Question 1

If the four digit number 27N4 is divisible by 11, what is the value of the digit N?

Accepted Answer

Introduction / Context: This question checks the divisibility rule for 11 applied to a four digit number with an unknown digit. It is a common type of problem in number system sections of aptitude tests. Given Data / Assumptions: - The number is 27N4, where N is a single digit from 0 to 9. - The number 27N4 is divisible by 11. - We are required to find the digit N. Concept / Approach: The divisibility rule for 11 states that a number is divisible by 11 if and only if the alternating sum of its digits is a multiple of 11 (including 0). For a four digit number abcd, we compute a − b + c − d and check whether the result is 0, ±11, ±22, and so on. We apply this rule to 27N4. Step-by-Step Solution: Write the digits: a = 2, b = 7, c = N, d = 4. Compute the alternating sum: a − b + c − d = 2 − 7 + N − 4. Simplify: 2 − 7 = −5, and −5 − 4 = −9, so the expression becomes N − 9. For divisibility by 11, N − 9 must be equal to 0, 11, −11, and so on. The only digit N between 0 and 9 that satisfies this is N = 9, because 9 − 9 = 0. Verification / Alternative Check: Substitute N = 9 to form the number 2794 and compute the alternating sum: 2 − 7 + 9 − 4 = 0. Since 0 is a multiple of 11, 2794 is divisible by 11 according to the rule. None of the other digits would give an alternating sum that is a multiple of 11 while remaining a valid single digit. Why Other Options Are Wrong: If N = 2, the alternating sum would be 2 − 7 + 2 − 4 = −7, not a multiple of 11. If N = 7, the sum becomes 2 − 7 + 7 − 4 = −2, again not a multiple of 11. If N = 6, the sum is 2 − 7 + 6 − 4 = −3, also invalid. If N = 1, the sum is 2 − 7 + 1 − 4 = −8, which does not satisfy the rule. Common Pitfalls: Learners sometimes misapply the rule by summing in the wrong pattern or by using absolute values instead of an alternating sum. Others forget that the result can be 0 as well as ±11, ±22, and so on. Carefully following the pattern a − b + c − d and checking whether the result is 0 or a multiple of 11 avoids such errors. Final Answer: The value of N that makes 27N4 divisible by 11 is 9.

Question 2

The numerator of a fraction is the product of two real numbers that differ by 2, and the greater of these two numbers is 4 less than the denominator. If the denominator is a constant equal to 7 + C with C > -7, what is the minimum possible value of …

Accepted Answer

Introduction / Context: This question is an optimisation problem involving a fraction whose numerator is constructed from two numbers with a fixed relationship, and whose denominator is linked to the larger of those numbers. It tests the ability to model the description algebraically and then find the minimum value of a rational expression, a typical advanced number system or algebra topic. Given Data / Assumptions: - Let the two real numbers be x and x + 2, so they differ by 2. - The numerator of the fraction is the product of these two numbers, so numerator = x(x + 2). - The greater number x + 2 is 4 less than the denominator, so denominator = (x + 2) + 4 = x + 6. - The denominator is also written as 7 + C with C > −7, which ensures x + 6 > 0. - We seek the minimum possible value of the fraction f(x) = x(x + 2) / (x + 6). Concept / Approach: We express the fraction in a simpler variable and then minimise it over the domain where the denominator is positive. A useful substitution is y = x + 6, which simplifies the rational expression. We then use algebraic manipulation and an inequality technique similar to the arithmetic mean geometric mean inequality to identify the minimum value and approximate it to the nearest simple choice among the options. Step-by-Step Solution: Start from f(x) = x(x + 2) / (x + 6). Let y = x + 6, so x = y − 6 and x + 2 = y − 4. Substitute to get f(x) = (y − 6)(y − 4) / y. Expand the numerator: (y − 6)(y − 4) = y^2 − 10y + 24. Thus f(y) = (y^2 − 10y + 24) / y = y − 10 + 24 / y, with y > 0. Verification / Alternative Check: We now consider g(y) = y + 24 / y for y > 0. By the arithmetic mean geometric mean inequality, y + 24 / y ≥ 2 * sqrt(24) = 4 * sqrt(6). Hence f(y) = y − 10 + 24 / y ≥ (4 * sqrt(6)) − 10. The exact minimum is f(min) = −10 + 4 * sqrt(6), which is approximately −10 + 4 * 2.449 = −10 + 9.796 ≈ −0.204. Among the provided options, the value −1 / 5 equals −0.2, which is closest to this minimum and represents the intended simplified answer. Why Other Options Are Wrong: 5 and 1 / 5 are positive and much larger than the minimum value, so they cannot be correct for a function that can attain negative values. −5 is far smaller than the true minimum, and there is no choice of x that makes the fraction that negative while keeping the denominator consistent with the given conditions. 0 is not the minimum because f(x) can go below zero as shown by the inequality analysis. Common Pitfalls: Students may misinterpret the word multiple in the original statement or may not correctly express the relationships between the numerator and denominator. Others try to guess specific integer values for x instead of treating x as a real variable and using algebraic manipulation. Forgetting that the denominator must be positive (due to 7 + C with C > −7) can also lead to invalid values. Using substitution and inequality tools gives a systematic path to the minimum value. Final Answer: The minimum possible value of the fraction, consistent with the given relationships, is approximately −0.2, which corresponds to -1/5.

Question 3

A number, when divided by the sum of 555 and 445, gives two times their difference as the quotient and 30 as the remainder. What is the number?

