Question 1

Expand and simplify (3x + 2)(2x - 5) into the form ax^2 + kx + n. What is the value of the expression a - n + k?

Accepted Answer

Introduction / Context: This is an algebra question involving expansion of a product of two binomials and identification of coefficients in the resulting quadratic expression. It checks familiarity with the distributive property and standard algebraic multiplication. Once the expression is written in the form ax^2 + kx + n, we are asked to compute a derived value, namely a - n + k. This type of question is common in algebra sections of aptitude exams. Given Data / Assumptions: The expression to expand is (3x + 2)(2x - 5). After expansion, it can be written in the form ax^2 + kx + n. a is the coefficient of x^2, k is the coefficient of x, and n is the constant term. We must find the value of a - n + k. Concept / Approach: We use the distributive property or the FOIL method to expand the product of two binomials. After expansion, we collect like terms in x^2, x and the constant term to match the standard quadratic form. Once the coefficients a, k and n are identified, we substitute them into the expression a - n + k and simplify to get the numeric answer. Step-by-Step Solution: Step 1: Start with the given expression (3x + 2)(2x - 5). Step 2: Multiply each term in the first bracket by each term in the second bracket. Step 3: First term: 3x * 2x = 6x^2. Step 4: Outer term: 3x * (−5) = −15x. Step 5: Inner term: 2 * 2x = 4x. Step 6: Last term: 2 * (−5) = −10. Step 7: Combine the like terms for x: −15x + 4x = −11x. Step 8: So the expanded expression is 6x^2 − 11x − 10. Step 9: Comparing with ax^2 + kx + n, we identify a = 6, k = −11 and n = −10. Step 10: Compute a − n + k = 6 − (−10) + (−11). Step 11: Simplify: 6 − (−10) = 6 + 10 = 16, and 16 + (−11) = 5. Step 12: Therefore, a − n + k = 5. Verification / Alternative check: Recheck the expansion: (3x + 2)(2x − 5) = 3x * 2x + 3x * (−5) + 2 * 2x + 2 * (−5) = 6x^2 − 15x + 4x − 10 = 6x^2 − 11x − 10. The coefficients are therefore confirmed as a = 6, k = −11 and n = −10. Substituting into a − n + k gives 6 − (−10) + (−11) = 6 + 10 − 11 = 16 − 11 = 5, confirming the answer. Why Other Options Are Wrong: Options (b) 8, (c) 9, and (d) 10 would correspond to miscalculations, either from incorrect expansion or incorrect handling of negative signs when computing a − n + k. Since the verified result is 5, these options do not match the correct computation. Common Pitfalls: Common errors include forgetting one of the terms in the FOIL expansion, adding −15x and 4x incorrectly, or misinterpreting the constant term when substituting into a − n + k. The negative signs in k and n can easily be mishandled. Carefully writing each intermediate step and paying attention to signs helps avoid these mistakes. Final Answer: The value of a − n + k is 5.

Question 2

In a shooting game a person takes 40 shots. He earns 50 paise for every successful hit and loses 10 paise for every miss. If his net gain at the end is Rs 5, how many hits did he make?

