simplify the numerator: 3k + 5k = 8k, so the mean is 8k / 2 = 4k. Step 4: We are told the average is 48, so set 4k = 48. Step 5: Solve for k: k = 48 / 4 = 12. Step 6: The two numbers are 3k = 3 * 12 = 36 and 5k = 5 * 12 = 60. Step 7: The product of these numbers is 36 * 60. Step 8: Compute the product: 36 * 60 = 2160. Step 9: Thus, the required product is 2160. Verification / Alternative check: We can verify by checking both the ratio and the average. The ratio 36 : 60 simplifies by dividing numerator and denominator by 12, giving 3 : 5, which matches the given ratio. The arithmetic mean of 36 and 60 is (36 + 60) / 2 = 96 / 2 = 48, which matches the given average. Since both conditions hold, the numbers 36 and 60 are correct, and their product of 2160 is confirmed. Why Other Options Are Wrong: Option (a) 3365, option (c) 1854, and option (d) 1632 do not equal 36 * 60. If we tried to back-calculate from these products, we would not get numbers in the ratio 3 : 5 with an average of 48. Therefore, these options do not satisfy the given conditions and are incorrect. Common Pitfalls: Some learners forget to divide by 2 when using the average formula or incorrectly set 3k + 5k equal to 48 instead of dividing by 2. Another typical error is to miscalculate the product after correctly finding the numbers. Being careful with the mean formula and performing arithmetic step by step helps avoid such mistakes. Final Answer: The product of the two numbers is 2160.

Question 10

The difference between two numbers is 25, and the smaller number is equal to one-sixth of the larger number. What is the value of the smaller number?

Accepted Answer

Introduction / Context: This is a straightforward algebraic word problem involving two numbers and their relationship. Many aptitude tests include such questions to assess basic equation formulation skills. We are told the difference between two numbers, and also that the smaller is a fixed fraction of the larger. Using this information, we can set up a simple system of equations in two variables and solve for the unknown numbers. Here, we are particularly interested in finding the smaller number. Given Data / Assumptions: There are two numbers, one larger and one smaller. The difference between the two numbers is 25. The smaller number is one-sixth of the larger number. We need to determine the smaller number. Concept / Approach: Let the larger number be L and the smaller number be S. We are given two conditions: L − S = 25 and S = (1/6) * L. By substituting S from the second equation into the first, we can form a single equation in L. Once we find L, we can use S = L / 6 to find the smaller number. This method is a standard approach for solving linear equations with two variables when we have exactly two relationships. Step-by-Step Solution: Step 1: Let the larger number be L and the smaller number be S. Step 2: From the problem, the difference between the numbers is 25, so L − S = 25. Step 3: Also, the smaller number is one-sixth of the larger number, so S = L / 6. Step 4: Substitute S = L / 6 into the equation L − S = 25. Step 5: This gives L − L / 6 = 25. Step 6: Combine like terms: L − L / 6 = (6L / 6 − L / 6) = 5L / 6. Step 7: So 5L / 6 = 25. Step 8: Multiply both sides by 6 to get 5L = 150. Step 9: Divide both sides by 5: L = 30. Step 10: Compute the smaller number S = L / 6 = 30 / 6 = 5. Verification / Alternative check: Check the conditions using L = 30 and S = 5. The difference L − S is 30 − 5 = 25, which matches the given difference. Also, S is one-sixth of L because 1/6 of 30 is 5. Since both statements in the problem are satisfied by these values, the solution is correct and the smaller number is 5. Why Other Options Are Wrong: Option (a) 10: If S were 10, L would be 60, giving a difference of 50, not 25. Option (c) 50: This is too large to be the smaller number given the difference and ratio conditions. Option (d) 25: If S were 25, L would be 150 with a difference of 125, not 25, so it fails both conditions. Common Pitfalls: A common mistake is to swap the roles of the larger and smaller numbers or incorrectly set up the equation as S − L = 25. Another frequent error is to mis-handle the fraction one-sixth when substituting into the equation. Carefully defining variables and systematically performing algebraic steps helps avoid such mistakes. Final Answer: The value of the smaller number is 5.