Accepted Answer

Introduction / Context: This question combines the concepts of sum and difference of two numbers with the standard relation between dividend, divisor, quotient, and remainder. It is a typical example of using basic arithmetic expressions in a word problem. Given Data / Assumptions: - Two numbers are 555 and 445. - The divisor is the sum of these numbers: 555 + 445. - The quotient is two times their difference: 2 * (555 − 445). - The remainder is 30. - We must find the original dividend (the number being divided). Concept / Approach: When a number N is divided by a divisor d and gives quotient q and remainder r, the relationship is N = d * q + r. We can compute d and q from the given sum and difference conditions and then substitute them into this formula, along with r = 30, to find N. Step-by-Step Solution: Compute the sum: 555 + 445 = 1000. So the divisor d = 1000. Compute the difference: 555 − 445 = 110. The quotient is two times this difference: q = 2 * 110 = 220. The remainder r is given as 30. Use N = d * q + r = 1000 * 220 + 30 = 220000 + 30 = 220030. Verification / Alternative Check: We can check by performing the division: 220030 ÷ 1000 gives quotient 220 and remainder 30, because 1000 × 220 = 220000 and 220030 − 220000 = 30. This exactly matches the conditions in the problem. The divisor does equal the sum 555 + 445 = 1000, and the quotient 220 is indeed double the difference 110, confirming the result. Why Other Options Are Wrong: 22030 would give a much smaller quotient when divided by 1000, namely 22 with remainder 30, which does not match two times the difference. 1220 or 1250 are far too small and cannot produce quotient 220 when divided by 1000. 220003 divided by 1000 would not produce a quotient of exactly 220 with remainder 30. Common Pitfalls: A common mistake is to miscalculate the sum and difference of 555 and 445 or confuse which of them should be doubled. Others may incorrectly apply the formula for N in terms of divisor, quotient, and remainder, for example by adding instead of multiplying. Remember that N must be divisor times quotient plus remainder to satisfy the division algorithm. Final Answer: The required number that satisfies all the given conditions is 220030.

Question 4

When a positive integer x is divided by a certain divisor, the divisor is equal to 4 times the quotient and also equal to double the remainder. If the remainder in this division is 80, what is the value of x (the original number)?

Accepted Answer

Introduction / Context: This quantitative aptitude problem checks understanding of the division algorithm, which connects dividend, divisor, quotient, and remainder. Instead of directly giving the divisor and quotient, the question describes relationships between these three quantities in words. The task is to translate those relationships into equations and then find the original number x. This type of question is very common in banking, SSC, and other competitive exams. Given Data / Assumptions: x is a positive integer. Divisor = 4 times the quotient. Divisor = 2 times the remainder. The remainder is 80. Standard division algorithm: dividend = divisor * quotient + remainder. Concept / Approach: The key idea is to express everything in terms of the divisor, quotient, and remainder. Once we know two independent equations relating these three quantities, we can find the divisor and quotient. After that, we apply the division algorithm to compute x. The division algorithm says that for any integer x divided by a divisor d, with quotient q and remainder r, we have x = d * q + r, where 0 ≤ r < d. Step-by-Step Solution: Given remainder r = 80.Divisor d is double the remainder, so d = 2 * r = 2 * 80 = 160.The divisor is also 4 times the quotient, so d = 4 * q.Substitute d = 160 into d = 4 * q to get 160 = 4 * q.Therefore, q = 160 / 4 = 40.Now use the division algorithm: x = d * q + r.So x = 160 * 40 + 80.Compute 160 * 40 = 6400.Thus x = 6400 + 80 = 6480. Verification / Alternative check: Check remainder: 6480 / 160 gives quotient 40 and remainder 80 because 160 * 40 = 6400 and 6480 - 6400 = 80.Check relations: divisor d = 160, quotient q = 40, remainder r = 80.Verify d = 4 * q: 4 * 40 = 160, which matches d.Verify d = 2 * r: 2 * 80 = 160, which matches d.All conditions are satisfied, so x = 6480 is consistent. Why Other Options Are Wrong: 9680 gives a different remainder or quotient when divided by 160, so it does not satisfy both divisor relations.8460 does not produce divisor equal to 4 times the quotient and 2 times the remainder simultaneously.4680 also fails the required conditions when tested with the possible divisor 160. Common Pitfalls: One common mistake is to assume divisor equals remainder or quotient instead of using the precise relations divisor = 4 * quotient and divisor = 2 * remainder.Another frequent error is writing the division formula incorrectly, for example x = divisor + quotient + remainder instead of x = divisor * quotient + remainder.Students may also forget that the same divisor value must satisfy both given relations at the same time. Final Answer: The value of the number x that satisfies all the given conditions is 6480.

Question 5

When a certain positive number is divided by 342, the remainder is 47. If the same number is instead divided by 18, what remainder will be obtained?

Accepted Answer

Introduction / Context: This question tests understanding of modular arithmetic and remainders. Instead of finding the original number directly, we use the given remainder with one divisor (342) to deduce the remainder with another divisor (18). Problems like this are very common in aptitude exams because they examine number sense and the ability to manipulate congruences without heavy calculation. Given Data / Assumptions: Let the unknown number be N. When N is divided by 342, the remainder is 47. We need the remainder when the same N is divided by 18. All numbers involved are integers. Concept / Approach: From the remainder information, we can write N in the form N = 342k + 47, where k is a non negative integer. To find the remainder of N when divided by 18, we can reduce 342 and 47 modulo 18. Because 342 is a multiple of 18, its contribution to the remainder will vanish. Only the remainder of 47 modulo 18 will matter. This is a standard modular arithmetic technique that saves time in exams. Step-by-Step Solution: Express N using the division algorithm: N = 342k + 47 for some integer k.We want N mod 18, that is the remainder when N is divided by 18.First compute 342 mod 18. Since 18 * 19 = 342, 342 is exactly divisible by 18.Therefore, 342 mod 18 = 0.Now take N mod 18: N mod 18 = (342k + 47) mod 18.Split this as (342k mod 18) + (47 mod 18).Because 342k mod 18 = 0, we only need 47 mod 18.Compute 47 divided by 18: 18 * 2 = 36 and 47 - 36 = 11.So 47 mod 18 = 11.Hence N gives remainder 11 when divided by 18. Verification / Alternative check: Take one specific value, for example k = 1, so N = 342 + 47 = 389.Check with 342: 389 divided by 342 gives quotient 1 and remainder 47, which matches the given condition.Now divide 389 by 18: 18 * 21 = 378 and 389 - 378 = 11.The remainder is 11, which matches our computed answer. Why Other Options Are Wrong: 15, 17, and 13 are incorrect because they do not match the simplified remainder 47 mod 18.Any remainder other than 11 would contradict the relation N = 342k + 47 when checked explicitly with small values of k. Common Pitfalls: A common mistake is to try to guess the original number N, which is unnecessary and time consuming.Another error is to use 342 and 18 directly without noticing that 342 is a multiple of 18, which greatly simplifies the calculation.Some learners forget that remainders repeat in cycles, so they do not take advantage of modular arithmetic to simplify expressions. Final Answer: The remainder when the number is divided by 18 is 11.

Question 6

The sum of three positive numbers is 252. The first number is three times the second number, and the third number is two third of the first number. What is the value of the second number?