Accepted Answer

Introduction / Context: This is a word problem involving linear equations and basic money calculations. The player earns money for each success and loses money for each failure across a fixed number of attempts. We are given the net gain and must determine how many of the attempts were successful. Problems like this are common in aptitude sections to test algebraic modeling skills using real-life style scenarios involving profit and loss per event. Given Data / Assumptions: Total number of shots taken is 40. The player earns 50 paise (Rs 0.50) for each hit. The player loses 10 paise (Rs 0.10) for each miss. His net gain from all shots is Rs 5, which is 500 paise. We must find the number of hits. Concept / Approach: Let the number of hits be h and the number of misses be m. Since there are 40 shots in total, m = 40 − h. The total earnings from hits are 50h paise, and the total losses from misses are 10m paise. The net gain is earnings minus losses and must equal 500 paise. So we set up the equation 50h − 10m = 500 and substitute m = 40 − h, reducing it to a single equation in h. Solving this equation gives the required number of successful hits. Step-by-Step Solution: Step 1: Let the number of hits be h. Step 2: Since there are 40 shots in total, the number of misses is m = 40 − h. Step 3: Earnings from hits are 50h paise. Step 4: Losses from misses are 10m paise = 10(40 − h) paise. Step 5: Net gain in paise is earnings minus losses: 50h − 10(40 − h). Step 6: The net gain is given as Rs 5, which is 500 paise. Step 7: Set up the equation: 50h − 10(40 − h) = 500. Step 8: Expand: 50h − 400 + 10h = 500. Step 9: Combine like terms: 60h − 400 = 500. Step 10: Add 400 to both sides: 60h = 900. Step 11: Divide both sides by 60: h = 900 / 60 = 15. Step 12: Therefore, the player made 15 successful hits. Verification / Alternative check: Compute net gain using h = 15. Then m = 40 − 15 = 25. Earnings from hits: 15 * 50 paise = 750 paise. Losses from misses: 25 * 10 paise = 250 paise. Net gain = 750 − 250 = 500 paise, which is Rs 5. This matches the given net gain and confirms that 15 hits is correct. Why Other Options Are Wrong: Option (b) 20: Would give earnings 20 * 50 = 1000 paise and losses 20 * 10 = 200 paise, net 800 paise, not 500. Option (c) 25: Would give earnings 25 * 50 = 1250 paise and losses 15 * 10 = 150 paise, net 1100 paise. Option (d) 30: Would give earnings 30 * 50 = 1500 paise and losses 10 * 10 = 100 paise, net 1400 paise. Common Pitfalls: Common errors include miscounting the number of misses, mixing rupees and paise without converting, or wrongly adding earnings and losses instead of subtracting losses from earnings. Setting up the net gain equation carefully and keeping all values in the same units (here, paise) avoids these mistakes. Final Answer: The number of successful hits is 15.

Question 3

The sum of the digits of a two-digit number is 8. If the original number is subtracted from the number formed by reversing its digits, the result is 54. What is the original number?

Accepted Answer

Introduction / Context: This question is about a two-digit number defined by conditions on its digit sum and the effect of reversing its digits. Such problems are standard in number system topics and involve representing the number using algebra and place value concepts. By forming equations from the given conditions, we can solve for the digits and hence determine the original number. Given Data / Assumptions: Let the original two-digit number be 10a + b, where a is the tens digit and b is the units digit. The sum of the digits is 8: a + b = 8. The number formed by reversing its digits is 10b + a. When the original number is subtracted from the reversed number, the result is 54: (10b + a) − (10a + b) = 54. We must find the original number 10a + b. Concept / Approach: We use place value representation for the original and reversed numbers. The problem gives two equations: one for the digit sum and one for the difference between the reversed number and the original. Simplifying the second equation will give a relationship between a and b, which combined with a + b = 8 allows us to solve for both digits. Once the digits are known, the original number is easily constructed and checked against the options. Step-by-Step Solution: Step 1: Let the original number be 10a + b. Step 2: The sum of its digits is a + b = 8. Step 3: The reversed number is 10b + a. Step 4: According to the problem, reversed number minus original number equals 54: (10b + a) − (10a + b) = 54. Step 5: Simplify the left side: 10b + a − 10a − b = 9b − 9a. Step 6: So 9b − 9a = 54. Step 7: Factor out 9: 9(b − a) = 54, so b − a = 54 / 9 = 6. Step 8: Now we have a + b = 8 and b − a = 6. Step 9: Add the two equations: (a + b) + (b − a) = 8 + 6, giving 2b = 14. Step 10: Solve for b: b = 14 / 2 = 7. Step 11: Substitute b = 7 into a + b = 8 to get a + 7 = 8, so a = 1. Step 12: Therefore, the original number is 10a + b = 10 * 1 + 7 = 17. Verification / Alternative check: Check the digit sum: 1 + 7 = 8, which matches. The reversed number is 71. Compute the difference: 71 − 17 = 54, exactly as given. Hence, the number 17 satisfies both conditions, confirming it as the correct answer. Why Other Options Are Wrong: Option (a) 28: Digits sum to 2 + 8 = 10, not 8. Option (b) 19: Digit sum is 10, and difference with 91 is 72, not 54. Option (c) 37: Digit sum is 10, and reversed difference 73 − 37 = 36, not 54. Common Pitfalls: Some learners may reverse the subtraction order and consider original minus reversed, which leads to a negative value or wrong equation. Others may mis-handle the simplification of (10b + a) − (10a + b). Correctly setting up and simplifying to 9(b − a) is crucial. Also, care must be taken when solving the system of equations for a and b so that arithmetic errors do not occur. Final Answer: The original two-digit number is 17.