Question 11

A mathematics textbook costs Rs 2 more than a literature textbook. If the total cost of 5 literature textbooks is Rs 38 more than the total cost of 3 mathematics textbooks, what is the cost of one literature textbook?

Accepted Answer

Introduction / Context: This question is a typical linear equation word problem involving prices of two different types of textbooks. It tests basic algebra skills in setting up equations from real-life style statements. We are given a relationship between the cost of a mathematics textbook and a literature textbook, and also a comparison between the total costs of multiple copies of each. The goal is to find the price of one literature textbook using these relationships. Given Data / Assumptions: Let the cost of one literature textbook be L rupees. The cost of one mathematics textbook is 2 rupees more, so it is L + 2. The total cost of 5 literature textbooks is 38 rupees more than the total cost of 3 mathematics textbooks. We must find the value of L. Concept / Approach: We express both total costs in terms of L, then use the given difference of 38 rupees between these totals to form a linear equation. Specifically, the cost of 5 literature textbooks is 5L, and the cost of 3 mathematics textbooks is 3(L + 2). Given that 5L is 38 more than the latter, we write 5L = 3(L + 2) + 38. Solving this equation gives the price of one literature textbook. This approach is standard for word problems involving comparative costs. Step-by-Step Solution: Step 1: Let the cost of one literature textbook be L rupees. Step 2: Then the cost of one mathematics textbook is L + 2 rupees. Step 3: The total cost of 5 literature textbooks is 5L. Step 4: The total cost of 3 mathematics textbooks is 3(L + 2). Step 5: According to the problem, 5 literature textbooks cost Rs 38 more than 3 mathematics textbooks. Step 6: Form the equation: 5L = 3(L + 2) + 38. Step 7: Expand the right side: 3(L + 2) = 3L + 6, so the equation becomes 5L = 3L + 6 + 38. Step 8: Simplify: 5L = 3L + 44. Step 9: Subtract 3L from both sides: 2L = 44. Step 10: Divide both sides by 2: L = 22. Step 11: Therefore, one literature textbook costs Rs 22. Verification / Alternative check: Check using L = 22. Then one mathematics textbook costs 22 + 2 = 24 rupees. The cost of 5 literature textbooks is 5 * 22 = 110 rupees. The cost of 3 mathematics textbooks is 3 * 24 = 72 rupees. The difference is 110 − 72 = 38 rupees, which matches the given condition. This confirms that the cost of one literature textbook is correctly found as 22 rupees. Why Other Options Are Wrong: Option (a) 44: If literature cost were 44, numbers would not satisfy the given 38 rupee difference. Option (c) 24: This would make mathematics textbooks cost 26, and totals would not differ by 38. Option (d) 11: This would produce totals that differ by much less than 38 rupees. Common Pitfalls: Common mistakes include reversing the relationship (for example, assuming mathematics is cheaper than literature), misinterpreting the phrase "cost 38 rupees more," or forgetting to add 2 correctly when computing mathematics textbooks costs. Being careful with the language and assigning variables clearly helps prevent these issues. Final Answer: The cost of one literature textbook is Rs 22.

Question 12

What is the largest four-digit number that is exactly divisible by 88 without leaving any remainder?