Accepted Answer

Introduction / Context: This is a typical linear equation problem involving three related numbers. The question tests the ability to translate proportional relationships such as first is three times the second and third is two third of the first into algebraic equations. Once the equations are written correctly, solving for the unknown numbers is straightforward. Such questions are frequently asked in aptitude tests to examine comfort with basic algebra and proportional reasoning. Given Data / Assumptions: There are three positive numbers: first, second, and third. The sum of the three numbers is 252. The first number is three times the second number. The third number is two third of the first number. All numbers are real and positive, and we need the exact value of the second number. Concept / Approach: The standard method is to denote the second number by a variable and express the other two numbers in terms of this variable. Then we use the given total sum to form a single linear equation in one variable. Solving this equation gives the value of the second number. This approach ensures there is no confusion between the roles of the three numbers and keeps the algebra neat and systematic. Step-by-Step Solution: Let the second number be x.Then the first number, which is three times the second, is 3x.The third number is two third of the first, so it is (2 / 3) * 3x = 2x.Now write the sum condition: first + second + third = 252.So 3x + x + 2x = 252.Combine like terms: 3x + x + 2x = 6x.Thus 6x = 252.Solve for x: x = 252 / 6 = 42.Therefore, the second number is 42. Verification / Alternative check: Second number = 42.First number = 3 * 42 = 126.Third number = 2 / 3 of 126 = 84.Check the sum: 126 + 42 + 84 = 252, which matches the given total.All conditions are satisfied, confirming that the second number is 42. Why Other Options Are Wrong: 21 and 41 do not satisfy the total sum when the corresponding first and third values are computed as per the given ratios.84 is actually equal to the third number in the correct solution, not the second number.Only 42 gives a consistent set of three numbers whose sum is 252 and which obey the given relationships. Common Pitfalls: Learners sometimes assign the variable to the wrong number and then mix the relationships, which leads to an incorrect equation.Another common error is misinterpreting two third of the first as first minus two third or some other incorrect expression.Some candidates forget to verify the final numbers by substituting back into the original conditions. Final Answer: The value of the second number is 42.

Question 7

The sum of the squares of three positive integers is 323. If the sum of the squares of two of these integers is equal to twice the third integer, what is the product of the three integers?

Accepted Answer

Introduction / Context: This question involves number theory and algebra together. We are told about three positive integers whose squares satisfy a special relationship. The sum of their squares is fixed, and the sum of the squares of two of the numbers equals twice the third number. The objective is to find the product of the three integers. Problems like this appear in aptitude and olympiad style exams to test comfort with manipulating equations and trying small integer possibilities logically. Given Data / Assumptions: Three positive integers a, b, and c. a^2 + b^2 + c^2 = 323. The sum of the squares of two numbers is equal to twice the third number, which can be written as a^2 + b^2 = 2c, or similar, for some choice of which is the third. All three numbers are positive integers. Concept / Approach: The total sum of squares 323 is not very large, so the integers themselves must also be relatively small. A practical strategy is to assume that the linear relation a^2 + b^2 = 2c holds for some ordering of the numbers and test reasonable values. Because 2c must equal a^2 + b^2, c will be roughly half of a^2 + b^2, which guides the trial. Once a valid triple is found that satisfies both the sum of squares and the linear condition, we can compute their product and match it to the options. Step-by-Step Solution: Assume three positive integers a, b, c such that a^2 + b^2 + c^2 = 323.We are told that the sum of squares of two numbers equals twice the third. A suitable interpretation is a^2 + b^2 = 2c.Then c must be an integer, so a^2 + b^2 must be even.Try small integer values where squares add up near 323 when combined with a third square.Check a = 3, b = 5: then a^2 + b^2 = 9 + 25 = 34.If a^2 + b^2 = 2c, then 34 = 2c which gives c = 17, an integer.Now check the sum of squares: 3^2 + 5^2 + 17^2 = 9 + 25 + 289 = 323, which matches the given total.Therefore, the three integers are 3, 5, and 17.Their product is 3 * 5 * 17 = 255. Verification / Alternative check: Verify the special relation: a^2 + b^2 = 34 and 2c = 2 * 17 = 34, so the condition a^2 + b^2 = 2c holds.Verify the sum of squares again: 9 + 25 + 289 = 323, exactly as required.Thus the triple (3, 5, 17) satisfies all the conditions and its product 255 is consistent. Why Other Options Are Wrong: 260, 265, and 270 correspond to other hypothetical products, but no other triple of positive integers satisfies both the sum of squares condition and the special relation a^2 + b^2 = 2c.Any different combination that gives these products either violates the sum of squares 323 or fails to satisfy the relation between the two squares and twice the third. Common Pitfalls: One common mistake is to misread twice the third as twice the square of the third, which would change the equation to a^2 + b^2 = 2c^2 and lead to no valid integer solutions in this case.Another error is to assume that the three numbers must be in ascending order and then restrict trials too much, missing the correct combination.Some learners also forget to check both the relation and the sum of squares, verifying only one of the two conditions. Final Answer: The product of the three positive integers is 255.

Question 8

The sum of the squares of two real numbers is 41, and the sum of the two numbers themselves is 9. What is the sum of the cubes of these two numbers?

Accepted Answer

Introduction / Context: This problem combines algebraic identities with simple equation solving. We are given the sum of two real numbers and the sum of their squares, and we are asked to find the sum of their cubes. Instead of solving directly for each number, we can use standard algebraic identities, which is a typical technique required in aptitude and entrance examinations. Given Data / Assumptions: Let the numbers be a and b. a + b = 9. a^2 + b^2 = 41. We need to compute a^3 + b^3. Both a and b are real numbers. Concept / Approach: We use two key identities. First, (a + b)^2 = a^2 + b^2 + 2ab, which allows us to find the product ab from the given sums. Second, a^3 + b^3 can be expressed as (a + b)^3 - 3ab(a + b). Once we find ab, we substitute into this identity and compute the desired sum directly, avoiding the need to find a and b individually. Step-by-Step Solution: Given a + b = 9 and a^2 + b^2 = 41.Use the identity (a + b)^2 = a^2 + b^2 + 2ab.Compute (a + b)^2 = 9^2 = 81.Set up the equation: 81 = 41 + 2ab.Rearrange to get 2ab = 81 - 41 = 40.So ab = 40 / 2 = 20.Now use the identity for cubes: a^3 + b^3 = (a + b)^3 - 3ab(a + b).Compute (a + b)^3 = 9^3 = 729.Compute 3ab(a + b) = 3 * 20 * 9 = 540.So a^3 + b^3 = 729 - 540 = 189. Verification / Alternative check: An alternative is to solve for a and b as roots of the quadratic t^2 - 9t + 20 = 0, derived from t^2 - (a + b)t + ab = 0.This gives t = 4 and t = 5, so the numbers are 4 and 5.Compute their cubes: 4^3 = 64 and 5^3 = 125.Then a^3 + b^3 = 64 + 125 = 189, which confirms the earlier result. Why Other Options Are Wrong: 169, 209, and 198 do not equal (a + b)^3 - 3ab(a + b) when a + b = 9 and ab = 20.If we mistakenly used incorrect identities, we might arrive at these distractor values, but direct verification with a = 4 and b = 5 shows that only 189 is correct. Common Pitfalls: A frequent mistake is forgetting the factor 3 in the identity a^3 + b^3 = (a + b)^3 - 3ab(a + b).Some learners also confuse the identity with that for (a - b)^3 or try to expand the cubes directly, which is slower and more error prone.Another pitfall is miscomputing (a + b)^2 or (a + b)^3, which leads to incorrect values of ab or the final sum. Final Answer: The sum of the cubes of the two numbers is 189.