Question 4

In this basic arithmetic question, one number is 25% of another number, and the larger number is 12 more than the smaller number. What is the value of the larger number?

Accepted Answer

Introduction / Context: This aptitude question tests your understanding of percentages and simple linear equations involving two related numbers. Many exams use this style of problem to check whether you can translate a verbal relationship such as one number being 25 percent of another into algebra, and then combine that with a difference condition to find the exact values. Given Data / Assumptions: One number is 25 percent of another number. The larger number is 12 more than the smaller number. We must find the value of the larger number. Both numbers are assumed to be real positive numbers, typically integers in this type of aptitude question. Concept / Approach: The key ideas are percentage conversion and forming simultaneous equations. Saying that one number is 25 percent of another means it is equal to 0.25 times the other number, or one fourth of it. We also know the difference between the larger and smaller number. By naming the numbers with variables, we can write algebraic equations and solve them systematically. Once the smaller number is found, the larger number follows directly from the relationship between them. Step-by-Step Solution: Step 1: Let the smaller number be x and the larger number be y. Step 2: The statement "one number is 25 percent of another" becomes x = 0.25 * y, which is the same as x = (1 / 4) * y. Step 3: The difference condition "the larger number is 12 more than the smaller" becomes y = x + 12. Step 4: Substitute x = (1 / 4) * y into y = x + 12 to express everything in terms of y. Step 5: This gives y = (1 / 4) * y + 12. Step 6: Move terms involving y to one side: y - (1 / 4) * y = 12. Step 7: Simplify the left side: (3 / 4) * y = 12. Step 8: Solve for y: y = 12 * (4 / 3) = 16. Step 9: Therefore the larger number is 16 and the smaller number is x = (1 / 4) * 16 = 4, which is consistent with the difference of 12. Verification / Alternative check: You can verify the answer quickly without full algebra by trying the options and checking the relationships. If the larger number is 16, then one fourth of 16 is 4, so the smaller number is 4. The difference between 16 and 4 is 12, which matches the condition. Also, 4 is exactly 25 percent of 16. None of the other options will satisfy both the 25 percent relation and the difference of 12 at the same time, so 16 is the only consistent solution. Why Other Options Are Wrong: 16 is correct, so check the others briefly. If the larger number were 20, then 25 percent of 20 is 5, and the difference 20 - 5 is 15, not 12. If the larger number were 24, then 25 percent of 24 is 6, and the difference is 18, not 12. If it were 12, then 25 percent of 12 is 3, and 12 - 3 is only 9. If it were 8, then 25 percent of 8 is 2, and 8 - 2 is 6. None of these satisfy the stated difference of 12. Common Pitfalls: A common mistake is to misread the statement and treat 25 percent incorrectly, either as 0.25 added to the number or as 25 divided by the number. Another error is to assume the smaller number is 25 percent less than the larger number rather than exactly one fourth of it. Students also sometimes mix up which number is larger and which is smaller, leading to reversed equations that do not match the conditions. Carefully defining variables and writing the equations directly from the words helps avoid these issues. Final Answer: The larger number that satisfies both conditions is 16.

Question 5

In this number theory question, you must count how many numbers from 2000 to 7000 are both perfect squares and perfect cubes at the same time.

