Question 1

The coordinates of the centroid of triangle ABC are (−1, −2). If the coordinates of vertices A and B are (6, −4) and (−2, 2) respectively, what are the coordinates of vertex C?

Accepted Answer

Introduction / Context: This coordinate geometry problem involves the centroid of a triangle. The centroid is the point where the three medians intersect and is known to be the average of the coordinates of the three vertices. We are given the centroid and two vertices and must find the coordinates of the third vertex. Given Data / Assumptions: Centroid G has coordinates (−1, −2).Vertex A has coordinates (6, −4).Vertex B has coordinates (−2, 2).Vertex C has coordinates (x, y), which we must determine. Concept / Approach: The centroid of a triangle with vertices A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃) is given by G = ((x₁ + x₂ + x₃) / 3, (y₁ + y₂ + y₃) / 3). We know G, A and B, so we can set up equations for the x and y coordinates of G and solve for x and y, the coordinates of C. Step-by-Step Solution: Step 1: Let C be (x, y).Step 2: Use the centroid formula for the x coordinate: (xA + xB + xC) / 3 = xG.Step 3: Substitute values: (6 + (−2) + x) / 3 = −1.Step 4: Simplify the numerator: (4 + x) / 3 = −1, so 4 + x = −3.Step 5: Solve for x: x = −3 − 4 = −7.Step 6: Now use the centroid formula for the y coordinate: (yA + yB + yC) / 3 = yG.Step 7: Substitute values: (−4 + 2 + y) / 3 = −2.Step 8: Simplify the numerator: (−2 + y) / 3 = −2, so −2 + y = −6.Step 9: Solve for y: y = −6 + 2 = −4.Step 10: Therefore, C has coordinates (−7, −4). Verification / Alternative check: Check the centroid using A(6, −4), B(−2, 2) and C(−7, −4). The x coordinate of the centroid is (6 + (−2) + (−7)) / 3 = (−3) / 3 = −1. The y coordinate is (−4 + 2 + (−4)) / 3 = (−6) / 3 = −2. This matches the given centroid (−1, −2), confirming the correctness of C(−7, −4). Why Other Options Are Wrong: The other options change signs or swap coordinates, giving centroids different from (−1, −2) when plugged into the formula. For example, using (7, −4) or (7, 4) would yield a positive x centroid, which contradicts the given centroid. Only (−7, −4) produces the correct average. Common Pitfalls: Some learners incorrectly divide by 2 instead of 3 or forget to include all three vertices when averaging. Others may make sign errors when adding the coordinates. Carefully applying the centroid formula and checking both coordinates prevents these mistakes. Final Answer: The coordinates of vertex C are (−7, −4).

Question 2

Two numbers are 10% and 20% less than a third number respectively. By what percentage must the second (smaller) number be increased to make it equal to the first number?

Accepted Answer

Introduction / Context: This percentage problem compares two numbers that are each given as a percentage less than a third number. We are asked how much the smaller of the two must be increased (in percentage terms) to reach the larger one. It tests comfort with percentage reduction and percentage increase, which are not symmetric operations. Given Data / Assumptions: Let the third (original) number be T.The first number is 10% less than T.The second number is 20% less than T.We must find the percentage increase needed to raise the second number up to the first. Concept / Approach: Express each of the two numbers as a multiple of T. A number that is 10% less than T equals 0.9T; a number that is 20% less equals 0.8T. Then we calculate how much the smaller number (0.8T) must be increased to reach 0.9T. The percentage increase is calculated as (difference / original smaller number) × 100. Step-by-Step Solution: Step 1: First number (10% less than T) = T − 0.10T = 0.90T.Step 2: Second number (20% less than T) = T − 0.20T = 0.80T.Step 3: The increase required to go from the second number to the first is 0.90T − 0.80T = 0.10T.Step 4: The percentage increase is measured relative to the second number, which is 0.80T.Step 5: Percentage increase = (increase / original smaller number) × 100.Step 6: Compute this: percentage increase = (0.10T / 0.80T) × 100.Step 7: Simplify the fraction: 0.10T / 0.80T = 1 / 8 = 0.125.Step 8: Multiply by 100 to get 0.125 × 100 = 12.5%. Verification / Alternative check: Take a simple value for T, such as T = 100. Then the first number is 90 and the second number is 80. The increase needed is 90 − 80 = 10. The percentage increase from 80 to 90 is (10 / 80) × 100 = 12.5%. This numerical check confirms the algebraic result. Why Other Options Are Wrong: A 10% increase from 80 would give 88, not 90. A 15% increase would produce 92, overshooting the first number. An 8% increase would give only 86.4. None of these match the required target of 90, so they are incorrect. Common Pitfalls: One common error is to subtract the percentage differences directly (20% − 10% = 10%) and assume that the answer is 10%, forgetting that percentage increase should be calculated relative to the smaller number. Another mistake is to compute the increase relative to T instead of the second number itself. Final Answer: The second number must be increased by 12.5% to equal the first number.

Question 3

In an arithmetic progression, the 2nd term is 8 and the 6th term is 20. What is the value of the 20th term of this A.P.?

Accepted Answer

Introduction / Context: This question asks for a specific term in an arithmetic progression (A.P.) when two earlier terms are known. By using the general formula for the n-th term, we can set up equations to find the first term and common difference, and then compute the required term. Given Data / Assumptions: Let the first term of the A.P. be a and the common difference be d.The 2nd term T2 is 8.The 6th term T6 is 20.We are asked to find the 20th term T20. Concept / Approach: The n-th term of an A.P. is given by Tn = a + (n − 1)d. Using this formula for T2 and T6 gives us a pair of linear equations in a and d. Once we compute a and d, we can use the same formula to find T20. Step-by-Step Solution: Step 1: Write T2 and T6 using Tn = a + (n − 1)d.Step 2: T2 = a + d = 8.Step 3: T6 = a + 5d = 20.Step 4: Subtract the equation for T2 from that for T6: (a + 5d) − (a + d) = 20 − 8.Step 5: This simplifies to 4d = 12, so d = 3.Step 6: Substitute d = 3 into a + d = 8: a + 3 = 8, so a = 5.Step 7: Now find the 20th term: T20 = a + 19d = 5 + 19 × 3.Step 8: Compute 19 × 3 = 57, so T20 = 5 + 57 = 62. Verification / Alternative check: We can write several terms to confirm the pattern: 5, 8, 11, 14, 17, 20, ... The 2nd term is indeed 8 and the 6th is 20. With d = 3, every step increases by 3. Counting forward, the 20th term works out to be 62, matching our calculation. Why Other Options Are Wrong: Values like 56 or 65 might arise from miscounting the number of steps (using 18d or 20d instead of 19d) or from an incorrect d. The value 69 is too large and indicates an error in computing a or d. Only 62 is consistent with the given terms and the formula. Common Pitfalls: One frequent error is using Tn = a + nd instead of a + (n − 1)d, which shifts all positions by one. Another is rearranging the linear equations incorrectly when solving for a and d. Keeping track of indices carefully and checking earlier terms against the given values can catch such mistakes. Final Answer: The 20th term of the arithmetic progression is 62.

