Q: Solve the linear inequality 3b - 7 < 32. Choose the correct description of the solution set for the real number b.

Introduction: This question tests solving a one-variable linear inequality. Unlike an equation, an inequality describes a range of values. The main rules are: you can add or subtract the same number from both sides without changing the inequality sign, and you can divide by a positive number without flipping the sign. Here we isolate b by first adding 7 and then dividing by 3. Given Data / Assumptions: Inequality: 3b - 7 < 32 b is a real number Operations used: addition/subtraction and division by a positive number Concept / Approach: Rearrange the inequality step-by-step to isolate b. Since 3 is positive, dividing by 3 will not reverse the inequality sign. The final result should be expressed as a comparison like b < some number, representing all valid solutions. Step-by-Step Solution: Start: 3b - 7 13 give values that make the left side 32 or more. b = 13 is not valid because equality is not allowed here. Common Pitfalls: Confusing < with ≤, or mistakenly reversing the inequality sign (which would happen only if dividing by a negative number, not in this problem). Final Answer: The solution set is b < 13.

Q: Simplify the equation 48x + 32x = 320 by combining like terms. What is the value of x?

Introduction: This question checks basic algebra: combining like terms and solving a linear equation. Terms like 48x and 32x are “like terms” because both contain x to the same power (x^1). You add their coefficients, then isolate x by dividing both sides by the combined coefficient. This is a foundational simplification skill used in equations, inequalities, and word problems. Given Data / Assumptions: Equation: 48x + 32x = 320 x is a real number Like terms can be combined by adding coefficients Concept / Approach: Combine the x-terms first: (48 + 32)x. Then solve (80x = 320) by dividing both sides by 80. This keeps the work clean and prevents arithmetic mistakes. Step-by-Step Solution: 48x + 32x = (48 + 32)x = 80x So the equation becomes 80x = 320 Divide both sides by 80: x = 320/80 x = 4 Verification / Alternative check: Substitute x = 4 into the original equation: 48*4 + 32*4 = 192 + 128 = 320. The left side equals the right side, so x = 4 is correct. Why Other Options Are Wrong: If x = 8, left side becomes 640. If x = 16, it becomes 1280. If x = 32, it becomes 2560. If x = 2, it becomes 160. Only x = 4 satisfies the equation exactly. Common Pitfalls: Adding 48 and 32 incorrectly, or dividing by the wrong coefficient after combining like terms. Final Answer: The value of x is 4.

Q: Given that 10^0.48 = x and 10^0.70 = y, and x raised to the power z equals y squared (x^z = y^2), the value of z is closest to which of the following?

Introduction: This question tests exponent rules and using logarithms (or exponent comparison) to solve for an unknown power. Because x and y are both written as powers of 10, we can rewrite x^z and y^2 as powers of 10 as well. When two powers of 10 are equal, their exponents must be equal. That converts the problem into a simple linear equation in z. The final step is choosing the closest option, because the exact value may not match an option exactly. Given Data / Assumptions: x = 10^0.48 y = 10^0.70 Relation: x^z = y^2 Rule: (a^m)^n = a^(m*n) If 10^A = 10^B, then A = B Concept / Approach: Rewrite both sides using base 10. Since x = 10^0.48, x^z = (10^0.48)^z = 10^(0.48z). And y^2 = (10^0.70)^2 = 10^(1.40). Equate exponents: 0.48z = 1.40, then solve for z and compare to the choices. Step-by-Step Solution: x^z = (10^0.48)^z = 10^(0.48z) y^2 = (10^0.70)^2 = 10^(0.70*2) = 10^1.40 So 10^(0.48z) = 10^1.40 Equate exponents: 0.48z = 1.40 z = 1.40 / 0.48 = 140/48 = 35/12 ≈ 2.9167 Closest option to 2.9167 is 2.9 Verification / Alternative check: If z ≈ 2.9167, then 0.48z ≈ 0.48*2.9167 ≈ 1.40, so x^z ≈ 10^1.40 which matches y^2 exactly in exponent form. Why Other Options Are Wrong: 2.7 gives exponent 0.48*2.7 = 1.296, too low. 3.6 and 3.7 give exponents above 1.7, too high. 3.1 gives 1.488, still too high. Only 2.9 is closest to 2.9167. Common Pitfalls: Multiplying exponents incorrectly, treating 10^0.70 squared as 10^0.49, or forgetting that equal bases imply equal exponents. Final Answer: The value of z is closest to 2.9.

Q: The average number of visitors to a library on the first four days of a week is 58. The average number of visitors on the second, third, fourth, and fifth days is 60. The number of visitors on the first and fifth days are in the ratio 8 : 9. How man…

Introduction: This question tests using averages to form equations about sums, and then combining that with a ratio condition. Averages are just total sum divided by number of days, so two different averages over overlapping day ranges allow you to isolate the difference between two specific days. After finding the difference between day 5 and day 1, the ratio 8:9 converts that difference into exact values. This is a classic “overlapping averages + ratio” aptitude pattern. Given Data / Assumptions: Average visitors on days 1 to 4 = 58 Average visitors on days 2 to 5 = 60 Days 1 and 5 are in ratio 8 : 9 Let visitors on day 1 = D1 and on day 5 = D5 Concept / Approach: Convert averages to sums: Sum(1..4) = 58*4 and Sum(2..5) = 60*4. Subtract the sums to eliminate common days 2,3,4 and obtain D5 - D1. Then apply the ratio D1:D5 = 8:9 to solve for D5. Step-by-Step Solution: Sum of days 1 to 4 = 58 * 4 = 232 Sum of days 2 to 5 = 60 * 4 = 240 Subtract: (days 2 to 5) - (days 1 to 4) = D5 - D1 So D5 - D1 = 240 - 232 = 8 Given D1 : D5 = 8 : 9, let D1 = 8k and D5 = 9k Then D5 - D1 = k = 8 So k = 8 and D5 = 9k = 72 Verification / Alternative check: D1 = 64 and D5 = 72 satisfy the ratio. Also D5 - D1 = 8 matches the difference derived from sums, so the result is consistent. Why Other Options Are Wrong: Any value other than 72 breaks either the required difference D5 - D1 = 8 or the ratio D1:D5 = 8:9 (or both). Common Pitfalls: Using 5 days in the average (instead of 4), subtracting in the wrong order (getting D1 - D5), or applying the ratio incorrectly as 9:8 instead of 8:9. Final Answer: 72 visitors came to the library on the fifth day.

Q: The square of a positive integer exceeds twenty times the integer by 96. What is that positive integer?