Accepted Answer

Introduction / Context: This problem tests understanding of perfect squares and perfect cubes, and how to recognize numbers that are both. Such numbers are called perfect sixth powers because a number that is simultaneously a square and a cube must be a power of an integer raised to the sixth power. The question asks you to identify how many such numbers lie between 2000 and 7000, inclusive of the endpoints if they qualify. Given Data / Assumptions: The range of interest is from 2000 to 7000. We are looking for numbers that are both perfect squares and perfect cubes. There are no additional restrictions such as odd, even, or prime. All numbers are understood to be positive integers. Concept / Approach: A number that is both a perfect square and a perfect cube must be of the form n^6, where n is a positive integer. This is because being a perfect square means the exponent on the prime factorization is even, being a cube means it is a multiple of 3, and the least common multiple of 2 and 3 is 6. So the central idea is to find all integers n such that n^6 lies between 2000 and 7000. Then we count how many such n exist. Step-by-Step Solution: Step 1: Express the condition as 2000 ≤ n^6 ≤ 7000. Step 2: Estimate powers of small integers. Compute 2^6 = 64, which is far below 2000. Step 3: Compute 3^6 = 729, which is still below 2000, so not in the range. Step 4: Compute 4^6. First 4^3 = 64, then square that to get 64^2 = 4096, which lies between 2000 and 7000. Step 5: Compute 5^6. First 5^3 = 125, then square that to get 125^2 = 15625, which is greater than 7000. Step 6: For n bigger than 5, n^6 will be even larger, so no further values will fall inside the range. Step 7: Check n = 4 carefully: 4096 is indeed both a perfect square (64^2) and a perfect cube (16^3), so it is valid. Step 8: Conclude that only one number, 4096, satisfies all the conditions. Verification / Alternative check: You can double check with the perfect square and cube perspective. 4096 is 64^2 so it is a square. It is also 16^3 so it is a cube. There is no other perfect cube between 2000 and 7000 that is also a perfect square. For instance, 10^3 = 1000 and 20^3 = 8000, so only 11^3 to 18^3 lie in the numeric range, and none of those are perfect squares. This reinforces that 4096 is the unique candidate. Why Other Options Are Wrong: Option 0 would suggest there is no such number, but 4096 contradicts that. Options 2, 3, and 4 would require at least two or more distinct sixth powers inside the interval, but we have seen that moving from 4^6 to 5^6 already passes beyond 7000. Since only 4^6 lies in the given range, any count higher than 1 is impossible. Common Pitfalls: Students sometimes confuse perfect squares and perfect cubes and look for any square or any cube rather than numbers that are both. Another common mistake is to test random numbers between 2000 and 7000 instead of using the sixth power idea, which wastes time and can lead to errors. Some learners also compute powers incorrectly, especially 4^6 and 5^6, so careful calculation is important. Recognizing that being both a square and a cube immediately implies a sixth power makes this question much easier and more systematic. Final Answer: There is exactly 1 number between 2000 and 7000 which is both a perfect square and a perfect cube.

Question 6

In a two digit number, the digit in the tens place is 8 and the digit in the units place is z. What is the correct algebraic expression for this two digit number?