Accepted Answer

Introduction / Context: This problem is about finding the greatest four-digit number that is exactly divisible by a given divisor, here 88. Such questions appear frequently in number system topics and are useful to test understanding of multiples and division. The key idea is to find the largest multiple of 88 that does not exceed the largest four-digit number, which is 9999. Given Data / Assumptions: The largest four-digit number is 9999. We need a number of the form 88 * k that is less than or equal to 9999. The number must be exactly divisible by 88, meaning remainder 0. We must find the largest such multiple. Concept / Approach: To find the largest four-digit number divisible by 88, we divide 9999 by 88 and consider the integer part of the quotient. That quotient multiplied by 88 gives the largest multiple of 88 not exceeding 9999. We do not need the remainder, except to confirm that 9999 itself is not divisible by 88. This approach avoids checking each option individually and directly uses division to locate the required multiple. Step-by-Step Solution: Step 1: Take the largest four-digit number, which is 9999. Step 2: Divide 9999 by 88 to find how many full multiples of 88 fit into 9999. Step 3: Perform the division: 9999 / 88 gives a quotient of 113 and some remainder. Step 4: Compute 88 * 113 = 9944. Step 5: Next multiple would be 88 * 114 = 10032, which exceeds 9999 and is therefore not a four-digit number. Step 6: Hence, 9944 is the largest four-digit number divisible by 88. Step 7: Verify that 9944 is indeed divisible by 88 by dividing: 9944 / 88 = 113 exactly, with remainder 0. Verification / Alternative check: We can also look at the options. Option (a) 8888 divided by 88 equals 101, which is a valid multiple but smaller than 9944. Option (b) 9999 divided by 88 gives a non-integer quotient with a remainder, so it is not exactly divisible. Option (d) 9988 divided by 88 equals 113 with remainder ... wait, computing clearly shows that 88 * 113 is 9944, so 9988 is not exactly divisible by 88. Therefore, among the options, only 9944 is both a four-digit number and the largest that is a multiple of 88 below 9999. Why Other Options Are Wrong: Option (a) 8888: A correct multiple of 88 but not the largest possible four-digit multiple. Option (b) 9999: Not divisible by 88, as the division leaves a remainder. Option (d) 9988: Not equal to 88 times an integer, so it is not exactly divisible by 88. Common Pitfalls: A common mistake is to choose 9999 or the numerically largest option without checking divisibility. Some students also miscalculate 88 times the quotient or misread the remainder. Using accurate division and verifying one or two key multiples helps ensure the correct result. Final Answer: The largest four-digit number exactly divisible by 88 is 9944.

Question 13

A three-digit number has digits whose sum is 10. The middle digit is equal to the sum of the other two digits, and the number increases by 99 when its digits are reversed. What is this three-digit number?

Accepted Answer

Introduction / Context: This question combines digit sums and digit reversal concepts in a three-digit number. Problems of this type are common in number system sections to test reasoning about place values, digit relationships and simple algebra. We are given conditions on the sum of digits, a relationship between the digits, and the effect of reversing the digits on the value of the number. Using these clues, we can form equations to determine the exact number. Given Data / Assumptions: The number is a three-digit number with digits a, b and c. The sum of the digits is 10, so a + b + c = 10. The middle digit b equals the sum of the other two digits, so b = a + c. Reversing the digits increases the number by 99. We must find the original three-digit number. Concept / Approach: Let the three-digit number be written as 100a + 10b + c. Reversing the digits gives 100c + 10b + a. We are told that the reversed number is 99 more than the original number. This gives an equation linking a and c. Along with the condition on the digit sum and the relation b = a + c, we can solve for a, b and c. Finally, we match the resulting number against the options. Step-by-Step Solution: Step 1: Let the original number be 100a + 10b + c. Step 2: The sum of the digits is a + b + c = 10. Step 3: The middle digit equals the sum of the others, so b = a + c. Step 4: Substitute b into the sum: a + (a + c) + c = 10, so 2a + 2c = 10, giving a + c = 5. Step 5: Thus b = a + c = 5, so the middle digit is 5. Step 6: When the digits are reversed, the number becomes 100c + 10b + a. Step 7: It is given that the reversed number is 99 more than the original, so (100c + 10b + a) − (100a + 10b + c) = 99. Step 8: Simplify the left side: (100c − 100a) + (a − c) = 99c − 99a = 99(c − a). Step 9: So 99(c − a) = 99, which implies c − a = 1. Step 10: We already have a + c = 5 and now c − a = 1. Adding the two equations gives 2c = 6, so c = 3, and then a = 2. Step 11: The digits are a = 2, b = 5, c = 3, giving the number 253. Verification / Alternative check: Check the conditions for 253. The sum of the digits is 2 + 5 + 3 = 10. The middle digit 5 is indeed equal to 2 + 3. When we reverse the digits, we get 352. The difference 352 − 253 is 99, exactly as stated in the problem. This confirms that 253 satisfies all the given conditions and is therefore the correct answer. Why Other Options Are Wrong: Option (b) 263: The sum of digits is 11, not 10. Option (c) 273: The middle digit 7 is not equal to 2 + 3, and the reversal condition fails. Option (d) 283: The digit relationships and the 99 difference condition are not satisfied. Common Pitfalls: Learners may forget that b must be exactly the sum of a and c or may mis-handle the place value expressions for the original and reversed numbers. It is also easy to make arithmetic errors when simplifying 99(c − a). Writing each equation carefully and double-checking arithmetic helps ensure the correct result. Final Answer: The three-digit number is 253.