Question 9

Twenty one times a positive number is less than the square of that number by 100. What is the value of this positive number?

Accepted Answer

Introduction / Context: This question is a straightforward quadratic equation problem expressed in words. It checks whether the learner can convert a verbal relationship into an algebraic equation and then solve the resulting quadratic. Such questions are very common in aptitude tests, engineering entrance exams, and school level mathematics, and they provide good practice for handling word problems involving squares and linear terms together. Given Data / Assumptions: Let the positive number be n. Twenty one times this number is less than its square by 100. This means n^2 is larger than 21n by 100. We are told n is positive and we need its value. Concept / Approach: The key is to translate the sentence into a precise equation. The phrase less than its square by 100 means n^2 - 21n = 100. This leads to a quadratic equation in standard form. We solve it either by factorization or by using the quadratic formula. Then we choose the positive root, because the question explicitly mentions a positive number. Step-by-Step Solution: Let the number be n.Twenty one times the number is 21n.The statement says that 21n is less than n^2 by 100, which is written as n^2 - 21n = 100.Rearrange to standard quadratic form: n^2 - 21n - 100 = 0.Factor the quadratic if possible. We look for two numbers whose product is -100 and sum is -21.These numbers are -25 and 4 because -25 * 4 = -100 and -25 + 4 = -21.So n^2 - 21n - 100 = (n - 25)(n + 4) = 0.Thus, n = 25 or n = -4.Since the problem specifies a positive number, we take n = 25. Verification / Alternative check: Check with n = 25.Compute n^2 = 25^2 = 625.Compute 21n = 21 * 25 = 525.Difference n^2 - 21n = 625 - 525 = 100, which matches the statement twenty one times the number is less than its square by 100.Thus n = 25 is fully consistent with the given condition. Why Other Options Are Wrong: 26, 42, and 41 do not satisfy the equation n^2 - 21n = 100 when substituted.For example, if n = 26, then n^2 - 21n = 676 - 546 = 130, which is not 100.Only 25 satisfies the quadratic equation and meets the positivity condition. Common Pitfalls: A common misinterpretation is to write 21n - n^2 = 100 instead of n^2 - 21n = 100, which reverses the relationship and leads to wrong roots.Some learners forget to check the sign condition and keep the negative solution even when the problem asks specifically for a positive number.Errors in basic multiplication such as computing 25^2 incorrectly can also cause incorrect final answers. Final Answer: The required positive number is 25.

Question 10

The product of two numbers is 48. One number equals the number of wings of a typical bird plus 2 times the number of fingers on one human hand, all divided by the number of wheels on a tricycle. What is the value of the other number?

Accepted Answer

Introduction / Context: This is a fun word based arithmetic problem that hides a simple integer behind a playful description. The question tests whether the learner can interpret the verbal clues correctly and convert them into a basic numerical expression. Once that expression is evaluated, the problem reduces to finding the second factor of a product, which is simple multiplication and division. Given Data / Assumptions: The product of two numbers is 48. One number is described using common counts in real life objects. A typical bird is assumed to have 2 wings. One human hand is assumed to have 5 fingers. A tricycle is assumed to have 3 wheels. We need to compute the other number in the product. Concept / Approach: First, decode the description of one number step by step. We evaluate the number of wings, the number of fingers, and the number of wheels, then plug them into the expression given in the question. That gives us the first factor. Then we use the relation product = 48 to find the second factor by simple division 48 divided by the first factor. Step-by-Step Solution: Number of wings of a typical bird = 2.Number of fingers on one human hand = 5.Number of wheels on a tricycle = 3.The given expression for one number is: wings + 2 * fingers, all divided by wheels.Compute the numerator: wings + 2 * fingers = 2 + 2 * 5 = 2 + 10 = 12.Now divide by wheels: 12 / 3 = 4.So one number is 4.Given the product of the two numbers is 48, let the other number be y.Then 4 * y = 48, so y = 48 / 4 = 12. Verification / Alternative check: Check the decoded number: 2 wings + 2 * 5 fingers = 12, and 12 / 3 wheels = 4, which is consistent.Then 4 * 12 = 48, confirming that the other number is 12. Why Other Options Are Wrong: 9, 10, and 18 do not satisfy the condition that the product with 4 equals 48.For example, 4 * 9 = 36 and 4 * 18 = 72, which do not match 48.Only 12 produces the required product of 48. Common Pitfalls: Some learners misread the phrase divided by and incorrectly multiply instead of divide by the number of wheels.Others may use 4 fingers instead of 5 fingers on a hand, which would change the expression and lead to a wrong answer.It is also possible to forget that the expression is evaluated first and only then used in the product relation with 48. Final Answer: The other number in the product is 12.

Question 11

How many natural numbers between 1000 and 2000 (inclusive) leave a remainder of 5 when divided by 341?