Accepted Answer

Introduction / Context: This question checks your understanding of how to represent a two digit number using place value and algebraic notation. Many aptitude and algebra problems use a variable for one digit and ask you to express the entire number. Understanding that the tens place contributes a multiple of ten and the units place contributes a single unit is the key concept here. Given Data / Assumptions: The number is a two digit number. The tens digit is 8. The units digit is represented by the variable z. We are asked to write the numerical value of the number as an algebraic expression in terms of z. Concept / Approach: Any two digit number can be written as 10 multiplied by the tens digit plus the units digit. For example, 37 is 3 * 10 + 7. Here, the tens digit is fixed at 8, and the units digit is the variable z. So the number must be of the form 10 * 8 plus z. We simply need to express that clearly and avoid confusing digit multiplication with place value. Step-by-Step Solution: Step 1: Recognize that a general two digit number with tens digit a and units digit b can be written as 10 * a + b. Step 2: In this problem, the tens digit is 8, so a = 8. Step 3: The units digit is represented by z, so b = z. Step 4: Substitute these into the generic form to obtain 10 * 8 + z. Step 5: Compute 10 * 8 = 80, so the number is 80 + z. Step 6: Hence, the correct algebraic expression for the number is 80 + z. Verification / Alternative check: To verify, choose a simple value for z, for example z = 3. Then the actual number should be 83, since the tens digit is 8 and the units digit is 3. If we use the expression 80 + z, we get 80 + 3 = 83, which matches. If we try an incorrect expression such as 8z + 8 with z = 3, we get 8 * 3 + 8 = 32, which does not have tens digit 8 and units digit 3, so it is wrong. This quick substitute check helps confirm that 80 + z is the only correct representation. Why Other Options Are Wrong: The option 8z + 8 treats the tens digit as a coefficient multiplied by z, which misrepresents place value. The expression 80z + z factors the number as a multiple of z, which makes the tens digit depend on z as well, which is not true. The expression 80z + 8 is far too large for reasonable values of z and again confuses digit position with multiplication. The option 8 + z would only represent a one or two digit number formed by adding the digits, not arranging them in tens and units places. None of these incorrect forms match the place value rule. Common Pitfalls: Students sometimes mix up the idea of multiplying by ten with the use of variables and write forms like 8z or z8 instead of 10 * 8 + z. Others may try to directly concatenate symbols and think of 8z as a two digit number rather than a product. Remember that algebraic expressions follow arithmetic rules and digits in decimal notation correspond to powers of ten, not to coefficients on a variable unless clearly defined that way. Carefully separating place value from variable notation avoids these misunderstandings. Final Answer: The correct expression for the two digit number is 80 + z.

Question 7

The sum of three numbers is 2. The first number is one half of the second number, and the third number is one fourth of the second number. What is the value of the second number?

Accepted Answer

Introduction / Context: This question is a classic example of solving a system of equations involving fractions and a total sum. You are given relationships between three numbers and their total, and you must find one of the numbers. Such problems are very common in quantitative aptitude tests because they check both your comfort with fractions and your ability to set up algebraic equations correctly. Given Data / Assumptions: The sum of three numbers is 2. The first number is one half of the second number. The third number is one fourth of the second number. We are required to find the exact value of the second number. All numbers are assumed to be real, and fractional answers are allowed. Concept / Approach: The standard approach is to introduce a variable for the second number and then express the first and third numbers in terms of this same variable. Using the sum condition, we obtain a single equation in one variable, which can be solved by ordinary fraction manipulation. The key idea is to keep the fractions organized and combine like terms carefully to avoid arithmetic mistakes. Step-by-Step Solution: Step 1: Let the second number be y. Step 2: The first number is one half of the second, so the first number is (1 / 2) * y. Step 3: The third number is one fourth of the second, so the third number is (1 / 4) * y. Step 4: The sum of the three numbers is 2, so write the equation (1 / 2) * y + y + (1 / 4) * y = 2. Step 5: Combine like terms by adding the fractions: (1 / 2) + 1 + (1 / 4) = (2 / 4) + (4 / 4) + (1 / 4) = 7 / 4. Step 6: Therefore, the equation becomes (7 / 4) * y = 2. Step 7: Solve for y by multiplying both sides by 4 / 7, obtaining y = 2 * (4 / 7) = 8 / 7. Step 8: Hence, the second number is 8 / 7. Verification / Alternative check: Verify by computing all three numbers explicitly. The second number is 8 / 7. The first number is half of this, so it is (1 / 2) * (8 / 7) = 4 / 7. The third number is one fourth of the second, so it is (1 / 4) * (8 / 7) = 2 / 7. Add them: 4 / 7 + 8 / 7 + 2 / 7 = 14 / 7 = 2, which matches the given total. This confirms that 8 / 7 is correct. Why Other Options Are Wrong: If the second number were 7 / 6, the first and third numbers would be 7 / 12 and 7 / 24, and their sum would not equal 2. Similar mismatches occur for 9 / 8, 10 / 9, and 4 / 3, where the computed total sum of the three numbers will not be exactly 2. Only 8 / 7 leads to a consistent set of three numbers that satisfy both the relationship conditions and the sum condition. Common Pitfalls: Many learners make errors when adding fractions or when converting them to a common denominator. Another frequent issue is incorrectly interpreting phrases like one half of the second number or one fourth of the second number, which must be multiplied, not added. Being patient with fraction arithmetic and writing each step clearly helps prevent these mistakes. It is also good practice to verify the final result by substituting back into the original conditions, which can catch small calculation errors. Final Answer: The second number that satisfies all the conditions is 8/7.