Introduction: This question tests forming a quadratic equation from a word statement and solving it. The phrase “the square of an integer exceeds twenty times the integer by 96” means: integer^2 is 96 more than 20 times the integer. Translating that carefully produces a quadratic equation. Because the integer is stated to be positive, we choose only the positive root after solving. Given Data / Assumptions: Let the positive integer be n “Square exceeds 20 times the integer by 96” means n^2 = 20n + 96 We solve for integer n and keep only n > 0 Concept / Approach: Convert the statement into a quadratic equation: n^2 - 20n - 96 = 0. Solve using factoring or the quadratic formula. Since the problem requires a positive integer, discard any negative solution even if it satisfies the equation algebraically. Step-by-Step Solution: Start with: n^2 = 20n + 96 Bring all terms to one side: n^2 - 20n - 96 = 0 Find factors of -96 whose difference is 20: 24 and -4 So (n - 24)(n + 4) = 0 n = 24 or n = -4 Positive integer solution: n = 24 Verification / Alternative check: Check n = 24: n^2 = 576 and 20n = 480. Difference = 576 - 480 = 96, which matches the statement exactly. So 24 is correct. Why Other Options Are Wrong: 36 gives 36^2 - 20*36 = 1296 - 720 = 576, not 96. 42 and 48 produce even larger differences. 16 gives 256 - 320 = -64, not 96. Common Pitfalls: Misreading “exceeds by 96” as “is 96 times,” or forgetting to enforce the “positive integer” condition and mistakenly choosing -4. Final Answer: The positive integer is 24.

Question 1

In a group of animals consisting of some buffaloes and some ducks, the total number of legs is 24 more than twice the total number of heads. Each buffalo has 4 legs and each duck has 2 legs. How many buffaloes are there in the group?

Accepted Answer

Introduction / Context: This is a classic heads and legs problem that is frequently seen in school level mathematics and competitive exams. It involves forming linear equations using the number of animals, the number of heads, and the number of legs. The question combines logical reasoning with simple algebra to determine how many animals of each type are present. Here the focus is on the number of buffaloes in a mixed group of buffaloes and ducks. Given Data / Assumptions: The group contains buffaloes and ducks only. Each animal has exactly 1 head. Each buffalo has 4 legs. Each duck has 2 legs. Total number of legs is 24 more than twice the total number of heads. All heads and legs are counted correctly. Concept / Approach: Let the number of buffaloes and ducks be variables. The total number of heads is simply the sum of these numbers, since each animal contributes one head. The total number of legs can be expressed as 4 times the number of buffaloes plus 2 times the number of ducks. The key relationship is that the total legs count exceeds twice the number of heads by exactly 24. This condition leads to a single linear equation in the number of buffaloes, which can then be solved directly. Step-by-Step Solution: Let the number of buffaloes be b. Let the number of ducks be d. Total number of heads = b + d. Total number of legs = 4b + 2d. According to the problem, total legs = 2 * (total heads) + 24. So, 4b + 2d = 2 * (b + d) + 24. Expand the right side: 4b + 2d = 2b + 2d + 24. Subtract 2b and 2d from both sides: 4b + 2d - 2b - 2d = 24. This simplifies to 2b = 24. So, b = 24 / 2 = 12. Therefore, there are 12 buffaloes in the group. Verification / Alternative check: We can verify the result by expressing d in terms of b. The equation 2b = 24 gives b = 12, but it does not restrict d directly. For any non negative number of ducks d, the relation 4b + 2d = 2(b + d) + 24 will hold only when b = 12. For example, if we assume there are 12 buffaloes and 6 ducks, then total heads = 18 and total legs = 12 * 4 + 6 * 2 = 48 + 12 = 60. Twice the number of heads is 36 and 36 + 24 = 60, so the condition is satisfied. The exact number of ducks is not required to answer the question, because the equation fixes only the number of buffaloes. Why Other Options Are Wrong: 6 buffaloes: If b = 6, then 2b = 12 and 4b + 2d cannot be equal to 2(b + d) + 24 for any integer d that makes sense, so the relation fails. 8 buffaloes or 10 buffaloes: Substituting these values into the equation 4b + 2d = 2(b + d) + 24 shows that the left and right sides cannot be made equal for non negative integer d in a realistic animal group. 14 buffaloes: If b = 14, then 2b = 28, which forces a mismatch in the legs condition. Only 12 buffaloes satisfy the given relationship. Common Pitfalls: One common mistake is to forget that each animal contributes exactly one head, which makes the total heads easy to express. Some learners also confuse the statement and try to equate total legs directly to 24 or to twice the heads without the plus 24 term. Another error is manipulating the algebra incorrectly when simplifying 4b + 2d = 2b + 2d + 24. Careful step by step algebra helps avoid losing terms or cancelling incorrectly. Final Answer: The number of buffaloes in the group is 12 buffaloes.

Question 2

By how much is three fifths of 350 greater than four sevenths of 210? Compute the exact numerical difference between these two quantities.

Accepted Answer

Introduction / Context: This question tests your ability to work with fractions applied to whole numbers and to compare two resulting values. It looks very simple, but such questions are common in speed tests where quick and accurate calculation is required. The key is to compute three fifths of 350 and four sevenths of 210 separately and then subtract to find how much one quantity is greater than the other. Given Data / Assumptions: We need to find 3/5 of 350. We also need to find 4/7 of 210. We must compute how much the first quantity is greater than the second. All numbers are real and standard arithmetic rules apply. Concept / Approach: The concept here is straightforward fractional multiplication. To find a fraction of a number, multiply the number by the numerator and divide by the denominator. After obtaining both values, we compare them by subtraction. Because both numbers are multiples of the denominators, the calculations are especially easy, which is typical for exam style questions designed to be done mentally or with very little working. Step-by-Step Solution: Compute three fifths of 350: (3/5) * 350. First, 350 / 5 = 70. Then, 3 * 70 = 210. So, three fifths of 350 is 210. Now compute four sevenths of 210: (4/7) * 210. First, 210 / 7 = 30. Then, 4 * 30 = 120. So, four sevenths of 210 is 120. Now find how much the first is greater than the second. Difference = 210 - 120 = 90. Verification / Alternative check: We can verify by checking that the fraction calculations are correct. Since 3/5 means 60 percent, 60 percent of 350 is indeed 210, because 10 percent is 35 and 6 * 35 = 210. For 4/7 of 210, note that 1/7 of 210 is 30, so 4/7 is 4 * 30 = 120. The difference 210 - 120 is clearly 90. The computations are consistent using both the fraction method and the percentage interpretation, which confirms the result. Why Other Options Are Wrong: 100 and 110: These would result from miscalculating one of the fractions or from incorrect subtraction such as 220 minus 120 or 210 minus 100. 120: This would suggest confusing the result of four sevenths of 210 with the difference value itself. 80: This comes from a mistaken subtraction such as 200 minus 120. Careful arithmetic eliminates these incorrect options. Common Pitfalls: A common mistake is to miscompute the fraction by multiplying directly without simplifying first, which may cause errors. For example, some students might try 3 * 350 / 5 in one step and miscalculate. Another pitfall is to subtract in the wrong order and answer how much the smaller is less than the larger without reading the question carefully. Always compute each fraction step by step and then subtract larger minus smaller as requested. Final Answer: Three fifths of 350 is greater than four sevenths of 210 by 90.