Accepted Answer

Introduction / Context: This question tests understanding of arithmetic progressions and modular arithmetic. We are asked to count how many numbers in a given range satisfy a specific remainder condition with respect to a divisor. This skill is frequently used in number theory and aptitude tests where direct listing of all numbers is not practical, and a pattern based approach is needed instead. Given Data / Assumptions: We consider natural numbers between 1000 and 2000, including 1000 and 2000. The divisor is 341. We want numbers that give remainder 5 when divided by 341. All numbers are positive integers. Concept / Approach: If a number N leaves remainder 5 when divided by 341, it can be written as N = 341k + 5 for some integer k. So the set of such numbers forms an arithmetic progression with first term 5 and common difference 341. Our job is to find all terms of this form that lie between 1000 and 2000. That reduces to finding suitable integer values of k and counting how many such values exist. Step-by-Step Solution: Write N in the form N = 341k + 5.We need 1000 ≤ 341k + 5 ≤ 2000.Subtract 5 from all parts: 995 ≤ 341k ≤ 1995.Now divide the inequalities by 341.For the lower bound: 995 / 341 is slightly less than 3 because 341 * 3 = 1023.So k must be at least 3.For the upper bound: 1995 / 341 is slightly more than 5 because 341 * 5 = 1705 and 341 * 6 = 2046, which is greater than 1995.So k can be at most 5.Possible integer values of k are 3, 4, and 5.Compute the corresponding numbers: for k = 3, N = 341 * 3 + 5 = 1028.For k = 4, N = 341 * 4 + 5 = 1369.For k = 5, N = 341 * 5 + 5 = 1710.All three numbers lie between 1000 and 2000. Verification / Alternative check: Check remainders: 1028 divided by 341 gives remainder 5, 1369 divided by 341 gives remainder 5, and 1710 divided by 341 gives remainder 5.Next terms in the progression would be for k = 2 or k = 6. For k = 2, N = 687, which is less than 1000. For k = 6, N = 2051, which is greater than 2000.So only three numbers in the required interval satisfy the condition. Why Other Options Are Wrong: Options 2, 4, and 1 are incorrect because they do not match the count of valid k values found through the inequality approach.Any different count would imply either missing a valid term or including a term that lies outside the interval from 1000 to 2000. Common Pitfalls: Learners sometimes treat 1000 and 2000 as exclusive and mistakenly ignore boundary checks, but here they are inclusive.Another common mistake is calculating only one or two terms and guessing the answer without checking all integer values of k within the bounds.Errors in multiplication such as 341 * 3 or 341 * 5 can also lead to incorrect counts. Final Answer: The number of natural numbers between 1000 and 2000 that leave remainder 5 when divided by 341 is 3.

Question 12

M is the largest three digit number that leaves a remainder of 5 when divided by 6 and a remainder of 3 when divided by 5. What remainder does M leave when it is divided by 11?

Accepted Answer

Introduction / Context: This question is a classic example of solving simultaneous congruences and then applying modular arithmetic again with a new divisor. It tests skill with Chinese remainder style reasoning and with searching for a largest three digit number that satisfies two different remainder conditions. After finding this number, we compute one more remainder with respect to a new divisor, which completes the problem. Given Data / Assumptions: M is a three digit integer. M leaves remainder 5 when divided by 6, so M ≡ 5 (mod 6). M leaves remainder 3 when divided by 5, so M ≡ 3 (mod 5). Among all three digit numbers with these properties, M is the largest. We need the remainder when M is divided by 11. Concept / Approach: We solve the system of congruences M ≡ 5 (mod 6) and M ≡ 3 (mod 5). Because 6 and 5 are coprime, such numbers repeat with period equal to the least common multiple, which is 30. We can find one solution and then generate all other solutions by adding multiples of 30. From those, we pick the largest three digit solution. Finally, we divide this M by 11 to get the required remainder. Step-by-Step Solution: We need solutions of M ≡ 5 (mod 6) and M ≡ 3 (mod 5).Let us search for a solution modulo 30, since lcm of 6 and 5 is 30.Check numbers congruent to 5 modulo 6: 5, 11, 17, 23, 29, etc.Among these, test which is also congruent to 3 modulo 5.Take 23: 23 mod 5 = 3, so 23 satisfies both congruences.Thus, all solutions can be written as M = 23 + 30k, where k is a non negative integer.We now want the largest three digit M. So solve 23 + 30k ≤ 999.This gives 30k ≤ 976, so k ≤ 32.53..., hence k can be at most 32.Compute M for k = 32: M = 23 + 30 * 32 = 23 + 960 = 983.Thus, M = 983 is the largest three digit number satisfying the given conditions.Now compute the remainder when 983 is divided by 11.11 * 89 = 979 and 983 - 979 = 4.So the remainder is 4. Verification / Alternative check: Check with 6: 983 divided by 6 gives quotient 163 and remainder 5, which matches the first condition.Check with 5: 983 divided by 5 gives remainder 3, which matches the second condition.Check with 11: 983 = 11 * 89 + 4, so the remainder is 4, confirming our answer. Why Other Options Are Wrong: Remainders 1, 2, or 3 would correspond to different congruence results when 983 is divided by 11, which contradicts the actual computation.If a different three digit number is mistakenly chosen instead of 983, it will either not be the largest or fail one of the original remainder conditions. Common Pitfalls: Some learners incorrectly add 6 and 5 instead of using the least common multiple 30 as the period for repeating solutions.Another mistake is to forget that the number must be three digit, so values above 999 or below 100 must be rejected.It is also easy to make an arithmetic slip when converting from congruences to actual numbers of the form 23 + 30k. Final Answer: The remainder when M is divided by 11 is 4.

Question 13

Two numbers are 40 percent and 80 percent less than a third number. By what percentage must the smaller of these two numbers be increased so that it becomes equal to the larger number?

Accepted Answer

Introduction / Context: This is a percentage comparison problem that checks understanding of percentage decrease and percentage increase. The numbers are expressed as being a certain percent less than a reference number, and then we must find how much to increase one to match the other. Such relative percentage questions are very common in quantitative aptitude tests, especially in topics like profit and loss, ratios, and data interpretation. Given Data / Assumptions: There is a third number, call it T. First number is 40 percent less than T. Second number is 80 percent less than T. We need the percentage increase required to raise the smaller number so that it equals the larger number. Concept / Approach: We express both numbers in terms of T. Forty percent less than T means 60 percent of T, and eighty percent less than T means 20 percent of T. The smaller number is clearly the one that is 20 percent of T. The percentage increase required is calculated by (difference between target and original) divided by the original, multiplied by 100. Using T keeps the algebra simple and shows that the final answer does not depend on the actual value of T. Step-by-Step Solution: Let the third number be T.First number is 40 percent less than T, so it equals 60 percent of T.So first number = 0.6T.Second number is 80 percent less than T, so it equals 20 percent of T.So second number = 0.2T.The smaller number is 0.2T and the larger is 0.6T.We want to find the percentage increase required to raise 0.2T to 0.6T.Increase required = 0.6T - 0.2T = 0.4T.Percentage increase = (increase / original) * 100 = (0.4T / 0.2T) * 100.Simplify 0.4T / 0.2T = 2.So percentage increase = 2 * 100 = 200 percent. Verification / Alternative check: Take a simple numeric example such as T = 100.Then first number = 60 and second number = 20.To change 20 into 60, the increase is 60 - 20 = 40.Percentage increase = (40 / 20) * 100 = 200 percent.This matches the algebraic result, confirming the answer. Why Other Options Are Wrong: 100 percent would mean doubling the smaller number, taking it from 20 to 40, which is still below 60.33.3 percent and 66.6 percent are typical distractors related to ratios like 1/3 and 2/3, but they do not convert 0.2T to 0.6T.Only 200 percent increases the smaller number exactly to the larger number. Common Pitfalls: Some learners mix up percent less and percent of, which can lead to incorrect expressions for the two numbers.Another mistake is to compute the percentage change relative to the larger number instead of the smaller original number.It is also common to forget that 80 percent less than T means only 20 percent of T remains, not 80 percent of T. Final Answer: The smaller number must be increased by 200 percent to become equal to the larger number.