Question 8

Find the smallest number that must be added to 756896 so that the resulting sum is exactly divisible by 11.

Accepted Answer

Introduction / Context: This problem focuses on divisibility by 11, a common topic in number system questions in aptitude exams. Instead of performing long division directly, it is usually more efficient to use the divisibility rule for 11 or modular arithmetic. You must find the smallest non negative integer that, when added to 756896, gives a number divisible by 11. Given Data / Assumptions: The starting number is 756896. We want to add the smallest possible number k so that 756896 + k is divisible by 11. The answer should be a non negative integer, and usually a small one. All calculations use standard integer arithmetic. Concept / Approach: The key concept here is working with remainders modulo 11. If a number leaves a remainder r when divided by 11, then we need to add (11 - r) to reach the next multiple of 11, unless the remainder is 0. There is also a digit based divisibility rule for 11 that uses alternating sums of digits, but calculating the remainder directly is often simpler when you are comfortable with modular arithmetic. Step-by-Step Solution: Step 1: Let k be the smallest number to add to 756896 so that 756896 + k is divisible by 11. Step 2: First find the remainder when 756896 is divided by 11. Step 3: Compute 756896 mod 11. Using division or a calculator, we find that 756896 leaves a remainder of 8 when divided by 11. Step 4: We want 756896 + k to have remainder 0 modulo 11, which means (8 + k) mod 11 must be 0. Step 5: So k must satisfy 8 + k = 11 or 8 + k = 22 and so on. The smallest positive solution is k = 3. Step 6: Check: 756896 + 3 = 756899. Step 7: Verify that 756899 divided by 11 has no remainder. Since 756896 had remainder 8, adding 3 increases the remainder to 11, which is equivalent to 0, confirming divisibility. Verification / Alternative check: As an alternative method, you can use the divisibility rule for 11 which states that the difference between the sum of digits in odd positions and the sum of digits in even positions should be a multiple of 11. However, because the number is large, modular arithmetic is more straightforward. Another verification is to try adding small values 1, 2, 3 in order and check remainders. Adding 1 gives remainder 9, adding 2 gives remainder 10, and adding 3 gives remainder 0, which confirms that 3 is indeed the smallest required addition. Why Other Options Are Wrong: If we add 1, the new remainder is 9, so the sum is not divisible by 11. Adding 2 results in remainder 10. Adding 4 yields remainder 1, and adding 5 yields remainder 2. All of these still leave a non zero remainder when divided by 11. Only adding 3 converts the remainder from 8 to 0, making 756899 the nearest multiple of 11 greater than 756896. Common Pitfalls: A common mistake is to attempt full division by 11, which is slower and more error prone, especially under exam pressure. Another pitfall is confusing the direction of the adjustment and trying to subtract a number instead of adding one. Some students also mistakenly assume that the added number must be a multiple of 11 itself. Remember that we only care about changing the remainder to zero, which is often achievable with a small adjustment like this case. Practicing remainder based thinking greatly simplifies such questions. Final Answer: The smallest number that must be added to 756896 to obtain a multiple of 11 is 3.

Question 9

Of three numbers whose average is 22, the first number is three eighths of the sum of the other two numbers. What is the value of the first number?