Question 4

Out of a group of three numbers, the second number is three times the first and also twice the third. If the sum of the three numbers is 55, what is the largest number in the group?

Accepted Answer

Introduction / Context: This algebraic word problem describes relationships among three numbers and their total sum. The second number is related both to the first and to the third, and we must use these relationships to form equations. The goal is to find which number is the largest and its value. Given Data / Assumptions: Let the three numbers be x (first), y (second) and z (third).The second number y is three times the first: y = 3x.The second number y is also twice the third: y = 2z.The sum x + y + z equals 55. Concept / Approach: We express all three numbers in terms of a single variable. From y = 3x and y = 2z, we can write y and z in terms of x, then substitute into the sum equation. This produces a single equation in x, which we solve. After that, we compute y and z and identify the largest number among them. Step-by-Step Solution: Step 1: From y = 3x, we have y expressed in terms of x.Step 2: From y = 2z, solve for z in terms of x: y = 2z implies 3x = 2z, so z = (3x) / 2.Step 3: Write the sum equation: x + y + z = 55.Step 4: Substitute y = 3x and z = 3x / 2 into the sum: x + 3x + 3x / 2 = 55.Step 5: Combine like terms: x + 3x = 4x, so we have 4x + 3x / 2 = 55.Step 6: Express 4x as 8x / 2 to get a common denominator: 8x / 2 + 3x / 2 = 55.Step 7: Add the fractions: (11x / 2) = 55.Step 8: Multiply both sides by 2: 11x = 110, so x = 10.Step 9: Then y = 3x = 3 × 10 = 30 and z = 3x / 2 = 30 / 2 = 15. Verification / Alternative check: Check the relationships: y is three times x (30 = 3 × 10) and y is twice z (30 = 2 × 15). The sum x + y + z = 10 + 30 + 15 = 55 matches the given total. These confirm that the numbers 10, 30 and 15 satisfy all conditions and that 30 is indeed the largest. Why Other Options Are Wrong: Options such as 26, 29 or 32 do not satisfy the equations when treated as the largest number y. For instance, if y were 26, then x would be about 26 / 3 and would not combine with a consistent z to give a sum of 55 while preserving the relationships. Only y = 30 satisfies all conditions. Common Pitfalls: Students may confuse which number is twice the other or incorrectly set up the relationships (for example, writing x = 3y instead of y = 3x). Fraction handling in the step z = 3x / 2 also sometimes causes errors. Always clearly assign variables and recheck that each equation reflects the wording of the problem. Final Answer: The largest number in the group is 30.

Question 5

What is the average (arithmetic mean) of the first 11 multiples of 11?

Accepted Answer

Introduction / Context: This question involves finding the average of the first 11 multiples of 11. These numbers form an arithmetic progression, and there is a simple way to find their average without listing every term. This tests understanding of averages and the structure of arithmetic sequences. Given Data / Assumptions: The first 11 multiples of 11 are 11, 22, 33, ..., 121.We must compute their average (sum divided by 11).This is an arithmetic progression with common difference 11. Concept / Approach: In an arithmetic progression, the average of the first n terms is equal to the average of the first and last term, that is, (first term + last term) / 2. Since these are consecutive multiples of 11, we can identify the first multiple (11) and the 11th multiple (11 × 11 = 121) and use this property directly. Step-by-Step Solution: Step 1: List enough terms to see the pattern: 11, 22, 33, ..., 121.Step 2: Recognize that this is an arithmetic progression with first term a = 11 and common difference d = 11.Step 3: The 11th term is 11 × 11 = 121.Step 4: Use the fact that the average of an arithmetic series is (first term + last term) / 2.Step 5: Compute average = (11 + 121) / 2.Step 6: Add the numerator: 11 + 121 = 132.Step 7: Divide by 2: 132 / 2 = 66. Verification / Alternative check: We can also compute the sum explicitly. The sum S of n terms of an A.P. is S = n * (first term + last term) / 2. Here n = 11, a = 11 and last term l = 121. So S = 11 * (11 + 121) / 2 = 11 * 132 / 2 = 11 * 66 = 726. The average is S / 11 = 726 / 11 = 66, confirming the result. Why Other Options Are Wrong: The value 22 is the second multiple of 11, not the average. The values 44 and 55 are mid range but not the exact center of the sequence. Because the sequence runs from 11 to 121, the true middle value, and hence the average, must be halfway between 11 and 121, which is 66. Common Pitfalls: Some learners mistakenly think the average is simply the 6th multiple of 11 (which is 66 here by coincidence) without realizing why this works. While that gives the correct numerical answer in this case, the more general and reliable method is using (first term + last term) / 2 or the full sum formula. Final Answer: The average of the first 11 multiples of 11 is 66.

Question 6

A number is greater than 58 times its reciprocal by 3/4. If x is the number, then x − 58/x = 3/4. What is the value of x?

Accepted Answer

Introduction / Context: This algebraic problem involves a number and its reciprocal. We are told that the number exceeds 58 times its reciprocal by 3/4. Translating this into an equation in x leads to a quadratic equation, which we can solve to find the value(s) of the number. Given Data / Assumptions: Let the number be x (x ≠ 0).The number is greater than 58 times its reciprocal by 3/4.This means x − 58/x = 3/4.We must solve for x and choose the correct value from the options. Concept / Approach: The equation x − 58/x = 3/4 contains both x and 1/x. To clear the fraction in the denominator, we multiply both sides by x, converting the equation into a quadratic form. Then we clear the remaining fraction by multiplying through by 4 and solve the quadratic equation using the discriminant method. Step-by-Step Solution: Step 1: Start with x − 58/x = 3/4.Step 2: Multiply both sides by x (x ≠ 0) to eliminate the denominator: x^2 − 58 = (3/4) x.Step 3: Multiply both sides by 4 to clear the fractional coefficient: 4x^2 − 232 = 3x.Step 4: Bring all terms to one side: 4x^2 − 3x − 232 = 0.Step 5: This is a quadratic equation with a = 4, b = −3 and c = −232.Step 6: Compute the discriminant D = b^2 − 4ac = (−3)^2 − 4 × 4 × (−232) = 9 + 3712 = 3721.Step 7: Note that 61^2 = 3721, so √D = 61.Step 8: Solve for x: x = [3 ± 61] / (2 × 4) = (3 ± 61) / 8.Step 9: The two roots are x = (3 + 61) / 8 = 64 / 8 = 8 and x = (3 − 61) / 8 = −58 / 8 = −29 / 4. Verification / Alternative check: Check x = 8 in the original equation. The reciprocal is 1/8, so 58 times the reciprocal is 58 × 1/8 = 7.25. Then x − 58/x = 8 − 7.25 = 0.75 = 3/4, which satisfies the condition. The other root, −29/4, is not among the answer options. The closest option is −8 or 12, but neither satisfies the original equation. Why Other Options Are Wrong: Substituting x = 12 gives 12 − 58/12, which is not equal to 3/4. Similarly, x = −8 or x = −12 do not satisfy the equation x − 58/x = 3/4. Only x = 8 fulfills the relationship exactly. Common Pitfalls: Typical errors include misinterpreting the phrase “greater than 58 times its reciprocal by 3/4” (which must be written as x − 58/x = 3/4), or making algebraic mistakes when clearing denominators and fractions. Another common issue is miscomputing the discriminant or its square root. Final Answer: The number that satisfies the condition is 8.