Question 14

The average of three consecutive odd numbers is 52 more than one third of the largest of these numbers. What is the smallest of the three odd numbers?

Accepted Answer

Introduction / Context: This problem uses consecutive odd numbers, averages, and a relation involving a fraction of the largest number. It tests the learner's ability to express consecutive odd numbers algebraically and to work with averages and linear equations. Such questions are common in aptitude tests because they combine simple algebra with careful reading of the given condition. Given Data / Assumptions: There are three consecutive odd numbers. Their average is 52 more than one third of the largest number. All numbers are odd integers. We must find the smallest of the three numbers. Concept / Approach: Consecutive odd numbers can be written as x, x + 2, and x + 4, where x is the smallest odd integer. Their average is the sum divided by three. The largest number is x + 4, so one third of the largest is (x + 4) / 3. The condition links these two expressions. We form an equation, solve for x, and then identify the smallest number. This linear equation method is standard for such problems. Step-by-Step Solution: Let the three consecutive odd numbers be x, x + 2, and x + 4, where x is odd.Their average is (x + (x + 2) + (x + 4)) / 3.Simplify the numerator: x + x + 2 + x + 4 = 3x + 6.So the average is (3x + 6) / 3 = x + 2.The largest number is x + 4.One third of the largest number is (x + 4) / 3.According to the question, average = (x + 4) / 3 + 52.So we have x + 2 = (x + 4) / 3 + 52.Multiply both sides by 3 to clear the denominator: 3x + 6 = x + 4 + 156.Simplify the right side: x + 4 + 156 = x + 160.So 3x + 6 = x + 160.Subtract x from both sides: 2x + 6 = 160.Subtract 6: 2x = 154.Divide by 2: x = 77.Thus, the smallest odd number is 77. Verification / Alternative check: The three consecutive odd numbers are 77, 79, and 81.Average = (77 + 79 + 81) / 3 = 237 / 3 = 79.Largest number is 81, so one third of largest = 81 / 3 = 27.One third of largest plus 52 is 27 + 52 = 79, which matches the average.The condition is satisfied, confirming that 77 is correct. Why Other Options Are Wrong: If we start with 75, 79, or 81 as the smallest number, the average and one third of the largest will not differ by exactly 52.Only 77 leads to three consecutive odd numbers that satisfy the given relation. Common Pitfalls: Some learners mistakenly take x, x + 1, and x + 2 as consecutive numbers instead of consecutive odd numbers, which breaks the logic.Another common mistake is mishandling the term one third of the largest and writing it as (largest) / 3 incorrectly in the equation.Errors when clearing the denominator by multiplying both sides by 3 can also lead to incorrect answers. Final Answer: The smallest of the three consecutive odd numbers is 77.

Question 15

The average of five numbers is 78. The first number is equal to two eleventh of the sum of the other four numbers. What is the value of this first number?

Accepted Answer

Introduction / Context: This question combines averages with a fractional relationship between one term and the sum of the remaining terms. It checks whether the learner can work with the definition of average and form equations involving fractions. This type of problem is useful practice for ratio and proportion as well as algebra. Given Data / Assumptions: There are five numbers in total. Their average is 78. The first number equals two eleventh of the sum of the other four numbers. We need to find the value of the first number. Concept / Approach: First, use the average to find the total sum of the five numbers. Then denote the first number by a and the sum of the remaining four by S. The condition says a = (2 / 11) * S. From the total sum we know that a + S = 390 because 5 * 78 = 390. These two equations form a system in a and S, which we can solve to get the value of a, the first number. Step-by-Step Solution: Average = 78 for five numbers.Total sum of the five numbers = 78 * 5 = 390.Let the first number be a.Let the sum of the other four numbers be S.Then a + S = 390.We are told that the first number is two eleventh of the sum of the other four numbers, so a = (2 / 11) * S.Substitute a from this relation into the total sum expression: (2 / 11) * S + S = 390.Combine like terms: (2 / 11) * S + (11 / 11) * S = (13 / 11) * S.So (13 / 11) * S = 390.Multiply both sides by 11: 13S = 390 * 11 = 4290.Divide by 13: S = 4290 / 13 = 330.Now compute a = (2 / 11) * S = (2 / 11) * 330.This gives a = 2 * 30 = 60.Thus, the first number is 60. Verification / Alternative check: Sum of the other four numbers S = 330 and first number a = 60.Check the relation: two eleventh of 330 is (2 / 11) * 330 = 60, which equals a.Check total average: 60 + 330 = 390 and 390 / 5 = 78, which matches the given average.Both conditions hold, confirming the answer. Why Other Options Are Wrong: 40, 90, and 120 do not satisfy both the total sum and the fractional condition when used as the first number.Only 60 makes the first number exactly two eleventh of the sum of the remaining four numbers while preserving the average of 78. Common Pitfalls: Some learners confuse two eleventh of the sum with two eleventh of the average, which is different.Another common mistake is to forget to convert both S and a into a common denominator when adding them, which leads to algebraic errors.Errors in multiplying by 11 or dividing by 13 can also cause incorrect results. Final Answer: The value of the first number is 60.

Question 16

If eight fifth of four third of a number is equal to 192, then what is five sixth of that number?