Accepted Answer

Introduction / Context: This question tests your understanding of averages and algebraic relationships among three numbers. You are given the average of the three numbers and a specific relationship between the first number and the sum of the other two numbers. By converting the average into a total sum and using the proportional relationship, you can solve for the first number using simple algebra. Given Data / Assumptions: The average of three numbers is 22. The first number is three eighths of the sum of the second and third numbers. We must find the value of the first number. All numbers are real, and most likely integers, given the typical style of aptitude questions. Concept / Approach: The average of three numbers is the total sum divided by 3. So, if the average is 22, the sum of the three numbers must be 3 * 22. By assigning variables to the three numbers, we can express the given fractional relationship and the total sum as algebraic equations. Then we solve the equations to find the first number. This is a classic linear equation problem involving fractions and sums. Step-by-Step Solution: Step 1: Let the three numbers be a, b, and c. Step 2: The average is 22, so the sum is a + b + c = 3 * 22 = 66. Step 3: The first number a is three eighths of the sum of the other two, so a = (3 / 8) * (b + c). Step 4: From the total sum, we know that b + c = 66 - a. Step 5: Substitute b + c into the relationship for a to get a = (3 / 8) * (66 - a). Step 6: Multiply both sides by 8 to clear the denominator: 8a = 3 * (66 - a). Step 7: Expand the right side: 8a = 198 - 3a. Step 8: Add 3a to both sides to combine like terms: 11a = 198. Step 9: Solve for a: a = 198 / 11 = 18. Step 10: Therefore, the first number is 18. Verification / Alternative check: Verify by computing possible values of b and c. From a + b + c = 66 and a = 18, we get b + c = 48. The condition a = (3 / 8) * (b + c) becomes 18 = (3 / 8) * 48. Compute (3 / 8) * 48 = 3 * 6 = 18, which checks out perfectly. This confirms that the first number being 18 satisfies both the average and the fractional relationship. Why Other Options Are Wrong: If the first number were 16, then b + c would be 50, and (3 / 8) * 50 would be 18.75, which does not match 16. For 20, b + c would be 46, and (3 / 8) * 46 would be 17.25, again not equal to 20. Similar mismatches occur for 22 and 24. Only 18 satisfies the equation a = (3 / 8) * (66 - a), so only that option is correct. Common Pitfalls: Students sometimes confuse the phrase three eighths of the sum of the other two with three eighths of the total sum, which would lead to the wrong equation. Another common error is to mishandle the fraction when clearing denominators or adding terms, resulting in incorrect arithmetic. Writing each step clearly, especially when moving between the sum and the relationship, helps avoid such mistakes. Always verify the final answer by substituting back into the original conditions. Final Answer: The first number is 18.

Question 10

How many natural numbers lie strictly between the square root of 261 and the square root of 45109?