Question 14

If two painters working together can paint two rooms in two hours, how many painters working at the same constant rate are required to paint 18 rooms in 6 hours?

Accepted Answer

Introduction / Context: This problem is a classic work and time question. It involves understanding how work rate scales with the number of workers and the time available. By determining the rate at which painters complete rooms, we can then scale up to find how many painters are required to complete a larger amount of work within a given time. These questions help in understanding proportional reasoning and basic work rate concepts. Given Data / Assumptions: Two painters can paint two rooms in two hours. All painters work at the same constant rate. We need to paint 18 rooms in 6 hours. We must find the number of painters required. Concept / Approach: The key idea is to compute the work rate per painter per hour. First, calculate the total painter-hours used in the initial situation and divide by the number of rooms completed to get rooms per painter-hour. Then, determine the total number of painter-hours needed to paint 18 rooms and divide this by the time allowed to find how many painters are required. This approach treats work as proportional to the product of number of workers and time. Step-by-Step Solution: Step 1: Two painters paint two rooms in two hours. Step 2: Total painter-hours used = number of painters * time = 2 * 2 = 4 painter-hours. Step 3: In these 4 painter-hours, 2 rooms are completed. Step 4: Therefore, the rate is 2 rooms / 4 painter-hours = 0.5 rooms per painter-hour. Step 5: Now we need to paint 18 rooms. Step 6: Total painter-hours required for 18 rooms = 18 / 0.5 = 36 painter-hours. Step 7: The time available is 6 hours. Step 8: Number of painters needed = total painter-hours / time = 36 / 6 = 6 painters. Step 9: Hence, 6 painters can complete 18 rooms in 6 hours at the same rate. Verification / Alternative check: With 6 painters working for 6 hours, total painter-hours are 6 * 6 = 36. At a rate of 0.5 rooms per painter-hour, the number of rooms completed is 36 * 0.5 = 18 rooms. This exactly matches the target number of rooms, confirming that 6 painters are sufficient and correctly calculated. Why Other Options Are Wrong: Option (a) 2 painters: They would provide only 2 * 6 = 12 painter-hours, resulting in 12 * 0.5 = 6 rooms, which is too few. Option (c) 4 painters: They provide 4 * 6 = 24 painter-hours, giving 24 * 0.5 = 12 rooms, still less than 18. Option (d) None: This suggests no option is correct, but we have shown that 6 painters works exactly, so this is incorrect. Common Pitfalls: A common mistake is to assume a direct multiple without computing the rate or to misinterpret how to scale workers and time. Some learners may incorrectly divide rooms by painters or time without relating everything through a consistent rate. Clearly computing painter-hours and using them as a unit of work helps avoid such confusion. Final Answer: The required number of painters is 6 painters.

Question 15

A person walks at only 6/7 of his usual walking speed and therefore reaches his destination 30 minutes late. What is his usual time to cover the same distance?