Question 7

What is the sum of the first 13 terms of an arithmetic progression if the first term is −10 and the 13th (last given) term is 26?

Accepted Answer

Introduction / Context: This question asks for the sum of a finite arithmetic progression (A.P.) when the first term and the last term of the sequence are known. The number of terms is 13. This is a direct application of the sum formula for an A.P. Given Data / Assumptions: The first term a is −10.The last (13th) term l is 26.The number of terms n is 13.We must find S13, the sum of the first 13 terms. Concept / Approach: For an arithmetic progression with n terms, first term a and last term l, the sum Sn is given by Sn = n * (a + l) / 2. Here we already know n, a and l, so there is no need to calculate the common difference. We simply substitute into the formula. Step-by-Step Solution: Step 1: Recall the sum formula Sn = n * (a + l) / 2.Step 2: Substitute n = 13, a = −10 and l = 26.Step 3: Compute a + l = −10 + 26 = 16.Step 4: Compute S13 = 13 * 16 / 2.Step 5: Simplify 16 / 2 = 8, so S13 = 13 * 8.Step 6: Multiply 13 by 8 to get S13 = 104. Verification / Alternative check: We can also confirm that 26 is indeed the 13th term. The common difference d can be found from the relation l = a + (n − 1)d: 26 = −10 + 12d. Then 26 + 10 = 36, so 12d = 36 and d = 3. The sequence is −10, −7, −4, −1, 2, 5, 8, 11, 14, 17, 20, 23, 26. Summing these manually gives the same result: pairs (−10, 26), (−7, 23), (−4, 20), (−1, 17), (2, 14), (5, 11) each add to 16 (six pairs), and the middle term 8. So total is 6 × 16 + 8 = 96 + 8 = 104. Why Other Options Are Wrong: Values like 140 or 98 could arise from miscalculating a + l or forgetting to divide by 2 in the sum formula. The value 84 would correspond to a shorter or different sequence. None of these obey Sn = n * (a + l) / 2 with the given values. Common Pitfalls: Some learners accidentally use (n − 1) instead of n in the sum formula or mix up a and l. Others forget that the formula already accounts for all n terms and attempt to add an extra term. Carefully substituting the values and simplifying step by step prevents such errors. Final Answer: The sum of the first 13 terms of the A.P. is 104.

Question 8

If the number 583_437 is completely divisible by 9, then what is the smallest whole digit that can be placed in the blank position (represented by _) to make the number divisible by 9?

Accepted Answer

Introduction / Context: This question tests the divisibility rule for 9 and asks you to determine a missing digit in a large number so that the entire number is divisible by 9. Among all such digits that work, you must choose the smallest one. Given Data / Assumptions: The number is 583_437, where the blank _ is a single digit from 0 to 9.The complete number must be divisible by 9.We must find the smallest digit that can replace the blank. Concept / Approach: A number is divisible by 9 if and only if the sum of its digits is a multiple of 9. We first compute the sum of the known digits in 583_437, then add the unknown digit x. We then find values of x that make the total a multiple of 9, and select the smallest such value. Step-by-Step Solution: Step 1: List the digits of 583_437: 5, 8, 3, x, 4, 3, 7.Step 2: Compute the sum of the known digits: 5 + 8 + 3 + 4 + 3 + 7.Step 3: Add these: 5 + 8 = 13, 13 + 3 = 16, 16 + 4 = 20, 20 + 3 = 23, 23 + 7 = 30.Step 4: The total digit sum is 30 + x.Step 5: For divisibility by 9, 30 + x must be a multiple of 9.Step 6: Multiples of 9 close to 30 are 27 and 36. Since 30 + x must be at least 30, the first possible multiple is 36.Step 7: Set 30 + x = 36 to get x = 6.Step 8: Another possible multiple would be 45, giving x = 15, which is not a single digit, so it is invalid.Step 9: Therefore, the only valid choice for x is 6, which is also the smallest possible. Verification / Alternative check: Check the full number 5836437. The digit sum is 5 + 8 + 3 + 6 + 4 + 3 + 7 = 36. Since 36 is a multiple of 9, the number is divisible by 9. No smaller digit x from 0 to 5 gives a total digit sum that is a multiple of 9. Why Other Options Are Wrong: If x = 3, the sum would be 33, which is not a multiple of 9. If x = 4, the sum would be 34, and if x = 5, the sum would be 35, neither of which is divisible by 9. Only x = 6 gives sum 36, satisfying the divisibility rule. Common Pitfalls: Some learners mistakenly look for divisibility by 3 instead of 9, or forget that the blank must be a single digit. Others incorrectly add the digits or stop at the first number they find without checking whether it is a multiple of 9. Being systematic with the digit sum and checking nearest multiples of 9 avoids these errors. Final Answer: The smallest digit that can replace the blank is 6.

Question 9

The sum of a non-zero number and three times its reciprocal is 52/7. What is the larger of the possible values of this number?