Question 3

The total price of 2 oranges, 3 bananas and 4 apples is Rs 15. The total price of 3 oranges, 2 bananas and 1 apple is Rs 10. Assuming each fruit of the same type has the same price, what is the total price of 4 oranges, 4 bananas and 4 apples togeth…

Accepted Answer

Introduction / Context: This problem is a linear equations question disguised as a fruit price puzzle. The cost of oranges, bananas, and apples is unknown, but we know the cost of two different combinations of these fruits. Using these combinations, we can form equations and then reason about the cost of another combination. Importantly, we are not asked to find the individual prices, only the total price of a specific bundle, which allows a clever shortcut. Given Data / Assumptions: 2 oranges + 3 bananas + 4 apples cost Rs 15. 3 oranges + 2 bananas + 1 apple cost Rs 10. All oranges have the same price, all bananas have the same price, and all apples have the same price. We need the cost of 4 oranges + 4 bananas + 4 apples. Concept / Approach: Let the price of one orange be O, one banana be B, and one apple be A, all in rupees. The two given conditions give us two linear equations in three variables. Normally, this is not enough to find unique values of O, B, and A. However, we are asked only for O + B + A multiplied by 4, that is 4(O + B + A). By manipulating the two equations, we can directly find the value of O + B + A without solving for each price separately. This is a good example of using algebraic reasoning to find the required expression rather than all individual unknowns. Step-by-Step Solution: Let the price of 1 orange be O rupees. Let the price of 1 banana be B rupees. Let the price of 1 apple be A rupees. From the first statement: 2O + 3B + 4A = 15. From the second statement: 3O + 2B + A = 10. We need the total price of 4 oranges, 4 bananas, and 4 apples. That total cost = 4O + 4B + 4A = 4(O + B + A). We will find O + B + A from the given equations. Observe that the second equation can be rearranged to express O + B + A. From 3O + 2B + A = 10, note that there are infinitely many solutions, but we can combine equations. A more direct approach: subtract the second equation from the first plus the second multiplied suitably to isolate O + B + A. However, there is a simpler relation: algebraic manipulation shows that O + B + A must equal 5. Therefore, 4(O + B + A) = 4 * 5 = 20. Verification / Alternative check: To confirm O + B + A = 5, we can treat O as A in the second equation and express B in terms of A. An algebraic manipulation gives families of solutions, but in every valid solution set consistent with both equations, the sum O + B + A remains 5. For example, choose a convenient value for A that satisfies both equations, then solve for O and B and confirm that O + B + A = 5. Regardless of the individual prices, the combination 4O + 4B + 4A is fixed at Rs 20, so the method and result are reliable. Why Other Options Are Wrong: Rs 10 and Rs 15: These values are too low and would correspond to mistaken assumptions, such as simply adding 10 and 15 and dividing by something incorrectly. Rs 25 and Rs 30: These are too high and would require that O + B + A be greater than 6, which does not satisfy both given equations simultaneously. Common Pitfalls: Learners sometimes try to solve for each variable individually, which is not possible uniquely with two equations and three unknowns. Others make the mistake of assuming arbitrary values for two variables without ensuring that both equations remain consistent. The right strategy is to focus on the specific expression required, here O + B + A, and manipulate the equations accordingly. This reinforces the idea that in algebra you often need not find every unknown, just the combination you are asked for. Final Answer: The total price of 4 oranges, 4 bananas, and 4 apples together is Rs 20.

Question 4

38% of 7500 plus an unknown percentage of 375 is equal to 50% of 6000. If ?% of 375 is added to 38% of 7500 to give exactly 50% of 6000, what is the value of the unknown percentage ?

Accepted Answer

Introduction / Context: This question checks your comfort with percentage calculations and forming a simple algebraic equation using an unknown percentage. It appears in percentage and simplification sections of aptitude tests. The key idea is to convert each percentage expression into a numerical value, express the unknown percentage as a variable, and then solve the resulting linear equation. Given Data / Assumptions: We have 38 percent of 7500. We add ? percent of 375. The sum is equal to 50 percent of 6000. We must find the value of the unknown percentage ?. Concept / Approach: Percent means per hundred, so x percent of a number N is (x/100) * N. Let the unknown percentage be p. Then p percent of 375 is (p/100) * 375. We compute 38 percent of 7500 and 50 percent of 6000 directly, and then set up an equation: 38 percent of 7500 plus (p percent of 375) equals 50 percent of 6000. Solving this equation for p gives the required percentage. Step-by-Step Solution: Let the required percentage be p percent. Compute 38 percent of 7500: (38/100) * 7500. First, 1 percent of 7500 is 75, so 38 percent is 38 * 75 = 2850. Compute 50 percent of 6000: (50/100) * 6000 = 0.5 * 6000 = 3000. p percent of 375 is (p/100) * 375. According to the problem, 2850 + (p/100) * 375 = 3000. Subtract 2850 from both sides: (p/100) * 375 = 3000 - 2850 = 150. So, (p/100) = 150 / 375. Compute 150 / 375: divide numerator and denominator by 75 to get 2 / 5 = 0.4. Thus, p/100 = 0.4, so p = 40. Therefore, the unknown percentage ? is 40 percent. Verification / Alternative check: Check by substituting p = 40. Then 40 percent of 375 is (40/100) * 375 = 0.4 * 375 = 150. We already have 38 percent of 7500 = 2850. Add them: 2850 + 150 = 3000. Compute 50 percent of 6000: 0.5 * 6000 = 3000. Both sides match exactly, so the solution p = 40 is verified as correct. Why Other Options Are Wrong: 45 percent: 45 percent of 375 is 168.75, giving a total of 3018.75, not 3000. 50 percent: 50 percent of 375 is 187.5, giving a total of 3037.5, which is too large. 60 percent: 60 percent of 375 is 225, giving 3075, again too high. 35 percent: 35 percent of 375 is 131.25, giving a total of 2981.25, which is less than 3000. Common Pitfalls: Some learners incorrectly calculate 38 percent of 7500 by doing 7500 / 38 or mixing up multiplication and division. Others forget to convert the unknown percentage into a fraction of 100, writing 375 * p instead of 375 * p / 100. Another common mistake is to shift terms incorrectly when solving the equation, leading to a wrong value for p. Working carefully with the percentage formula and keeping track of each step avoids these issues. Final Answer: The value of the unknown percentage ? is 40%.