Accepted Answer

Introduction / Context: This is a classical speed and time problem where a change in speed causes a change in time taken. The key relationship is that for a fixed distance, speed and time are inversely proportional. When the person walks slower than usual, the time taken increases, and we are given this increase as 30 minutes. Using the ratio of speeds, we can find the ratio of times and then determine the usual time for the journey. Given Data / Assumptions: The person walks at 6/7 of his usual speed. Because of this reduced speed, he is 30 minutes late. The distance remains the same in both scenarios. We must find his usual time to cover the distance. Concept / Approach: For a fixed distance, speed * time is constant, so time is inversely proportional to speed. If the usual speed is S and usual time is T, then distance = S * T. When speed changes to 6S/7, the new time T' must satisfy S * T = (6S/7) * T', so T' = (7/6) * T. The extra time taken is T' − T, which is given as 30 minutes. Setting (7/6) * T − T equal to 30 minutes allows us to solve for T and find the usual travel time. Step-by-Step Solution: Step 1: Let the usual speed be S and the usual time be T hours. Step 2: The distance is D = S * T. Step 3: At the reduced speed, the speed becomes 6S / 7. Step 4: Let the new time taken be T' hours. Then D = (6S / 7) * T'. Step 5: Since D is the same, S * T = (6S / 7) * T'. Step 6: Cancel S on both sides to get T = (6 / 7) * T'. Step 7: Rearranging gives T' = (7 / 6) * T. Step 8: The person is 30 minutes late, so T' − T = 30 minutes. Step 9: Express 30 minutes in hours: 30 minutes = 0.5 hours. Step 10: Substitute T' = (7 / 6) * T: (7 / 6) * T − T = 0.5. Step 11: Simplify the left side: (7T / 6 − 6T / 6) = T / 6. Step 12: So T / 6 = 0.5, giving T = 0.5 * 6 = 3 hours. Verification / Alternative check: With a usual time of 3 hours, at reduced speed the time becomes T' = (7 / 6) * 3 = 3.5 hours, that is 3 hours and 30 minutes. The difference is 30 minutes, exactly matching the given delay. This confirms that the usual time is 3 hours. Why Other Options Are Wrong: Option (a) 2 hrs 50 min: This value would not produce an exact 30 minute delay under the given speed ratio. Option (b) 3 hrs 25 min: Using this as T would yield a different delay than 30 minutes. Option (d) 2 hrs 5 min: This is significantly smaller and inconsistent with the specified ratio and delay. Common Pitfalls: A common error is to treat speed and time as directly proportional instead of inversely proportional, leading to wrong equations. Some learners also forget to convert 30 minutes into hours or mis-handle the fractions involved in (7 / 6). Carefully setting up the proportional relationship and using consistent units helps avoid mistakes. Final Answer: The person's usual time to cover the distance is 3 hours.

Question 16

Find the number such that when it is multiplied by 15, the result is greater than the original number by 56.

Accepted Answer

Introduction / Context: This question is a simple algebraic equation problem involving an unknown number and a relationship between its multiple and itself. The statement describes how much bigger the product of the number and 15 is compared to the original number. From this, we can form a linear equation and solve for the unknown. Such questions are meant to test basic equation setup skills in aptitude exams. Given Data / Assumptions: Let the unknown number be x. When x is multiplied by 15, the result is 56 more than x. We need to determine the value of x. Concept / Approach: The phrase "when it is multiplied by 15, the result is 56 greater than the original number" can be expressed algebraically as 15x = x + 56. This is a simple one-variable linear equation. Solving this equation will give the value of x directly. Once x is found, we check whether it matches one of the options and whether it satisfies the original condition stated in the problem. Step-by-Step Solution: Step 1: Let the required number be x. Step 2: According to the problem, when x is multiplied by 15, we get a value that is 56 more than x. Step 3: Translate this into an equation: 15x = x + 56. Step 4: Subtract x from both sides to gather like terms: 15x − x = 56. Step 5: This simplifies to 14x = 56. Step 6: Divide both sides by 14: x = 56 / 14. Step 7: Compute the division: 56 / 14 = 4. Step 8: Therefore, the required number is 4. Verification / Alternative check: Check the condition with x = 4. The product when multiplied by 15 is 15 * 4 = 60. The original number is 4. The difference 60 − 4 = 56, which matches the statement that the product is 56 greater than the original number. This confirms that x = 4 is correct. Why Other Options Are Wrong: Option (a) 6: If x = 6, then 15x = 90 and 90 − 6 = 84, not 56. Option (c) 3: If x = 3, then 15x = 45 and 45 − 3 = 42, not 56. Option (d) 12: If x = 12, then 15x = 180 and 180 − 12 = 168, not 56. Common Pitfalls: Some learners misinterpret the phrase and write 15x + 56 = x or x = 15x + 56, which reverses or distorts the relationship. Others may make arithmetic errors when subtracting x from 15x. Carefully translating the wording into a correct equation and handling the subtraction correctly avoids these mistakes. Final Answer: The required number is 4.