Accepted Answer

Introduction / Context: This problem tests your ability to set up and solve an equation that involves both a number and its reciprocal. Such questions are common in algebra and quantitative aptitude because they combine fractions, quadratic equations and careful interpretation of the worded condition. Here, the expression uses three times the reciprocal of the number, and the total is given as 52/7. We are specifically asked for the larger possible value of the number. Given Data / Assumptions: - Let the non-zero number be x. - Its reciprocal is 1 / x. - The sum x + 3 * (1 / x) is equal to 52 / 7. - x is real and non zero so that the reciprocal is defined. - We are asked to report the larger of the possible values of x. Concept / Approach: The key idea is to translate the verbal description into an algebraic equation and then clear the denominator by multiplying through by x. This produces a quadratic equation in standard form. Solving the quadratic using factoring or the quadratic formula yields two possible values for x. Finally, we compare these values and select the larger one as required by the question. Careful simplification of fractions is important to avoid errors. Step-by-Step Solution: Step 1: Write the equation from the statement: x + 3 * (1 / x) = 52 / 7. Step 2: Combine the terms: x + 3 / x = 52 / 7. Step 3: Multiply both sides by x to eliminate the denominator: x^2 + 3 = (52 / 7) * x. Step 4: Multiply both sides by 7 to clear the fraction: 7x^2 + 21 = 52x. Step 5: Bring all terms to one side: 7x^2 - 52x + 21 = 0. Step 6: This is a quadratic equation in standard form ax^2 + bx + c = 0. Step 7: Compute the discriminant: D = b^2 - 4ac = 52^2 - 4 * 7 * 21 = 2704 - 588 = 2116. Step 8: The square root of 2116 is 46, since 46 * 46 = 2116. Step 9: Use the quadratic formula: x = [52 ± 46] / (2 * 7) = [52 ± 46] / 14. Step 10: First root: x = (52 + 46) / 14 = 98 / 14 = 7. Step 11: Second root: x = (52 - 46) / 14 = 6 / 14 = 3 / 7. Step 12: The two possible values of x are 7 and 3 / 7. The larger of these is 7. Verification / Alternative check: Substitute x = 7 back into the original expression. The reciprocal is 1 / 7. Compute x + 3 * (1 / x) = 7 + 3 / 7 = (49 / 7) + (3 / 7) = 52 / 7, which matches the given value. For completeness, if x = 3 / 7, its reciprocal is 7 / 3, and x + 3 * (1 / x) again simplifies to 52 / 7. Both values satisfy the equation, so choosing the larger one, 7, is correct according to the question. Why Other Options Are Wrong: Values 6, 8 and 9 do not satisfy x + 3 / x = 52 / 7 when substituted. For example, with x = 8, we get 8 + 3 / 8, which is clearly far from 52 / 7. These values come from guessing rather than solving the equation and therefore cannot be correct answers. Common Pitfalls: Some learners forget to multiply the entire equation by x or 7 when clearing denominators, which leads to incomplete or incorrect equations. Others might mistakenly think there is only one solution and ignore the second root. It is important to remember that quadratic equations typically have two solutions and then use the wording of the problem to decide which one is required. Careful algebraic manipulation and verification by substitution help avoid such mistakes. Final Answer: The larger possible value of the number is 7, which matches option C.

Question 10

Among three numbers, the first is twice the second and three times the third. If the average of the three numbers is 429, what is the difference between the first and the third number?

Accepted Answer

Introduction / Context: This question checks your understanding of averages and linear relationships between numbers. Three numbers are linked by simple multiplier relationships: the first is twice the second and three times the third. The average of the three numbers is given, and you are asked to find the difference between the first and the third number. This is a typical algebraic word problem where forming the correct equations is the key step. Given Data / Assumptions: - There are three numbers: first, second and third. - First number = 2 * (second number). - First number = 3 * (third number). - The average of the three numbers is 429. - We need the value of (first number) minus (third number). Concept / Approach: Averages relate the sum of numbers to the number of terms. If the average of three numbers is known, their total sum is simply 3 times the average. Using the given multiplier relationships, we can express all three numbers in terms of a single variable representing the first number. Substituting into the average formula produces an equation that we solve for this first number. Then we compute the third number and finally their difference. Expressing everything in one variable keeps the algebra straightforward. Step-by-Step Solution: Step 1: Let the first number be a. Step 2: The first is twice the second, so the second number is a / 2. Step 3: The first is three times the third, so the third number is a / 3. Step 4: The average of the three numbers is 429, so (a + a / 2 + a / 3) / 3 = 429. Step 5: Combine terms in the numerator using a common denominator. The coefficients are 1, 1/2 and 1/3. Their sum is 1 + 1/2 + 1/3 = 6/6 + 3/6 + 2/6 = 11/6. Step 6: So the numerator is (11/6) * a, and the average is [(11/6) * a] / 3 = (11/18) * a. Step 7: Set this equal to 429: (11/18) * a = 429. Step 8: Solve for a: a = 429 * 18 / 11. Step 9: Compute 429 / 11 = 39, because 11 * 39 = 429. Step 10: Then a = 39 * 18 = 702. Step 11: The first number is 702. The third number is a / 3 = 702 / 3 = 234. Step 12: The required difference is first minus third = 702 - 234 = 468. Verification / Alternative check: Compute all three numbers explicitly. First = 702. Second = a / 2 = 351. Third = 234. Check the relationships: 702 is indeed twice 351 and three times 234. Now check the average: (702 + 351 + 234) / 3 = 1287 / 3 = 429, which matches the given average. This confirms that the numbers are correct and the difference 468 is accurate. Why Other Options Are Wrong: Values 412, 517 and 427 come from incorrect algebra or arithmetic. For example, a mistake in summing the fractional coefficients or mishandling division by 11 can easily produce such wrong differences. None of these values correspond to the difference between 702 and 234, which is the only pair that satisfies all conditions of the problem. Common Pitfalls: Students sometimes mix up which number is multiple of which, for example treating the second as twice the first instead of the other way around. Another frequent error is in handling the fractions when adding 1, 1/2 and 1/3, or forgetting to divide the sum by 3 when using the average formula. Writing each step clearly, especially the conversion to a single variable, greatly reduces the chance of such errors. Final Answer: The difference between the first and the third number is 468, which matches option B.

Question 11

The digits 2, 5, 0, 6 and 8 are used to form five-digit numbers without repetition. What is the difference between the greatest and the least such five-digit numbers?

Accepted Answer

Introduction / Context: This problem is about forming numbers from given digits under specific rules and then comparing those numbers. You are given the digits 2, 5, 0, 6 and 8 and asked to form the greatest and least possible five digit numbers without repeating any digit. After that, you must compute the difference between these two extreme values. This tests place value understanding and logical arrangement of digits rather than heavy computation. Given Data / Assumptions: - Available digits are 2, 5, 0, 6 and 8. - Each digit can be used at most once in a number. - We must form valid five digit numbers, so the leading digit cannot be 0. - We need the greatest and least such numbers and their difference. Concept / Approach: To form the greatest five digit number, we place the largest available digit in the highest place value (ten thousand place), then the next largest in the thousand place and so on, making sure we do not put 0 in the first position. To get the least five digit number, we place the smallest non zero digit in the first place, then arrange the remaining digits in ascending order from left to right, including 0 after the first digit. Once we have both numbers, we subtract the smaller from the larger to find the required difference. Step-by-Step Solution: Step 1: List the digits in increasing order: 0, 2, 5, 6, 8. Step 2: To form the greatest five digit number, choose the largest digit for the first place. That is 8. Step 3: For the remaining places, use the remaining digits in descending order: 6, 5, 2 and 0. Step 4: The greatest five digit number is therefore 86520. Step 5: To form the least five digit number, the first digit cannot be 0, so choose the smallest non zero digit, which is 2, for the ten thousand place. Step 6: Arrange the remaining digits 0, 5, 6 and 8 in ascending order for the remaining places: 0, 5, 6, 8. Step 7: The least five digit number is 20568. Step 8: Now compute the difference: 86520 - 20568. Step 9: Subtract 20568 from 86520. First subtract 20000 to get 66520, then subtract 568 to get 65952. Step 10: Thus, the required difference is 65952. Verification / Alternative check: You can quickly verify the subtraction using column method. Writing 86520 minus 20568 and borrowing as needed will yield the same result of 65952. Also, check that no other arrangement can produce a larger or smaller five digit number under the conditions. Any attempt to place a digit smaller than 8 in the first position of the greatest number or a digit larger than 2 in the first position of the least number will make the numbers less extreme, so the chosen arrangements are indeed optimal. Why Other Options Are Wrong: Values like 69552, 65925 and 63952 represent subtracting from slightly different largest or smallest numbers, often due to misplacing 0 or mixing up the digit order. For example, using 85620 as the greatest or 25068 as the least will alter the difference. Only 65952 corresponds to correctly formed extreme numbers that respect place value and the no repetition rule. Common Pitfalls: A typical mistake is putting 0 in the ten thousand place when forming the smallest number, which would create a four digit number rather than a valid five digit one. Another error is not strictly sorting digits for the largest or smallest arrangements, especially when 0 is involved. Remember that the leading digit of a multi digit number can never be 0, and always order remaining digits carefully once the first position is fixed. Final Answer: The difference between the greatest and least such five digit numbers is 65952, which corresponds to option C.