Question 5

Find the exact value of 1397 multiplied by 1397, that is, compute 1397 * 1397 without using a calculator.

Accepted Answer

Introduction / Context: This problem asks for the square of a four digit number, 1397. Although it looks like a direct multiplication question, it also tests your familiarity with algebraic identities and your ability to simplify calculations using nearby convenient numbers. Such questions appear in mental math and simplification sections, where using identities instead of raw multiplication can save significant time. Given Data / Assumptions: We need to compute 1397 * 1397. All calculations are in standard decimal arithmetic. No calculator is to be used in typical exam conditions. Concept / Approach: Instead of directly multiplying 1397 by 1397 using long multiplication, we can view 1397 as a number close to 1400. Then we can apply the identity (a - b)^2 = a^2 - 2ab + b^2. Here, a = 1400 and b = 3, because 1397 = 1400 - 3. This method makes the squaring much easier because squaring 1400 and computing the remaining terms is straightforward when compared to full long multiplication. Step-by-Step Solution: Write 1397 as 1400 - 3. We need (1400 - 3)^2. Use the identity (a - b)^2 = a^2 - 2ab + b^2. Here, a = 1400 and b = 3. Compute a^2 = 1400^2 = 1400 * 1400. 1400 * 1400 = 14 * 14 * 100 * 100 = 196 * 10000 = 1960000. Compute 2ab = 2 * 1400 * 3 = 2800 * 3 = 8400. Compute b^2 = 3^2 = 9. Now, (1400 - 3)^2 = 1960000 - 8400 + 9. First subtract: 1960000 - 8400 = 1951600. Then add 9: 1951600 + 9 = 1951609. Therefore, 1397 * 1397 = 1951609. Verification / Alternative check: A long multiplication method would also confirm the same result, though it is more laborious. Breaking 1397 into 1400 - 3 and squaring via the identity reduces the risk of arithmetic mistakes. Additionally, a quick digital sum check (casting out nines) can be used. The sum of digits of 1397 is 1 + 3 + 9 + 7 = 20, and 20 reduces to 2 + 0 = 2. The square of a number whose digit sum is 2 has digit sum 4 (since 2^2 = 4). The digits of 1951609 sum to 1 + 9 + 5 + 1 + 6 + 0 + 9 = 31, which reduces to 3 + 1 = 4, consistent with the expected check, supporting the correctness of 1951609. Why Other Options Are Wrong: The other options differ significantly from 1951609 and would arise from mistakes such as forgetting to add b^2, miscalculating 2ab, or mixing up intermediate subtraction steps. For instance, using 1960000 - 8300 instead of 8400 or misplacing digits when adding the final 9 could lead to some of these incorrect values. Common Pitfalls: The main pitfall is attempting full long multiplication under time pressure and making digit alignment errors. Another common error is to misremember the identity as a^2 + 2ab + b^2 instead of a^2 - 2ab + b^2 for a minus sign, which gives a completely different result. Also, while simplifying 1960000 - 8400, it is easy to subtract incorrectly. Using neat stepwise calculation and checking intermediate steps helps avoid these issues. Final Answer: The exact value of 1397 multiplied by 1397 is 1951609.

Question 6

Simplify the sum of the mixed fractions 4 1/3, 2 1/5 and 4 1/4, and express your final answer as a mixed fraction in simplest form.

Accepted Answer

Introduction / Context: This question tests basic fraction skills, especially handling mixed fractions. You must convert mixed numbers to improper fractions, add them, and then convert back to a mixed fraction if required. Operations like this are common in school mathematics and form the foundation for more advanced fraction and ratio calculations in aptitude exams. Given Data / Assumptions: The three mixed fractions are 4 1/3, 2 1/5, and 4 1/4. We are required to find their sum. The final answer must be expressed as a mixed fraction in simplest form. Concept / Approach: Mixed fractions are easier to add if we first convert them to improper fractions. An improper fraction has the form (whole * denominator + numerator) over the denominator. Once converted, we add the fractions by finding a common denominator. After obtaining the final improper fraction, we simplify it if possible and then convert it back to a mixed fraction for a more intuitive interpretation. Step-by-Step Solution: First convert each mixed fraction to an improper fraction. 4 1/3 = (4 * 3 + 1) / 3 = 13/3. 2 1/5 = (2 * 5 + 1) / 5 = 11/5. 4 1/4 = (4 * 4 + 1) / 4 = 17/4. We now add 13/3 + 11/5 + 17/4. The denominators are 3, 5, and 4. A convenient common denominator is 60. Convert each fraction to denominator 60. 13/3 = (13 * 20) / 60 = 260/60. 11/5 = (11 * 12) / 60 = 132/60. 17/4 = (17 * 15) / 60 = 255/60. Now add the numerators: 260 + 132 + 255 = 647. So the sum is 647/60. Convert 647/60 to a mixed fraction. Divide 647 by 60: 60 * 10 = 600, remainder 47. So, 647/60 = 10 47/60. Verification / Alternative check: We can also add the whole number parts and fractional parts separately as a quick check. Whole parts: 4 + 2 + 4 = 10. Fractional parts: 1/3 + 1/5 + 1/4. Convert to denominator 60: 1/3 = 20/60, 1/5 = 12/60, 1/4 = 15/60. Sum of fractional parts = 20/60 + 12/60 + 15/60 = 47/60. Thus the total is 10 and 47/60, confirming the result obtained earlier. Why Other Options Are Wrong: 11 45/73 and 11 47/36: These fractions have incorrect denominators and do not come from any reasonable common denominator of 3, 5, and 4. 10 35/48: The denominator 48 does not match the least common multiple of 3, 5, and 4, which is 60, and the numerator 35 is also inconsistent with the correct sum of fractional parts. 9 7/8: This value is less than 10, whereas the sum of whole parts alone is already 10, so it cannot be correct. Common Pitfalls: Common mistakes include miscalculating the improper fraction representation, choosing an incorrect common denominator, or adding the numerators incorrectly. Some students also forget to simplify the final fraction or convert it back to a mixed number. Working in an organised way, either by converting everything to a common denominator or by separately summing whole and fractional parts, helps avoid these problems. Final Answer: The simplified sum of the three mixed fractions is 10 47/60.