Question 17

What is the common ratio of the geometric sequence 4, 2, 1, 0.5, 0.25, 0.125, ... ?

Accepted Answer

Introduction / Context: This question checks knowledge of geometric sequences, specifically the concept of the common ratio. In a geometric sequence, each term is obtained by multiplying the previous term by a constant factor called the common ratio. Recognizing this ratio is essential for working with geometric progressions in many aptitude and mathematics problems. The given sequence is 4, 2, 1, 0.5, 0.25, 0.125, and so on. Given Data / Assumptions: The sequence is 4, 2, 1, 0.5, 0.25, 0.125, ... The terms are in a geometric progression. We must find the common ratio r. The common ratio is constant between any consecutive pair of terms. Concept / Approach: For a geometric sequence, the common ratio r can be found by dividing any term by its previous term, provided we choose consecutive nonzero terms. That is, r = T(n+1) / T(n). In this sequence, we can use 2 / 4, or 1 / 2, or 0.5 / 1, etc. All these ratios should be equal. By evaluating this ratio and confirming it across several consecutive term pairs, we can identify the correct common ratio. Step-by-Step Solution: Step 1: Consider the first two terms, 4 and 2. Step 2: Compute the ratio of the second term to the first term: r = 2 / 4 = 0.5. Step 3: Check this ratio with the next pair of terms, 2 and 1. Step 4: Compute 1 / 2 = 0.5, which matches the earlier value. Step 5: Check one more pair, 1 and 0.5. Step 6: Compute 0.5 / 1 = 0.5, again the same value. Step 7: For 0.5 and 0.25, compute 0.25 / 0.5 = 0.5. Step 8: Since the ratio is consistently 0.5, the common ratio of this geometric sequence is 0.5. Verification / Alternative check: Starting from the first term 4 and using r = 0.5, the terms are generated as follows: second term = 4 * 0.5 = 2, third term = 2 * 0.5 = 1, fourth term = 1 * 0.5 = 0.5, fifth term = 0.5 * 0.5 = 0.25, and sixth term = 0.25 * 0.5 = 0.125. These match the given sequence exactly, verifying that the common ratio is correctly identified as 0.5. Why Other Options Are Wrong: Option (b) −1: Would cause the signs of terms to alternate between positive and negative, which is not observed here. Option (c) 1.5: This would make each term larger than the previous, whereas in the given sequence each term is smaller. Option (d) −0.5: This would again alternate signs, which is not the case in the given sequence. Common Pitfalls: A common mistake is to subtract terms instead of dividing, which would lead to confusing this with an arithmetic progression. Another error is to divide in the wrong order (previous over next), which still yields the reciprocal but can cause confusion about the ratio. Always take the ratio as next term divided by previous term for geometric sequences. Final Answer: The common ratio of the sequence is 0.5.

Question 18

A number is doubled and then 9 is added. If the resulting value is multiplied by 3, the final result is 75. What is the original number?