Question 12

The average of five numbers is 76. The first number is equal to 3/7 of the sum of the other four numbers. What is the value of the first number?

Accepted Answer

Introduction / Context: This question combines the concept of average with a proportional relationship between one term and the sum of the remaining terms. You are told that the average of five numbers is 76 and that the first number is three sevenths of the sum of the other four. From this information, you must find the value of the first number. Such problems are very common in aptitude tests and help reinforce algebraic thinking with averages. Given Data / Assumptions: - There are five numbers in total. - Their average is 76. - Let the first number be x. - The sum of the other four numbers is 3/7 related to x: specifically, x = (3 / 7) * (sum of the other four). - We need to determine the value of x. Concept / Approach: The average of five numbers being 76 means their total sum is 5 * 76. If we call the first number x, then the sum of the remaining four numbers is the total minus x. The condition that the first number is three sevenths of the sum of the other four gives a direct equation involving x and the remaining sum. We set up this equation, substitute the total sum in terms of the average and then solve for x. This approach uses only basic algebra and the definition of average. Step-by-Step Solution: Step 1: Let the five numbers be x, a, b, c and d, where x is the first number. Step 2: Their average is 76, so the total sum is 5 * 76 = 380. Step 3: The sum of the other four numbers is a + b + c + d = 380 - x. Step 4: The condition states that the first number x is three sevenths of the sum of the other four numbers: x = (3 / 7) * (380 - x). Step 5: Multiply both sides by 7 to clear the denominator: 7x = 3 * (380 - x). Step 6: Expand the right side: 7x = 1140 - 3x. Step 7: Bring all x terms to one side: 7x + 3x = 1140, so 10x = 1140. Step 8: Solve for x by dividing both sides by 10: x = 114. Step 9: Therefore, the first number is 114. Verification / Alternative check: Check the relationship using the calculated value x = 114. The sum of the other four numbers is 380 - 114 = 266. Compute (3 / 7) of 266: 266 / 7 = 38, and 3 * 38 = 114. This matches x, so the proportional condition is satisfied. The average also remains 76 because the total sum is still 380. These checks confirm that 114 is correct. Why Other Options Are Wrong: If the first number were 171, the sum of the remaining four would be 380 - 171 = 209, and 3/7 of 209 is not 171. Similar mismatches occur for 76 and 228. None of these values produce a consistent relationship between the first number and the sum of the other four, nor do they maintain the average of 76 in the correct way. Only 114 satisfies both conditions simultaneously. Common Pitfalls: One common mistake is to misread the condition and assume the first number is 3/7 of the total sum instead of the sum of the other four. Another error is to forget that the sum of the other four is total minus x, leading to an incorrect equation. Carefully interpreting the phrase and writing each step symbolically helps avoid these misunderstandings. Final Answer: The value of the first number is 114, which corresponds to option B.

Question 13

If the sum of a non-zero real number and its reciprocal is 2, what is the value of the number?

Accepted Answer

Introduction / Context: This is a classic algebra question involving a number and its reciprocal. You are told that the sum of a non-zero real number and its reciprocal equals 2, and you must determine the number. The problem leads naturally to a quadratic equation and tests basic equation solving skills and understanding of reciprocals. Given Data / Assumptions: - Let the non-zero real number be x. - Its reciprocal is 1 / x. - The equation given is x + 1 / x = 2. - x cannot be 0 because the reciprocal 1 / x would be undefined. Concept / Approach: The usual method is to clear the denominator by multiplying the equation by x, which results in a quadratic equation in x. We then solve this quadratic, either by factoring or using the quadratic formula. It is important to remember domain restrictions: x must be non zero. After finding all algebraic solutions, we check them against the options and the domain condition before choosing the final answer. Step-by-Step Solution: Step 1: Start from the equation x + 1 / x = 2. Step 2: Multiply both sides of the equation by x to remove the fraction: x^2 + 1 = 2x. Step 3: Rearrange to standard quadratic form: x^2 - 2x + 1 = 0. Step 4: Recognize that x^2 - 2x + 1 is a perfect square: x^2 - 2x + 1 = (x - 1)^2. Step 5: So the equation becomes (x - 1)^2 = 0. Step 6: Take the square root of both sides: x - 1 = 0. Step 7: Hence x = 1. Step 8: There is only one real solution, and it satisfies x not equal to 0. Verification / Alternative check: Substitute x = 1 back into the original equation to verify: x + 1 / x = 1 + 1 / 1 = 1 + 1 = 2. This matches the given sum, so x = 1 is indeed a valid solution. Since the quadratic factors as a perfect square, there are no other distinct real solutions to check, and no extraneous roots are introduced by clearing the denominator as x is non zero. Why Other Options Are Wrong: Option A (0) is invalid because the reciprocal 1 / 0 is undefined. Option C (-1) gives -1 + (-1) = -2, not 2. Option D (2) gives 2 + 1/2 = 2.5, which is also not equal to 2. Only x = 1 satisfies the given equation and respects the non-zero condition on the number. Common Pitfalls: Some learners may try to guess the answer and overlook the non-zero restriction, or they might attempt to square both sides directly, which is unnecessary and can create extraneous solutions. Others might incorrectly factor the quadratic or mis-handle signs when rearranging terms. Writing the equation carefully, using standard algebraic techniques and then verifying by substitution helps avoid these errors. Final Answer: The number whose sum with its reciprocal is 2 is 1, which corresponds to option B.

Question 14

What are the two consecutive natural numbers whose sum of squares is equal to 145?