Question 7

If a + b = 5 and 3a + 2b = 20, what is the value of the expression 3a + b?

Accepted Answer

Introduction / Context: This question is a simple example of solving simultaneous linear equations and then evaluating a particular expression in terms of the variables. Rather than asking for the individual values of a and b, the exam tests whether you can use the given equations efficiently to find 3a + b. Such questions are routine in algebra sections and help build skill in manipulating equations smartly. Given Data / Assumptions: a + b = 5. 3a + 2b = 20. We must find the value of 3a + b. We assume a and b are real numbers. Concept / Approach: There are two main approaches. One is to solve for a and b explicitly by using substitution or elimination, and then compute 3a + b. The other is to manipulate the equations directly to get 3a + b without fully solving for both variables. In either approach, the calculations are straightforward. Here we will solve for a and b explicitly to keep the logic transparent and then evaluate the required expression. Step-by-Step Solution: From the first equation, a + b = 5. Express a in terms of b: a = 5 - b. Substitute this into the second equation 3a + 2b = 20. 3(5 - b) + 2b = 20. Expand: 15 - 3b + 2b = 20. Combine like terms: 15 - b = 20. Rearrange to find b: -b = 20 - 15 = 5, so b = -5. Now substitute b = -5 back into a = 5 - b. a = 5 - (-5) = 10. So, a = 10 and b = -5. Now compute the required expression 3a + b. 3a + b = 3 * 10 + (-5) = 30 - 5 = 25. Verification / Alternative check: We can quickly verify that a = 10 and b = -5 satisfy both original equations. First equation: a + b = 10 + (-5) = 5, which matches. Second equation: 3a + 2b = 3 * 10 + 2 * (-5) = 30 - 10 = 20, which matches as well. Since the pair (10, -5) satisfies both equations, the computed value 3a + b = 25 is consistent and correct. Why Other Options Are Wrong: 15 and 20: These might result from partial substitution errors or from using only one equation without checking the second. For example, someone might incorrectly assume b is positive or miscalculate 3a + b after solving. 30 and 10: These could arise from using a + b directly, such as mistakenly treating 3a + b as 3(a + b) or mixing coefficients, both of which give wrong results. Common Pitfalls: Common errors include incorrect substitution, such as writing a = 5 + b instead of a = 5 - b, or mismanaging signs during simplification. Some learners rush and compute 3(a + b) instead of 3a + b, which gives 3 * 5 = 15 and leads to an incorrect option. Carefully distinguishing between expressions and using clean algebraic steps prevents these mistakes. Final Answer: The value of the expression 3a + b is 25.

Question 8

In a competitive test, Hemavathi scores +3 marks for each correct answer and loses 2 marks for each wrong answer. She attempts a total of 35 questions and her final score is 60 marks. How many questions does she answer correctly?

Accepted Answer

Introduction: This question tests forming and solving a simple linear system from a marks scheme. Many aptitude tests award positive marks for correct answers and deduct marks for wrong ones (negative marking). The key idea is that the total attempts split into correct and wrong, and the total score is a weighted sum using the given marks per correct and per wrong. Once you translate the words into two equations, the answer follows with basic algebra. Given Data / Assumptions: Marks per correct answer = +3 Marks per wrong answer = -2 Total questions attempted = 35 Total score obtained = 60 Let correct answers = c, wrong answers = w Concept / Approach: Use two facts: (1) c + w equals total attempts, and (2) 3c - 2w equals total score. Solve the pair of linear equations. A standard shortcut is to substitute w = 35 - c into the score equation to get a single equation in c. Step-by-Step Solution: c + w = 35 3c - 2w = 60 Substitute w = 35 - c into the score equation 3c - 2(35 - c) = 60 3c - 70 + 2c = 60 5c = 130 c = 26 Verification / Alternative check: If c = 26, then w = 35 - 26 = 9. Score = 3*26 - 2*9 = 78 - 18 = 60, which matches the given total score exactly. Why Other Options Are Wrong: 24 gives w = 11 and score 72 - 22 = 50. 25 gives score 75 - 20 = 55. 23 gives score 69 - 24 = 45. 27 gives score 81 - 16 = 65. Only 26 produces 60. Common Pitfalls: Forgetting negative marking (treating wrong as +2), using total questions instead of attempted questions, or mixing up c and w during substitution. Final Answer: Hemavathi answers 26 questions correctly.

Question 9

Evaluate the expression 217 * 217 + 183 * 183 using correct arithmetic. Choose the correct value of the final result.

Accepted Answer

Introduction: This problem tests quick simplification and accurate squaring. Expressions like a*a + b*b are easiest when you compute each square carefully using (u + v)^2 = u^2 + 2uv + v^2. That reduces mental mistakes compared to long multiplication. After finding both squares, add them to get the final value. Accuracy matters because the options are close-looking numbers. Given Data / Assumptions: Compute 217 * 217 Compute 183 * 183 Add the two results Use (u + v)^2 expansion to simplify squaring Concept / Approach: Write 217 as (200 + 17) and 183 as (180 + 3). Then square each using (u + v)^2 = u^2 + 2uv + v^2. Finally, add the two square values to obtain the required sum. Step-by-Step Solution: 217^2 = (200 + 17)^2 = 200^2 + 2*200*17 + 17^2 = 40000 + 6800 + 289 = 47089 183^2 = (180 + 3)^2 = 180^2 + 2*180*3 + 3^2 = 32400 + 1080 + 9 = 33489 Required sum = 47089 + 33489 = 80578 Verification / Alternative check: A quick reasonableness check: 217^2 is a bit less than 220^2 = 48400 and 183^2 is a bit more than 180^2 = 32400, so total near 80800. The exact 80578 fits this range well. Why Other Options Are Wrong: They come from common arithmetic slips: missing the 2uv term, mis-adding the final sum, or squaring 217/183 incorrectly by a few hundreds or thousands. Common Pitfalls: Using (u+v)^2 = u^2 + v^2 (forgetting 2uv), or making an addition error when combining the two squared results. Final Answer: The value of 217 * 217 + 183 * 183 is 80,578.

Question 10

A man can buy a certain number of notebooks for ₹300. If the price of each notebook increases by ₹5, he can buy 10 fewer notebooks for the same ₹300. What is the original price (in rupees) of each notebook?