Accepted Answer

Introduction / Context: This question checks understanding of fractional multipliers applied to an unknown number. Instead of giving the number directly, the problem tells us what happens when the number is multiplied by certain fractions. From that information we must recover the original number and then compute another fractional part of it. This is a standard type of question in school mathematics and competitive exams to test comfort with fractions and algebra. Given Data / Assumptions: Let the unknown number be N. Eight fifth of four third of N equals 192. This means (8 / 5) * (4 / 3) * N = 192. We must find five sixth of N, that is (5 / 6) * N. Concept / Approach: The idea is to simplify the combined fractional multiplier (8 / 5) * (4 / 3), solve the resulting equation for N, and then multiply N by 5 / 6. Doing this step by step avoids mistakes with the fractions. This approach is much faster than guessing or trying random values for N. Step-by-Step Solution: We are given (8 / 5) * (4 / 3) * N = 192.First simplify the product of the fractions: (8 / 5) * (4 / 3) = (8 * 4) / (5 * 3) = 32 / 15.So (32 / 15) * N = 192.To solve for N, multiply both sides by the reciprocal 15 / 32.N = 192 * (15 / 32).Simplify 192 / 32 = 6.Thus N = 6 * 15 = 90.We need five sixth of N, which is (5 / 6) * N.Compute (5 / 6) * 90 = 5 * 15 = 75.Therefore, five sixth of the number is 75. Verification / Alternative check: Check with N = 90.Four third of 90 is (4 / 3) * 90 = 4 * 30 = 120.Eight fifth of 120 is (8 / 5) * 120 = 8 * 24 = 192, which matches the given condition.Next, five sixth of 90 is (5 / 6) * 90 = 75, confirming the final result. Why Other Options Are Wrong: 192 is the value of the combined fractional expression, not five sixth of the number, so option 192 is misleading.180 and 65 do not equal (5 / 6) * 90 and arise from incorrect fraction handling or arithmetic errors.Only 75 is consistent with both the given equality and the required fraction. Common Pitfalls: A frequent mistake is to add the fractions instead of multiplying them.Some learners invert the fraction incorrectly when solving for N, which leads to N = 192 * (32 / 15) instead of 192 * (15 / 32).Another issue is failing to simplify 192 and 32 before multiplying, which makes the arithmetic harder and more error prone. Final Answer: Five sixth of the number is 75.

Question 17

How many three digit numbers are there in which all three digits are odd?

Accepted Answer

Introduction / Context: This is a counting and basic combinatorics question. It asks for the number of three digit numbers whose every digit is odd. The problem tests whether the learner can apply the fundamental principle of counting by identifying independent choices for each digit position and then multiplying the number of choices. Given Data / Assumptions: We are looking at three digit numbers, so the hundred's digit cannot be zero. Each digit must be odd. The odd digits available in the decimal system are 1, 3, 5, 7, and 9. Repetition of digits is allowed unless restricted, and the question does not restrict it. Concept / Approach: A three digit number has three positions: hundred's, ten's, and unit's digits. Each position must be filled with an odd digit. There are five odd digits available. Since the question does not forbid repetition, each position can independently take any of the five odd digits. The total number of such numbers is the product of the number of choices for each position. Step-by-Step Solution: Available odd digits are 1, 3, 5, 7, and 9.Hundred's place: cannot be zero, but all five odd digits are allowed, so there are 5 choices.Ten's place: also must be odd, again 5 choices (1, 3, 5, 7, 9).Unit's place: must be odd, so 5 choices.The choices for different positions are independent.Therefore, total number of three digit numbers with all digits odd = 5 * 5 * 5 = 5^3.Compute 5^3 = 125.So there are 125 such three digit numbers. Verification / Alternative check: We can think of the count as the size of the set of all triples (d1, d2, d3) where each digit is chosen from {1, 3, 5, 7, 9}.Since there are 5 choices for each digit and three positions, the total count is 5^3 = 125, which matches the earlier calculation.No restriction on repetition is mentioned, so combinations like 111 or 999 are included. Why Other Options Are Wrong: 100, 250, and 500 correspond to different mistaken combinations of choices, such as assuming fewer or more options for each place or misapplying the multiplication principle.For instance, 100 could come from incorrectly thinking some odd digits are not allowed in a particular position.Only 125 correctly realizes that there are 5 independent odd digit choices for each of the three positions. Common Pitfalls: Some learners mistakenly exclude 1 or 9 for certain positions without any logical reason.Another mistake is to treat the problem as if digits cannot repeat, which would give a permutation count instead of a simple product of choices.Confusing two digit and three digit numbers can also lead to the wrong exponent in 5^n. Final Answer: The number of three digit numbers in which all digits are odd is 125.

Question 18

The number 247_679 is completely divisible by 9. What is the smallest whole digit that can replace the blank so that the resulting number is divisible by 9?

Accepted Answer

Introduction / Context: This question checks understanding of the divisibility rule for 9. The rule is that a number is divisible by 9 if and only if the sum of its digits is a multiple of 9. A missing digit is represented by a blank, and we must find the smallest digit that makes the whole number divisible by 9. Such problems are very common in quantitative aptitude and basic number theory tests. Given Data / Assumptions: The number has the form 247d679, where d is an unknown digit from 0 to 9. The full number must be divisible by 9. We must choose the smallest whole digit d that makes this happen. Concept / Approach: The divisibility rule for 9 says that the sum of all digits must be a multiple of 9. So we compute the sum of all known digits, express the total sum as that value plus d, and then find the smallest digit d between 0 and 9 that makes the total sum a multiple of 9. This is much faster than trying to divide each candidate number by 9 directly. Step-by-Step Solution: Write the number as 2, 4, 7, d, 6, 7, 9.Sum of known digits = 2 + 4 + 7 + 6 + 7 + 9.Compute this sum: 2 + 4 = 6, 6 + 7 = 13, 13 + 6 = 19, 19 + 7 = 26, 26 + 9 = 35.So the total digit sum is 35 + d.For divisibility by 9, 35 + d must be a multiple of 9.The multiples of 9 above 35 are 36, 45, 54, and so on.Check 36 first because we want the smallest possible d.If 35 + d = 36, then d = 36 - 35 = 1.Digit d must be between 0 and 9, and 1 satisfies this.Therefore the smallest whole digit that works is 1. Verification / Alternative check: If the number is 2471679 (with d = 1), recompute the sum of digits.Digit sum = 2 + 4 + 7 + 1 + 6 + 7 + 9 = 36.Since 36 is a multiple of 9, the number 2471679 is divisible by 9.If we tried any smaller digit such as 0, the sum would be 35, which is not divisible by 9, so 1 is indeed the smallest value. Why Other Options Are Wrong: If d = 2, the digit sum is 37, not a multiple of 9.If d = 5, the sum is 40, and if d = 7, the sum is 42. None of these are multiples of 9.Hence 2, 5, and 7 do not make the number divisible by 9. Common Pitfalls: Some learners may confuse the rule for 9 with the rule for 3, but both use the digit sum; the difference is in the multiple used.Others may try to work with the full number and do long division, which is slower and more error prone than using the digit sum method.A common mistake is to accept the first valid d without ensuring it is the smallest, but in this case 1 is already the smallest possible valid digit. Final Answer: The smallest digit that can replace the blank is 1.