Accepted Answer

Introduction / Context: This question asks you to find two consecutive natural numbers whose squares add up to 145. It tests your ability to model consecutive numbers with algebra, set up an equation using their squares and then solve the resulting quadratic equation. Consecutive number problems appear frequently in aptitude tests and help build comfort with simple algebraic modeling. Given Data / Assumptions: - There are two consecutive natural numbers. - Let the smaller one be n, so the next is n + 1. - Their sum of squares is 145. - We are looking for natural number solutions, that is positive integers. Concept / Approach: Consecutive numbers can be easily expressed as n and n + 1. The condition on the sum of squares translates into the equation n^2 + (n + 1)^2 = 145. Expanding and simplifying this expression will give a quadratic equation in n. Solving this equation yields the possible values of n, and we then check which solutions are valid natural numbers. Finally, we report the pair that satisfies the original condition. Step-by-Step Solution: Step 1: Let the two consecutive natural numbers be n and n + 1. Step 2: Write the condition on their squares: n^2 + (n + 1)^2 = 145. Step 3: Expand (n + 1)^2: (n + 1)^2 = n^2 + 2n + 1. Step 4: Substitute into the equation: n^2 + n^2 + 2n + 1 = 145. Step 5: Combine like terms: 2n^2 + 2n + 1 = 145. Step 6: Subtract 145 from both sides: 2n^2 + 2n + 1 - 145 = 0, so 2n^2 + 2n - 144 = 0. Step 7: Divide the entire equation by 2 to simplify: n^2 + n - 72 = 0. Step 8: Factor the quadratic: n^2 + n - 72 = (n + 9)(n - 8) = 0. Step 9: Set each factor equal to zero: n + 9 = 0 or n - 8 = 0. Step 10: So n = -9 or n = 8. Step 11: Since we need natural numbers, we discard n = -9 and keep n = 8. Step 12: The consecutive numbers are therefore 8 and 9. Verification / Alternative check: Verify by direct substitution. Compute 8^2 + 9^2 = 64 + 81 = 145, which matches the required sum. None of the other option pairs have this property. For example, 6^2 + 7^2 = 36 + 49 = 85, and 9^2 + 10^2 = 81 + 100 = 181, both of which differ from 145. This confirms that 8 and 9 are the correct numbers. Why Other Options Are Wrong: The pairs 6 and 7, 13 and 14, and 9 and 10 do not satisfy the sum of squares condition. Their squared sums are 85, 365 and 181 respectively, none of which equal 145. These values may come from miscalculating the quadratic or guessing without checking. Only the pair 8 and 9 produces the correct total of 145 for the sum of squares. Common Pitfalls: Some learners may misinterpret the phrase "consecutive natural numbers" and treat the numbers as n and n + 2 or use only one equation and guess the second number. Others might miscalculate the square or forget to combine all terms correctly when simplifying the quadratic. Taking care with the algebraic expansion and factoring ensures a smooth path to the correct solution. Final Answer: The required consecutive natural numbers are 8 and 9, which matches option A.

Question 15

A number is greater than twice its reciprocal by 31/4. What is the larger of the possible values of this number?

Accepted Answer

Introduction / Context: This question involves a number and its reciprocal linked by a linear relationship. The statement says that the number is greater than twice its reciprocal by 31/4. From this, you are asked to find the larger of the possible values of the number. This leads to a quadratic equation and tests your ability to handle algebra involving reciprocals, fractions and multiple solutions. Given Data / Assumptions: - Let the non-zero number be x. - Its reciprocal is 1 / x. - The number x is greater than 2 * (1 / x) by 31 / 4. - This means x - 2 / x = 31 / 4. - We must find the larger of the possible values of x. Concept / Approach: The given relation directly translates into an equation that we can manipulate into a quadratic. Multiplying through by x clears the denominator. After rearranging into standard quadratic form, we solve using the quadratic formula or factoring. This will usually produce two real solutions. Because the question explicitly asks for the larger value, we will compare the solutions and report the greater one. Throughout, it is important to remember that x must be non zero so that the reciprocal is defined. Step-by-Step Solution: Step 1: Translate the relationship into an equation: x - 2 / x = 31 / 4. Step 2: Multiply both sides by x to eliminate the denominator: x^2 - 2 = (31 / 4) * x. Step 3: Multiply both sides by 4 to clear the fraction: 4x^2 - 8 = 31x. Step 4: Rearrange to standard quadratic form: 4x^2 - 31x - 8 = 0. Step 5: Identify coefficients: a = 4, b = -31, c = -8. Step 6: Compute the discriminant D = b^2 - 4ac = (-31)^2 - 4 * 4 * (-8) = 961 + 128 = 1089. Step 7: The square root of 1089 is 33, since 33 * 33 = 1089. Step 8: Use the quadratic formula: x = [31 ± 33] / (2 * 4) = [31 ± 33] / 8. Step 9: First root: x = (31 + 33) / 8 = 64 / 8 = 8. Step 10: Second root: x = (31 - 33) / 8 = -2 / 8 = -1 / 4. Step 11: The two possible values of x are 8 and -1 / 4, and the larger of these is 8. Verification / Alternative check: Verify the relation for x = 8. The reciprocal is 1 / 8. Compute twice the reciprocal: 2 * (1 / 8) = 1 / 4. Then x - 2 / x becomes 8 - 1 / 4 = (32 / 4) - (1 / 4) = 31 / 4, exactly as given in the problem statement. This confirms that x = 8 satisfies the condition. If desired, you can also check x = -1 / 4, which also satisfies the equation, but is smaller than 8 and therefore not the answer requested. Why Other Options Are Wrong: Values 6, 7 and 9 do not satisfy the relationship when substituted into x - 2 / x. For example, with x = 7, we get 7 - 2 / 7, which is far from 31 / 4. These values might look attractive as simple integers but do not arise from solving the quadratic equation correctly. Only 8 is the larger solution produced by the correct algebraic process. Common Pitfalls: Common errors include forgetting to multiply the entire equation by x or 4 when clearing denominators, which leads to an incomplete or incorrect quadratic. Some students may not consider that a quadratic can have two real solutions and thus fail to check both, or they might misinterpret the phrase "greater than" when forming the equation. Writing each step explicitly, including the non zero restriction, and verifying solutions in the original relation helps avoid these issues. Final Answer: The larger possible value of the number is 8, which corresponds to option B.

Question 16

The product of the digits of a two-digit number is 27. If 54 is added to the number, the resulting number has its digits interchanged. What is the original two-digit number?