Accepted Answer

Introduction: This question tests translating a real-life buying situation into an algebraic equation. The amount of money is fixed (₹300), so quantity equals total money divided by price per notebook. When price increases, the quantity decreases. The phrase “10 fewer notebooks” creates a relationship between the two quantities. Setting up the equation correctly is the main challenge; solving the resulting quadratic is straightforward. Given Data / Assumptions: Total money available = ₹300 Original price per notebook = p rupees New price per notebook = p + 5 rupees New quantity = original quantity - 10 Concept / Approach: Quantity purchased = 300/price. So original quantity is 300/p and new quantity is 300/(p+5). The condition says 300/(p+5) = 300/p - 10. Solve this equation for p (take only the positive solution, since price must be positive). Step-by-Step Solution: Original quantity = 300/p New quantity = 300/(p + 5) Condition: 300/(p + 5) = 300/p - 10 Multiply by p(p+5): 300p = (p+5)(300 - 10p) Expand: 300p = 300p - 10p^2 + 1500 - 50p Bring terms together: 10p^2 + 50p - 1500 = 0 Divide by 10: p^2 + 5p - 150 = 0 Factor: (p + 15)(p - 10) = 0 Positive solution: p = 10 Verification / Alternative check: If p = 10, original quantity = 300/10 = 30. New price = 15, new quantity = 300/15 = 20, which is 10 fewer. Works perfectly. Why Other Options Are Wrong: Any other price fails the exact “10 fewer” condition when you compute 300/p and 300/(p+5). Only ₹10 satisfies both quantities consistently. Common Pitfalls: Using 300 - 5 instead of p + 5, forgetting that quantity is division (not subtraction), or keeping the negative root (price cannot be negative). Final Answer: The original price of each notebook is ₹10.

Question 11

Solve the linear inequality 3b - 7 < 32. Choose the correct description of the solution set for the real number b.

Accepted Answer

Introduction: This question tests solving a one-variable linear inequality. Unlike an equation, an inequality describes a range of values. The main rules are: you can add or subtract the same number from both sides without changing the inequality sign, and you can divide by a positive number without flipping the sign. Here we isolate b by first adding 7 and then dividing by 3. Given Data / Assumptions: Inequality: 3b - 7 < 32 b is a real number Operations used: addition/subtraction and division by a positive number Concept / Approach: Rearrange the inequality step-by-step to isolate b. Since 3 is positive, dividing by 3 will not reverse the inequality sign. The final result should be expressed as a comparison like b < some number, representing all valid solutions. Step-by-Step Solution: Start: 3b - 7 < 32 Add 7 to both sides: 3b < 39 Divide both sides by 3: b < 13 Verification / Alternative check: Test a value less than 13, say b = 12: 3*12 - 7 = 36 - 7 = 29, and 29 < 32 is true. Test b = 13: 3*13 - 7 = 39 - 7 = 32, and 32 < 32 is false. So b must be strictly less than 13. Why Other Options Are Wrong: b ≤ 13 incorrectly includes 13 which fails the strict “<”. b ≥ 13 and b > 13 give values that make the left side 32 or more. b = 13 is not valid because equality is not allowed here. Common Pitfalls: Confusing < with ≤, or mistakenly reversing the inequality sign (which would happen only if dividing by a negative number, not in this problem). Final Answer: The solution set is b < 13.

Question 12

Simplify the equation 48x + 32x = 320 by combining like terms. What is the value of x?

Accepted Answer

Introduction: This question checks basic algebra: combining like terms and solving a linear equation. Terms like 48x and 32x are “like terms” because both contain x to the same power (x^1). You add their coefficients, then isolate x by dividing both sides by the combined coefficient. This is a foundational simplification skill used in equations, inequalities, and word problems. Given Data / Assumptions: Equation: 48x + 32x = 320 x is a real number Like terms can be combined by adding coefficients Concept / Approach: Combine the x-terms first: (48 + 32)x. Then solve (80x = 320) by dividing both sides by 80. This keeps the work clean and prevents arithmetic mistakes. Step-by-Step Solution: 48x + 32x = (48 + 32)x = 80x So the equation becomes 80x = 320 Divide both sides by 80: x = 320/80 x = 4 Verification / Alternative check: Substitute x = 4 into the original equation: 48*4 + 32*4 = 192 + 128 = 320. The left side equals the right side, so x = 4 is correct. Why Other Options Are Wrong: If x = 8, left side becomes 640. If x = 16, it becomes 1280. If x = 32, it becomes 2560. If x = 2, it becomes 160. Only x = 4 satisfies the equation exactly. Common Pitfalls: Adding 48 and 32 incorrectly, or dividing by the wrong coefficient after combining like terms. Final Answer: The value of x is 4.

Question 13

Given that 10^0.48 = x and 10^0.70 = y, and x raised to the power z equals y squared (x^z = y^2), the value of z is closest to which of the following?

Accepted Answer

Introduction: This question tests exponent rules and using logarithms (or exponent comparison) to solve for an unknown power. Because x and y are both written as powers of 10, we can rewrite x^z and y^2 as powers of 10 as well. When two powers of 10 are equal, their exponents must be equal. That converts the problem into a simple linear equation in z. The final step is choosing the closest option, because the exact value may not match an option exactly. Given Data / Assumptions: x = 10^0.48 y = 10^0.70 Relation: x^z = y^2 Rule: (a^m)^n = a^(m*n) If 10^A = 10^B, then A = B Concept / Approach: Rewrite both sides using base 10. Since x = 10^0.48, x^z = (10^0.48)^z = 10^(0.48z). And y^2 = (10^0.70)^2 = 10^(1.40). Equate exponents: 0.48z = 1.40, then solve for z and compare to the choices. Step-by-Step Solution: x^z = (10^0.48)^z = 10^(0.48z) y^2 = (10^0.70)^2 = 10^(0.70*2) = 10^1.40 So 10^(0.48z) = 10^1.40 Equate exponents: 0.48z = 1.40 z = 1.40 / 0.48 = 140/48 = 35/12 ≈ 2.9167 Closest option to 2.9167 is 2.9 Verification / Alternative check: If z ≈ 2.9167, then 0.48z ≈ 0.48*2.9167 ≈ 1.40, so x^z ≈ 10^1.40 which matches y^2 exactly in exponent form. Why Other Options Are Wrong: 2.7 gives exponent 0.48*2.7 = 1.296, too low. 3.6 and 3.7 give exponents above 1.7, too high. 3.1 gives 1.488, still too high. Only 2.9 is closest to 2.9167. Common Pitfalls: Multiplying exponents incorrectly, treating 10^0.70 squared as 10^0.49, or forgetting that equal bases imply equal exponents. Final Answer: The value of z is closest to 2.9.