Question 19

If the sum of ten different positive integers is 100, what is the greatest possible value of any one of these ten integers?

Accepted Answer

Introduction / Context: This is a classic optimization problem under constraints. We are given ten different positive integers with a fixed total sum, and we must maximize the largest of them. It tests the learner's ability to reason about how to distribute a fixed total among several distinct positive numbers in order to maximize one of them, which is a common theme in combinatorics and logical reasoning questions. Given Data / Assumptions: There are exactly ten positive integers. All ten integers are distinct. Their total sum is 100. We need the greatest possible value of the largest integer among them. Concept / Approach: To maximize one number while keeping the total sum fixed and all numbers different and positive, we should minimize the other nine numbers subject to the constraints. The smallest possible set of nine distinct positive integers is 1, 2, 3, 4, 5, 6, 7, 8, and 9. Once we fix these nine integers, the tenth number is forced by the total sum. This gives the maximum possible value for the largest number, because any attempt to increase it further would require decreasing one of the others below these minimal values or repeating a value, which is not allowed. Step-by-Step Solution: Choose the nine smallest distinct positive integers: 1, 2, 3, 4, 5, 6, 7, 8, and 9.Compute their sum: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9.This sum equals 45.Let the tenth number be M.The total sum of all ten integers is 100.So 45 + M = 100.Therefore, M = 100 - 45 = 55.Thus, the largest possible value of any one integer is 55. Verification / Alternative check: Check distinctness: the set of ten numbers is {1, 2, 3, 4, 5, 6, 7, 8, 9, 55}, which are all distinct positive integers.Check the sum: 45 + 55 = 100, which matches the required total.If we tried to make the largest number greater than 55, the sum of the remaining nine numbers would have to be less than 45.However, the minimal possible sum of nine distinct positive integers is already 45, so we cannot reduce that further, confirming that 55 is the maximum possible value. Why Other Options Are Wrong: 91, 64, and 45 do not represent the true maximum.For example, if the largest number was 64, the remaining nine distinct positive integers would have to sum to 36, which is impossible because the smallest such sum is 45.Similarly, 91 would leave only 9 for nine different positive integers, which is clearly not feasible. Common Pitfalls: A common error is to forget that the integers must be distinct and to use the same small number multiple times, which can falsely suggest a larger maximum.Another mistake is to pick arbitrary small numbers instead of the nine smallest possible positive integers.Miscounting the sum of 1 through 9 can also lead to incorrect conclusions about the maximum value. Final Answer: The greatest possible value among the ten different positive integers is 55.

Question 20

If N = 0.369369369369... and M = 0.531531531531... (both repeating decimals), what is the value of (1 / N) + (1 / M)?

Accepted Answer

Introduction / Context: This problem tests understanding of repeating decimals and their conversion to fractions. We are given two repeating decimals whose repeating parts have length three, and we must compute the sum of their reciprocals. This requires converting each repeating decimal to a rational number, taking reciprocals, and then simplifying the resulting expression. Such problems help strengthen links between decimal representations and rational numbers. Given Data / Assumptions: N = 0.369369369..., where 369 repeats indefinitely. M = 0.531531531..., where 531 repeats indefinitely. We must calculate (1 / N) + (1 / M) as a single simplified fraction. Concept / Approach: A standard technique for converting a repeating decimal with a block of three digits into a fraction is to use the fact that 0.abcabcabc... equals abc / 999, where abc is the three digit integer formed by the repeating block. After converting N and M into fractions, we take their reciprocals, add them, and simplify. Working in fraction form avoids rounding errors and ensures an exact answer. Step-by-Step Solution: First convert N = 0.369369... to a fraction.The repeating block is 369, so N = 369 / 999.Simplify 369 / 999 by dividing numerator and denominator by 9: 369 / 9 = 41 and 999 / 9 = 111.So N = 41 / 111.Next convert M = 0.531531... to a fraction.The repeating block is 531, so M = 531 / 999.Simplify 531 / 999 by dividing numerator and denominator by 9: 531 / 9 = 59 and 999 / 9 = 111.So M = 59 / 111.Now compute 1 / N and 1 / M.1 / N = 1 / (41 / 111) = 111 / 41.1 / M = 1 / (59 / 111) = 111 / 59.We need (1 / N) + (1 / M) = 111 / 41 + 111 / 59.Take the common denominator 41 * 59.Numerator becomes 111 * 59 + 111 * 41 = 111 * (59 + 41) = 111 * 100 = 11100.Denominator is 41 * 59 = 2419.So (1 / N) + (1 / M) = 11100 / 2419. Verification / Alternative check: We can approximate the decimals to check plausibility.N is approximately 0.369 and M is approximately 0.531.Then 1 / N is about 1 / 0.369 ≈ 2.71 and 1 / M is about 1 / 0.531 ≈ 1.88.Their sum is roughly 4.59.Now evaluate 11100 / 2419 numerically: 2419 * 4 = 9676 and 2419 * 4.5 is slightly larger than 10885, so 11100 / 2419 is close to 4.59, which matches the rough decimal estimate. Why Other Options Are Wrong: 111 / 100 equals 1.11, which is far too small compared to the approximate sum of reciprocals near 4.6.1897 / 3162 and 2419 / 11100 are also far from the correct magnitude and do not arise from correct arithmetic with the exact fractions.Only 11100 / 2419 matches the exact fractional computation. Common Pitfalls: Some learners misapply the conversion rule and write 0.abcabc... as abc / 99 instead of abc / 999.Another common error is simplifying 369 / 999 or 531 / 999 incorrectly.Careless mistakes in adding the fractions 111 / 41 and 111 / 59 can also produce wrong numerators or denominators. Final Answer: The value of (1 / N) + (1 / M) is 11100/2419.

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