Accepted Answer

Introduction / Context: This question combines number properties with a bit of algebra. You are told that a certain two digit number has digits whose product is 27. When 54 is added to this number, the digits of the new number are the reverse of the original digits. You must find the original two digit number. Problems like this test understanding of place value and how to represent a two digit number algebraically in terms of its tens and units digits. Given Data / Assumptions: - Let the two digit number be 10a + b, where a is the tens digit and b is the units digit. - a is a digit from 1 to 9, and b is a digit from 0 to 9. - The product of the digits is a * b = 27. - Adding 54 to the original number gives a new number whose digits are interchanged, that is 10b + a. Concept / Approach: We use two conditions: one for the product of the digits and another for the digit reversal after adding 54. The product condition restricts the possible pairs (a, b) to those whose product is 27. The reversal condition becomes a linear equation in a and b based on the decimal representation of the numbers. By solving this system of conditions, we identify the correct pair of digits and hence the original two digit number. This approach is systematic and avoids guesswork. Step-by-Step Solution: Step 1: From the product condition a * b = 27, list all possible integer digit pairs (a, b) with a from 1 to 9 and b from 0 to 9. Step 2: Factor 27. Possible pairs are (3, 9) and (9, 3). The combinations (1, 27) or (27, 1) are not valid digit pairs. Step 3: Now apply the reversal condition. The original number is 10a + b. After adding 54, the new number is 10b + a. Step 4: Write the equation: 10a + b + 54 = 10b + a. Step 5: Simplify: 10a + b + 54 = 10b + a implies 9a - 9b = -54. Step 6: Divide both sides by 9: a - b = -6, so a = b - 6. Step 7: Consider the first candidate pair from the product condition: (a, b) = (3, 9). Here a - b = 3 - 9 = -6, which matches the equation. Step 8: Consider the second candidate: (a, b) = (9, 3). Here a - b = 9 - 3 = 6, which does not match -6, so this pair is invalid. Step 9: Therefore, the digits must be a = 3 and b = 9, and the original number is 10a + b = 10 * 3 + 9 = 39. Verification / Alternative check: Check both conditions for the number 39. The digits are 3 and 9, whose product is 3 * 9 = 27, satisfying the first condition. Next, add 54: 39 + 54 = 93. The new number 93 has digits 9 and 3, which are indeed the reverse of the original digits. This confirms that 39 is the correct number and that no other candidate satisfies both conditions simultaneously. Why Other Options Are Wrong: Option B (93) is actually the result after adding 54, not the original number. Options C (63) and D (36) have digit products 18 and 18 respectively, not 27, so they fail the first condition. Even if one tried to apply the reversal condition, adding 54 to these numbers does not produce a number with digits reversed. Only 39 satisfies all the given requirements. Common Pitfalls: Some learners may forget that the tens digit cannot be zero or may incorrectly interpret "product of digits" as something involving the whole number. Others may assume any pair of digits with product 27 will work without testing the addition condition. Always turn the wording into clear algebraic equations and check each candidate pair carefully. This ensures a reliable solution without confusion. Final Answer: The original two digit number is 39, which corresponds to option A.

Question 17

The average of 29 consecutive even integers is 60. What is the largest (highest) integer in this sequence?

Accepted Answer

Introduction / Context: This problem tests your understanding of averages and properties of consecutive even integers. When you have an odd number of equally spaced values in a sequence (such as consecutive even integers), the average is equal to the middle term of the sequence. Here, the average of 29 consecutive even integers is given as 60, and you are asked to find the largest integer in the sequence. Recognizing the symmetry of such sequences makes this question much easier. Given Data / Assumptions: - There are 29 consecutive even integers. - Their average is 60. - The integers are evenly spaced with a common difference of 2. - We must determine the largest integer in the sequence. Concept / Approach: For any arithmetic progression with an odd number of terms, the average of the terms equals the middle term. Consecutive even integers form such an arithmetic progression with common difference 2. With 29 terms, the middle term is the 15th term. Since the average is 60, the 15th term is 60. Once we know the middle term, we can move forward or backward using the common difference to find any other term, including the largest, which is the 29th term in this case. Step-by-Step Solution: Step 1: Recognize that 29 is odd, so there is a single middle term in the sequence. Step 2: The number of terms is 29, so the middle term is the (29 + 1) / 2 = 15th term. Step 3: For an arithmetic progression, the average of all terms equals the middle term when the number of terms is odd. Step 4: Therefore, the 15th term is equal to the average, which is 60. Step 5: The sequence consists of consecutive even integers with common difference 2. So each step to the right adds 2. Step 6: The largest integer in the sequence is the 29th term. Step 7: The 29th term is 14 places after the 15th term, because 29 - 15 = 14. Step 8: Each step adds 2, so the increment from the 15th to the 29th term is 14 * 2 = 28. Step 9: Add this increment to the middle term: 60 + 28 = 88. Step 10: Hence, the largest integer in the sequence is 88. Verification / Alternative check: You can also construct part of the sequence around 60. The terms around the middle are 60 - 28, 60 - 26, ..., 60, ..., 60 + 26, 60 + 28. Counting 14 even steps below 60 and 14 even steps above gives 29 terms in total, with the highest being 60 + 28 = 88. This direct reasoning confirms the earlier calculation and shows that the structure is symmetric around 60. Why Other Options Are Wrong: Values 118, 176 and 120 correspond to using incorrect differences or misinterpreting how many steps away from the middle term the last term is. For example, taking 29 * 2 added to 60 instead of 14 * 2 leads to a much larger, incorrect value. Only 88 preserves the correct distance from the middle term in a set of 29 evenly spaced numbers. Common Pitfalls: Some learners forget that the average equals the middle term only when the number of terms is odd. Others miscount the number of steps between the middle and the last term, or mistakenly use a common difference of 1 instead of 2 for even integers. Always double check the step count and the common difference when moving from the middle to the endpoints in such sequences. Final Answer: The highest integer in the sequence is 88, which matches option A.

Question 18

Select the correct option: Convert the decimal number 101 (base 10) into its binary representation.

Accepted Answer

Introduction / Context: This question is from number systems and digital representation. You are asked to convert the decimal number 101 (base 10) into binary. Understanding how to convert between decimal and binary is important in computer science, digital electronics and aptitude tests involving number bases. The binary system is base 2, using only the digits 0 and 1, so we must express 101 as a sum of powers of 2 and then write the corresponding binary digits. Given Data / Assumptions: - The given number is 101 in decimal (base 10). - We must convert this number into base 2, that is, binary. - Binary digits (bits) are 0 and 1 only. - We are to select the correct binary representation from the options. Concept / Approach: The standard method of converting a decimal number to binary is repeated division by 2, keeping track of remainders. The binary digits are obtained by reading the remainders from last to first. Another viewpoint is to find which powers of 2 sum up to the decimal number, and then mark 1s in those positions and 0s elsewhere. For a number like 101, both methods are manageable and lead to the same binary representation. We will use the repeated division method here for clarity. Step-by-Step Solution: Step 1: Start with the decimal number 101. Step 2: Divide 101 by 2. The quotient is 50 and the remainder is 1. This remainder is the least significant bit (rightmost bit). Step 3: Divide the quotient 50 by 2. The new quotient is 25 and the remainder is 0. Step 4: Divide 25 by 2. The quotient is 12 and the remainder is 1. Step 5: Divide 12 by 2. The quotient is 6 and the remainder is 0. Step 6: Divide 6 by 2. The quotient is 3 and the remainder is 0. Step 7: Divide 3 by 2. The quotient is 1 and the remainder is 1. Step 8: Divide 1 by 2. The quotient is 0 and the remainder is 1. We stop when the quotient becomes 0. Step 9: Write the remainders from last to first: 1 1 0 0 1 0 1. Step 10: Hence, the binary representation of 101 is 1100101. Verification / Alternative check: We can verify by converting 1100101 back to decimal. From left to right, the bits correspond to 2^6, 2^5, 2^4, 2^3, 2^2, 2^1 and 2^0. The places with 1s are 2^6, 2^5, 2^2 and 2^0. Compute: 2^6 = 64, 2^5 = 32, 2^2 = 4 and 2^0 = 1. Summing these gives 64 + 32 + 4 + 1 = 101, which matches the original decimal number. This confirms that 1100101 is correct. Why Other Options Are Wrong: Option A (1101001) corresponds to 2^6 + 2^5 + 2^3 + 2^0 = 64 + 32 + 8 + 1 = 105. Option B (1100111) represents 64 + 32 + 4 + 2 + 1 = 103. Option C (1101011) represents 64 + 32 + 8 + 2 + 1 = 107. None of these equal 101 in decimal. Only option D (1100101) evaluates exactly to 101 and is therefore correct. Common Pitfalls: Some learners may read the remainders in the wrong order, starting from the first division instead of the last, which reverses the bits and produces an incorrect binary number. Others may miscalculate powers of 2 when verifying the answer. Remember that the binary equivalent is built by reading remainders from the last division back to the first, and that place values in binary double each step to the left: 1, 2, 4, 8, 16 and so on. Final Answer: The binary representation of decimal 101 is 1100101, which corresponds to option D.