Accepted Answer

Introduction / Context: This problem is a simple linear equation word problem involving a sequence of operations performed on an unknown number. The number is doubled, increased by a fixed amount, and then the result is multiplied by 3 to give 75. These kinds of questions are designed to test whether learners can translate multi-step verbal descriptions into algebraic equations and then solve them correctly. Given Data / Assumptions: Let the original number be x. The number is doubled to give 2x. Then 9 is added, giving 2x + 9. This result is multiplied by 3, giving 3(2x + 9). The final result equals 75. We need to find the value of x. Concept / Approach: We systematically reflect each step in the operations as algebraic expressions. Doubling corresponds to multiplying by 2, then adding 9 is a simple addition, and multiplying the result by 3 corresponds to distributing 3 across the expression. Setting the final expression equal to 75 yields a solvable linear equation. After solving for x, we verify by reapplying the steps to check whether the final result is indeed 75. Step-by-Step Solution: Step 1: Let the original number be x. Step 2: The number is doubled, giving 2x. Step 3: Add 9 to this result to obtain 2x + 9. Step 4: The problem says that if this result is multiplied by 3, the final answer is 75, so write 3(2x + 9) = 75. Step 5: Expand the left side: 3 * 2x + 3 * 9 = 6x + 27. Step 6: So we have the equation 6x + 27 = 75. Step 7: Subtract 27 from both sides to isolate the term with x: 6x = 75 − 27. Step 8: Compute the difference 75 − 27 = 48, so 6x = 48. Step 9: Divide both sides by 6: x = 48 / 6 = 8. Step 10: Hence, the original number is 8. Verification / Alternative check: Starting with x = 8, double it to get 16. Add 9 to obtain 25. Multiply 25 by 3 to get 75. This matches the final result given in the problem, confirming that our solution is correct and that the original number is 8. Why Other Options Are Wrong: Option (a) 2: Doubling gives 4, plus 9 is 13, and 13 * 3 = 39, not 75. Option (b) 4: Doubling gives 8, plus 9 is 17, and 17 * 3 = 51, not 75. Option (c) 6: Doubling gives 12, plus 9 is 21, and 21 * 3 = 63, not 75. Common Pitfalls: Learners may misread the phrase and incorrectly interpret the order of operations, for example adding 9 after multiplying by 3. It is important to follow the steps in the exact order described: double, then add 9, then multiply by 3. Another common error arises when expanding 3(2x + 9) incorrectly. Careful reading and correct expansion prevent such mistakes. Final Answer: The original number is 8.

Question 19

For a certain fraction, if 1 is added to its denominator the value of the fraction becomes 1/2, and if 1 is added to its numerator the value of the fraction becomes 1. What is the original fraction?

Accepted Answer

Introduction / Context: This question tests understanding of algebraic representation of fractions and how changes to the numerator or denominator affect the fraction's value. We are given two different transformations of the same fraction and the resulting values. By modeling these conditions using algebra, we can form two equations and solve for the numerator and denominator of the original fraction. Such fraction puzzles are common in aptitude and school-level mathematics. Given Data / Assumptions: Let the original fraction be x / y. When 1 is added to the denominator, the fraction becomes 1/2, so x / (y + 1) = 1/2. When 1 is added to the numerator, the fraction becomes 1, so (x + 1) / y = 1. We must find the original fraction x / y. Concept / Approach: We translate the two conditions into algebraic equations. From x / (y + 1) = 1/2, cross multiplication gives 2x = y + 1. From (x + 1) / y = 1, we obtain x + 1 = y. These two equations in x and y can be solved simultaneously. Once we find x and y, we can express the original fraction and match it with the options given. This approach uses elementary algebra and reasoning with fractions. Step-by-Step Solution: Step 1: Let the original fraction be x / y. Step 2: From the first condition, x / (y + 1) = 1/2. Step 3: Cross multiply: 2x = y + 1. Call this Equation (1). Step 4: From the second condition, (x + 1) / y = 1. Step 5: Cross multiply: x + 1 = y. Call this Equation (2). Step 6: Substitute y from Equation (2) into Equation (1). Step 7: Equation (2) gives y = x + 1. Step 8: Substitute into Equation (1): 2x = (x + 1) + 1. Step 9: Simplify the right side: (x + 1) + 1 = x + 2, so 2x = x + 2. Step 10: Subtract x from both sides: x = 2. Step 11: Substitute x = 2 into y = x + 1 to get y = 2 + 1 = 3. Step 12: The original fraction is x / y = 2 / 3. Verification / Alternative check: Check the first condition using 2/3. Adding 1 to the denominator gives 2 / (3 + 1) = 2 / 4 = 1/2, which matches. Check the second condition. Adding 1 to the numerator gives (2 + 1) / 3 = 3 / 3 = 1, which also matches. Thus, 2/3 satisfies both conditions and is the correct original fraction. Why Other Options Are Wrong: Option (a) 1/3: Adding 1 to numerator or denominator does not produce 1/2 and 1 as required. Option (c) 4/3 and option (d) 5/3: These are improper fractions and do not satisfy both transformation conditions when checked. Common Pitfalls: Common mistakes include mixing up numerator and denominator when applying the changes or writing the conditions as x + 1 / y + 1 instead of adjusting only one part at a time. Another error is failing to cross multiply correctly, leading to wrong equations. Careful translation of the word statement into algebraic form and systematic solving of the two equations avoids these problems. Final Answer: The original fraction is 2/3.