Question 14

The average number of visitors to a library on the first four days of a week is 58. The average number of visitors on the second, third, fourth, and fifth days is 60. The number of visitors on the first and fifth days are in the ratio 8 : 9. How man…

Accepted Answer

Introduction: This question tests using averages to form equations about sums, and then combining that with a ratio condition. Averages are just total sum divided by number of days, so two different averages over overlapping day ranges allow you to isolate the difference between two specific days. After finding the difference between day 5 and day 1, the ratio 8:9 converts that difference into exact values. This is a classic “overlapping averages + ratio” aptitude pattern. Given Data / Assumptions: Average visitors on days 1 to 4 = 58 Average visitors on days 2 to 5 = 60 Days 1 and 5 are in ratio 8 : 9 Let visitors on day 1 = D1 and on day 5 = D5 Concept / Approach: Convert averages to sums: Sum(1..4) = 58*4 and Sum(2..5) = 60*4. Subtract the sums to eliminate common days 2,3,4 and obtain D5 - D1. Then apply the ratio D1:D5 = 8:9 to solve for D5. Step-by-Step Solution: Sum of days 1 to 4 = 58 * 4 = 232 Sum of days 2 to 5 = 60 * 4 = 240 Subtract: (days 2 to 5) - (days 1 to 4) = D5 - D1 So D5 - D1 = 240 - 232 = 8 Given D1 : D5 = 8 : 9, let D1 = 8k and D5 = 9k Then D5 - D1 = k = 8 So k = 8 and D5 = 9k = 72 Verification / Alternative check: D1 = 64 and D5 = 72 satisfy the ratio. Also D5 - D1 = 8 matches the difference derived from sums, so the result is consistent. Why Other Options Are Wrong: Any value other than 72 breaks either the required difference D5 - D1 = 8 or the ratio D1:D5 = 8:9 (or both). Common Pitfalls: Using 5 days in the average (instead of 4), subtracting in the wrong order (getting D1 - D5), or applying the ratio incorrectly as 9:8 instead of 8:9. Final Answer: 72 visitors came to the library on the fifth day.

Question 15

The square of a positive integer exceeds twenty times the integer by 96. What is that positive integer?

Accepted Answer

Introduction: This question tests forming a quadratic equation from a word statement and solving it. The phrase “the square of an integer exceeds twenty times the integer by 96” means: integer^2 is 96 more than 20 times the integer. Translating that carefully produces a quadratic equation. Because the integer is stated to be positive, we choose only the positive root after solving. Given Data / Assumptions: Let the positive integer be n “Square exceeds 20 times the integer by 96” means n^2 = 20n + 96 We solve for integer n and keep only n > 0 Concept / Approach: Convert the statement into a quadratic equation: n^2 - 20n - 96 = 0. Solve using factoring or the quadratic formula. Since the problem requires a positive integer, discard any negative solution even if it satisfies the equation algebraically. Step-by-Step Solution: Start with: n^2 = 20n + 96 Bring all terms to one side: n^2 - 20n - 96 = 0 Find factors of -96 whose difference is 20: 24 and -4 So (n - 24)(n + 4) = 0 n = 24 or n = -4 Positive integer solution: n = 24 Verification / Alternative check: Check n = 24: n^2 = 576 and 20n = 480. Difference = 576 - 480 = 96, which matches the statement exactly. So 24 is correct. Why Other Options Are Wrong: 36 gives 36^2 - 20*36 = 1296 - 720 = 576, not 96. 42 and 48 produce even larger differences. 16 gives 256 - 320 = -64, not 96. Common Pitfalls: Misreading “exceeds by 96” as “is 96 times,” or forgetting to enforce the “positive integer” condition and mistakenly choosing -4. Final Answer: The positive integer is 24.

Question 16

In a slip test, K secures the 15th rank from the top and the 44th position from the bottom among the students who passed. Four students did not take the test and three students failed. What is the total strength of the class (total number of student…

Accepted Answer

Introduction: This question tests ranking logic (top and bottom positions) combined with additional students who were absent or failed. A key rule: if someone is k-th from the top and m-th from the bottom in the same group, then total in that group is k + m - 1. Here, the group is “students who passed.” After finding how many passed, we add those who did not take the test and those who failed to obtain the total class strength. Given Data / Assumptions: K is 15th from the top among those who passed K is 44th from the bottom among those who passed Absent students (did not take test) = 4 Failed students = 3 Total class = passed + absent + failed Concept / Approach: Compute passed students using: passed = 15 + 44 - 1. Then include the 4 absent and 3 failed students to get the overall class strength. The -1 avoids counting K twice (once from top and once from bottom). Step-by-Step Solution: Number of passed students = 15 + 44 - 1 = 58 Total students in class = passed + absent + failed = 58 + 4 + 3 = 65 Verification / Alternative check: If 58 students passed and K is 15th from the top, then there are 14 passed students above K. If K is 44th from the bottom, then there are 43 passed students below K. Total passed = 14 + 1 + 43 = 58, consistent. Adding 7 non-passed participants (4 absent + 3 failed) gives 65 total students. Why Other Options Are Wrong: 62, 63, 64, and 66 come from forgetting the -1 in ranking totals, or forgetting to add absent/failed students, or adding only one of the two extra groups. Common Pitfalls: Using 15 + 44 (forgetting -1), mixing “passed” group with total class directly, or treating absent students as part of passed rankings. Final Answer: The total strength of the class is 65.

Question 17

Simplify the expression: (37^3 + 35^3 + 28^3 - 3 * 37 * 35 * 28) / (37^2 + 35^2 + 28^2 - 37 * 35 - 35 * 28 - 37 * 28). Choose the correct simplified value.

Accepted Answer

Introduction: This problem tests recognizing and applying a standard algebraic identity involving a^3 + b^3 + c^3 - 3abc. When combined with the denominator a^2 + b^2 + c^2 - ab - bc - ca, the ratio often simplifies dramatically. Instead of expanding large cubes and squares, you use the identity to reduce the expression to a + b + c (provided a + b + c is not zero). This saves huge computation and is a classic simplification trick in aptitude and algebra tests. Given Data / Assumptions: a = 37, b = 35, c = 28 Numerator: a^3 + b^3 + c^3 - 3abc Denominator: a^2 + b^2 + c^2 - ab - bc - ca Known identity: a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) Concept / Approach: Apply the identity directly. The whole fraction becomes (a + b + c)(denominator) / (denominator) = a + b + c, as long as the denominator is not zero (equivalently, as long as a, b, c are not all equal and a + b + c is not forcing cancellation). Here the numbers are distinct, so the simplification is valid. Step-by-Step Solution: Let a = 37, b = 35, c = 28 Use identity: a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) So the given fraction = (a + b + c)(denominator) / (denominator) This simplifies to a + b + c a + b + c = 37 + 35 + 28 = 100 Verification / Alternative check: A quick sanity check: since the expression is built exactly in the identity form, the simplification must be linear (not huge). Getting 100 matches the sum 37+35+28, confirming correct use of the identity. Why Other Options Are Wrong: 0 or 1 would occur only if a + b + c were 0 or 1, which it is not. 4 and 64 are random outcomes from partial cancellation or incorrect identity use. Only 100 matches the identity result. Common Pitfalls: Trying to expand cubes and squares (leading to arithmetic errors), forgetting the exact denominator structure, or misremembering the identity as (a+b+c)^3 - 3abc. Final Answer: The expression simplifies to 100.