Question 19

In an arithmetic progression, the 2nd term is 17 and the 8th term is -1. What is the 14th term of this progression?

Accepted Answer

Introduction / Context: This problem is about arithmetic progressions (A.P.), where each term after the first is obtained by adding a constant common difference. You are given the 2nd and 8th terms of an arithmetic progression and asked to find the 14th term. The question checks your ability to use the general formula for the n-th term of an A.P. and to solve for the first term and common difference using the given terms. Given Data / Assumptions: - Let the first term of the A.P. be a. - Let the common difference be d. - The 2nd term T_2 is 17. - The 8th term T_8 is -1. - We must find the 14th term T_14. Concept / Approach: The n-th term of an arithmetic progression is given by T_n = a + (n - 1)d. Using this formula, we can write equations for T_2 and T_8 in terms of a and d. Solving these two linear equations gives us the values of a and d. Once we know a and d, we substitute them into the formula for T_14 to obtain the required term. This method is standard for A.P. problems where two terms are given and another term is required. Step-by-Step Solution: Step 1: Write T_2 in terms of a and d: T_2 = a + (2 - 1)d = a + d. Step 2: We are told T_2 = 17, so a + d = 17. This is Equation (1). Step 3: Write T_8 in terms of a and d: T_8 = a + (8 - 1)d = a + 7d. Step 4: We are told T_8 = -1, so a + 7d = -1. This is Equation (2). Step 5: Subtract Equation (1) from Equation (2) to eliminate a: (a + 7d) - (a + d) = -1 - 17. Step 6: This gives 6d = -18, so d = -3. Step 7: Substitute d = -3 back into Equation (1): a + (-3) = 17, so a = 20. span style="display:block;">Step 8: Now find T_14 using T_14 = a + (14 - 1)d = a + 13d. Step 9: Substitute a = 20 and d = -3: T_14 = 20 + 13 * (-3) = 20 - 39. Step 10: Compute 20 - 39 = -19, so T_14 = -19. Verification / Alternative check: We can generate a few terms to check consistency. With a = 20 and d = -3, the sequence begins 20, 17, 14, 11, 8, 5, 2, -1, ... The 2nd term is indeed 17 and the 8th term is -1. Continuing this pattern, each term decreases by 3. From the 8th term -1 to the 14th term, there are 6 steps, each subtracting 3, giving -1 - 18 = -19. This matches our formula based calculation. Why Other Options Are Wrong: Values such as -22, -25 and -28 correspond to using incorrect values for the common difference or miscounting the steps when moving from the known term to the 14th term. Only -19 is consistent with both T_2 = 17 and T_8 = -1 and follows the correct arithmetic progression pattern with d = -3. Common Pitfalls: Some learners mistakenly use the formula T_n = a + nd instead of a + (n - 1)d, which shifts all terms by one position. Others may incorrectly subtract equations or mis-handle signs, especially with negative terms. It is helpful to explicitly label the terms and carefully track the common difference when moving along the sequence. Final Answer: The 14th term of the arithmetic progression is -19, which corresponds to option C.

Question 20

What is the least number that must be subtracted from 3401 so that the resulting number is exactly divisible by 11?

Accepted Answer

Introduction / Context: This question checks your understanding of divisibility by 11 and your ability to adjust a given number so that it becomes a multiple of 11. You are asked for the least number that must be subtracted from 3401 in order to make the result divisible by 11. Such problems are common in aptitude exams and help reinforce modular arithmetic and divisibility rules. Given Data / Assumptions: - The original number is 3401. - We want to find a non negative integer k to subtract. - The number 3401 - k should be divisible by 11. - The value of k should be as small as possible. Concept / Approach: One straightforward approach is to find the remainder when 3401 is divided by 11. If a number N gives remainder r when divided by 11, then N - r is divisible by 11. Thus, the smallest non negative number that must be subtracted to make N divisible by 11 is exactly this remainder. We can compute the remainder through standard long division or by quick mental calculation using nearby multiples of 11. Step-by-Step Solution: Step 1: Divide 3401 by 11 to find the quotient and remainder. Step 2: Note that 11 * 300 = 3300. Subtracting from 3401 gives 3401 - 3300 = 101. Step 3: Now divide 101 by 11. Since 11 * 9 = 99, we have 101 - 99 = 2 as the remaining part. Step 4: Thus 3401 = 11 * 309 + 2. Step 5: The remainder when 3401 is divided by 11 is 2. Step 6: To make the number divisible by 11, we must subtract this remainder. Step 7: So, the required number to subtract is k = 2 and the resulting number is 3401 - 2 = 3399. Step 8: 3399 is divisible by 11 because 3399 / 11 = 309 exactly. Verification / Alternative check: As a second check, apply the divisibility rule for 11 to 3399. For 3399, sum of digits in odd positions (from left) is 3 + 9 = 12, and sum of digits in even positions is 3 + 9 = 12. The difference is 12 - 12 = 0, which is a multiple of 11, confirming that 3399 is divisible by 11. Doing the same for 3401 yields a non zero difference, so 3401 itself is not divisible by 11. This confirms that subtracting 2 is necessary and sufficient. Why Other Options Are Wrong: If we subtract 0, we stay at 3401, which is not divisible by 11. Subtracting 1 gives 3400, which upon division by 11 leaves a remainder. Subtracting 3 yields 3398, which also is not a multiple of 11. Only subtracting 2 brings the number to the nearest lower multiple of 11, namely 3399. Therefore the least required subtraction is 2. Common Pitfalls: Some learners may instead try to add numbers to 3401, or they may apply the divisibility rule incorrectly by miscounting positions of digits. Others might guess without checking the remainder carefully. Remember that checking the exact remainder is a quick and reliable method, and subtracting that remainder will always yield the nearest smaller multiple of the divisor. Final Answer: The least number that must be subtracted from 3401 is 2, which corresponds to option C.

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