Question 20

The sum of the digits of a two-digit number is 9 less than the number itself. Which digit must appear in the units place of this two-digit number?

Accepted Answer

Introduction / Context: This question is about forming an equation for a two-digit number using its digits and a condition relating the sum of the digits to the number itself. It tests understanding of place value representation and basic algebra. We are asked to determine whether a specific units digit can be identified from the given information or whether the data are insufficient to determine a unique digit. Given Data / Assumptions: Let the two-digit number be 10a + b, where a is the tens digit and b is the units digit. The sum of the digits is a + b. The sum of the digits is 9 less than the number: a + b = (10a + b) − 9. We must decide which digit is in the units place, or whether the data are inadequate. Concept / Approach: We represent the number using its tens and units digits, then use the given relation to form an equation. Simplifying the equation will reveal a relationship between a and b. If this relationship determines b uniquely, we can choose a specific digit. If the relationship leaves b free to take multiple values, then the information is insufficient to determine the units digit, and the correct conclusion is that the data are inadequate. Step-by-Step Solution: Step 1: Let the two-digit number be 10a + b. Step 2: The sum of its digits is a + b. Step 3: According to the problem, this sum is 9 less than the number itself: a + b = (10a + b) − 9. Step 4: Simplify the right side: (10a + b) − 9 = 10a + b − 9. Step 5: Set up the equation: a + b = 10a + b − 9. Step 6: Subtract a + b from both sides: 0 = 9a − 9. Step 7: This simplifies to 9a − 9 = 0, so 9a = 9. Step 8: Divide by 9 to get a = 1. Step 9: The tens digit is therefore fixed at 1, so the number is 10 + b. Step 10: However, there is no restriction on b from this equation alone. Any b from 0 to 9 gives a number whose sum of digits is 9 less than the number. Verification / Alternative check: Take examples. If b = 0, the number is 10. Sum of digits is 1 + 0 = 1, and 10 − 9 = 1, so it works. If b = 5, the number is 15. Sum of digits is 1 + 5 = 6, and 15 − 9 = 6, which also works. Clearly several choices of b satisfy the condition. Therefore, there is no unique digit that must occupy the units place, confirming that the data are inadequate to identify a single units digit. Why Other Options Are Wrong: Options (a) 1, (b) 2, and (c) 3 each suggest a specific units digit. However, as shown, the units digit can be 0, 1, 2, 3 and so on, as long as the tens digit is 1. Therefore, none of these specific digits is forced by the conditions, so these options are incorrect. Common Pitfalls: A frequent mistake is to assume that because the tens digit is fixed as 1, the units digit must also be determined, or to try to guess a particular b based on a single example. Another error is mishandling the algebraic simplification, leading to expressions that seem to involve b but actually do not. Recognizing that multiple valid numbers satisfy the condition is key to concluding that the data are inadequate. Final Answer: No unique units digit can be determined, so the correct conclusion is data inadequate.

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