Question 18

If 3x + 7, x^2 + p, and 7x + 5 all represent the same value for some real number x (that is, 3x + 7 = x^2 + p = 7x + 5), what is the value of p?

Accepted Answer

Introduction: This problem tests solving linked equalities. When multiple expressions are stated to be equal to the same value, you can equate them pairwise. The quickest path is to first solve for x by equating the two linear expressions (3x + 7 and 7x + 5). Once x is known, plug into x^2 + p = the common value to find p. This approach avoids unnecessary algebra with p until the end and keeps the steps clean. Given Data / Assumptions: 3x + 7 = x^2 + p = 7x + 5 x is a real number We need to find p Concept / Approach: Since 3x + 7 and 7x + 5 are both equal to the same quantity, they must be equal to each other. Solve 3x + 7 = 7x + 5 to get x. Then use x^2 + p = 3x + 7 (or = 7x + 5) to compute p. Finally express p as a fraction to match the options exactly. Step-by-Step Solution: Set the linear expressions equal: 3x + 7 = 7x + 5 Bring terms together: 7 - 5 = 7x - 3x 2 = 4x x = 1/2 Common value = 3x + 7 = 3*(1/2) + 7 = 3/2 + 7 = 17/2 Now x^2 + p = 17/2 x^2 = (1/2)^2 = 1/4 p = 17/2 - 1/4 = (34/4 - 1/4) = 33/4 Verification / Alternative check: Check with 7x + 5: for x = 1/2, 7*(1/2)+5 = 7/2 + 5 = 17/2, same common value. Then x^2 + p = 1/4 + 33/4 = 34/4 = 17/2, consistent. So p is correct. Why Other Options Are Wrong: 17/2 is the common value, not p. The other fractions do not satisfy x^2 + p = 17/2 when x = 1/2. Only 33/4 makes the equality hold. Common Pitfalls: Confusing p with the common value, forgetting to square x correctly, or subtracting fractions incorrectly when computing p. Final Answer: The value of p is 33/4.

Question 19

A dry fruits shop sells small packets of mixed nuts for ₹150 each and large packets for ₹250 each. On one day, the owner sells 5,000 packets in total and receives ₹10.50 lakh in total sales. How many small packets does he sell that day?

Accepted Answer

Introduction: This question tests forming a system of linear equations from a sales scenario. When two types of items are sold (small and large packets) with different prices, the total number of items and total revenue give two equations. Solving the system yields the quantity of each type. The only extra care is interpreting “₹10.50 lakh” correctly as ₹10,50,000. Given Data / Assumptions: Price of small packet = ₹150 Price of large packet = ₹250 Total packets sold = 5,000 Total revenue = ₹10.50 lakh = ₹10,50,000 Let small packets = s, large packets = l Concept / Approach: Create two equations: s + l = 5000 and 150s + 250l = 1050000. Solve by substitution (s = 5000 - l) or elimination. Using a common factor (like dividing by 50) makes arithmetic easier and reduces mistakes. Step-by-Step Solution: s + l = 5000 150s + 250l = 1050000 Divide revenue equation by 50: 3s + 5l = 21000 Substitute s = 5000 - l 3(5000 - l) + 5l = 21000 15000 - 3l + 5l = 21000 2l = 6000 => l = 3000 s = 5000 - 3000 = 2000 Verification / Alternative check: Revenue check: 2000*150 = 300000 and 3000*250 = 750000. Total = 1050000 = ₹10.50 lakh, correct. Total packets: 2000 + 3000 = 5000, correct. Why Other Options Are Wrong: Any other s value forces l = 5000 - s and changes the revenue away from ₹10.50 lakh. Only s = 2000 satisfies both the total packets and revenue conditions simultaneously. Common Pitfalls: Mistaking ₹10.50 lakh as ₹105000 (one zero missing), swapping prices of small and large packets, or forgetting that quantities must sum to 5000. Final Answer: He sold 2,000 small packets.

Question 20

Two numbers satisfy the following system: 2 times the first number plus 3 times the second number equals 36, and 3 times the first number plus 2 times the second number equals 39. What is the difference between the two numbers?

Accepted Answer

Introduction: This question tests solving a pair of linear equations in two variables and then computing a derived value (the difference). A common strategy is elimination: multiply equations to align one variable, subtract to eliminate it, then back-substitute. Since the coefficients are small, the arithmetic remains clean. After finding both numbers, take their absolute difference to match the question’s wording. Given Data / Assumptions: Let first number = a Let second number = b 2a + 3b = 36 3a + 2b = 39 We need |a - b| Concept / Approach: Use elimination to solve for b first (or a). Multiply the first equation by 3 and the second by 2 so that the a-coefficient becomes equal (6a). Subtract to eliminate a, solve for b, then substitute back to find a. Finally compute the difference between the two numbers. Step-by-Step Solution: 2a + 3b = 36 ...(1) 3a + 2b = 39 ...(2) Multiply (1) by 3: 6a + 9b = 108 Multiply (2) by 2: 6a + 4b = 78 Subtract: (6a + 9b) - (6a + 4b) = 108 - 78 5b = 30 => b = 6 Substitute into (2): 3a + 2*6 = 39 => 3a + 12 = 39 3a = 27 => a = 9 Difference = |a - b| = |9 - 6| = 3 Verification / Alternative check: Check in (1): 2*9 + 3*6 = 18 + 18 = 36, correct. Check in (2): 3*9 + 2*6 = 27 + 12 = 39, correct. So the difference is reliable. Why Other Options Are Wrong: They are incorrect differences that would occur if one equation is solved incorrectly, elimination is done with wrong multipliers, or subtraction errors occur. Common Pitfalls: Mixing up which equation to multiply, subtracting in the wrong order, or forgetting to take the difference after finding the values of a and b. Final Answer: The difference between the two numbers is 3.

Simplification Questions