Accepted Answer

Introduction / Context: This question checks your ability to work with square roots and count integers in a range described using inequalities. Instead of asking directly for numbers between two integers, the problem uses square roots of non perfect squares, which forces you to approximate or reason about their integer parts and then count natural numbers between them. Given Data / Assumptions: We consider the square root of 261, written as sqrt(261). We also consider the square root of 45109, written as sqrt(45109). We must find how many natural numbers lie strictly between these two square roots. Natural numbers are positive integers 1, 2, 3, and so on. Concept / Approach: The key idea is that if a real number t satisfies a < t < b, then the integers between a and b are counted by looking at the smallest integer greater than a and the largest integer less than b. In this case, since 261 and 45109 are not perfect squares, their square roots are not integers. We approximate or bound them to determine the first and last integers inside the open interval (sqrt(261), sqrt(45109)). Then we count how many integers fall in that interval. Step-by-Step Solution: Step 1: Observe that 16^2 = 256 and 17^2 = 289. Step 2: Since 261 is between 256 and 289, we know that 16 < sqrt(261) < 17. Step 3: Next, note that 212^2 = 44944 and 213^2 = 45369. Step 4: Since 45109 lies between 44944 and 45369, we have 212 < sqrt(45109) < 213. Step 5: We are asked to find natural numbers strictly between sqrt(261) and sqrt(45109), so we need integers n such that sqrt(261) < n < sqrt(45109). Step 6: From the bounds, this means we need integers strictly greater than 16 and strictly less than 213, but we have a tighter upper bound of less than 213, and sqrt(45109) is already between 212 and 213. Step 7: Therefore the smallest integer greater than sqrt(261) is 17. Step 8: The largest integer less than sqrt(45109) is 212. Step 9: The natural numbers satisfying the condition are 17, 18, 19, up to 212. Step 10: The count of integers from 17 to 212 is 212 - 17 + 1 = 196. Verification / Alternative check: One way to check is simply to compute the number of integers from 1 to 212, which is 212, and subtract the number from 1 to 16, which is 16. The result 212 - 16 = 196 again gives the count of integers from 17 to 212 inclusive. Since we have already shown that these are exactly the integers between the two square roots, this confirms that the answer is correct. No other integers can lie in the interval because any integer less than 17 would be less than sqrt(261), and any integer greater than 212 would exceed sqrt(45109). Why Other Options Are Wrong: An answer of 144, 168, 195, or 200 would imply a shorter or longer range of integers. Such numbers come from incorrect bounds, such as using 18 instead of 17, or 211 instead of 212, or from miscounting the length of the interval. Precise identification of the lower and upper integer bounds is essential, and small off by one errors easily lead to those incorrect options. Common Pitfalls: A frequent mistake is forgetting that the interval is strict, so students sometimes include 16 or 213, which should not be counted. Another common issue is miscalculating the squares of 16, 17, 212, or 213, which shifts the bounding integers. Some learners also forget to add one when counting integers from m to n inclusive, leading to answers that are exactly one less than the true count. Careful work with inequalities and attentive counting help prevent these errors. Final Answer: The number of natural numbers between the two square roots is 196.

Question 11

What is the least number that must be added to the product 684 × 686 so that the result becomes a perfect square?

Accepted Answer

Introduction / Context: This problem asks you to work with products of consecutive numbers and recognize a pattern that relates such products to perfect squares. The product 684 × 686 involves two numbers that are equally spaced around 685. This structure allows you to use the identity for (n − 1)(n + 1) and connect it directly to n^2. Many number system questions exploit this identity, so understanding it gives a fast and elegant solution. Given Data / Assumptions: The product considered is 684 × 686. We must find the least number k such that 684 × 686 + k is a perfect square. We are looking for the smallest non negative integer that works. All arithmetic is performed with integers. Concept / Approach: The key identity is (n − 1)(n + 1) = n^2 − 1. Here, if we take n = 685, then n − 1 is 684 and n + 1 is 686. So the product 684 × 686 is exactly 685^2 − 1. To turn this into a perfect square, we simply need to add 1, which makes the expression equal to 685^2. Recognizing this pattern avoids long multiplication and trial and error with squares. Step-by-Step Solution: Step 1: Observe that 684 and 686 are symmetric around 685. Step 2: Let n = 685, then 684 = n − 1 and 686 = n + 1. Step 3: Use the identity (n − 1)(n + 1) = n^2 − 1. Step 4: Substitute n = 685 into this identity to get 684 × 686 = 685^2 − 1. Step 5: We want 684 × 686 + k to be a perfect square. Step 6: From the expression above, 684 × 686 + 1 = 685^2 − 1 + 1 = 685^2. Step 7: Since 685^2 is clearly a perfect square, adding k = 1 makes the result a perfect square. Step 8: Any smaller non negative k must be 0, but 684 × 686 itself equals 685^2 − 1, which is not a perfect square. Step 9: Therefore the least number that must be added is 1. Verification / Alternative check: To verify, you can compute the product numerically. 684 × 686 equals 469224. Add 1 to obtain 469225. Now note that 685^2 is also 469225, which confirms that 469225 is a perfect square. If you try adding any other option, for instance 684 or 685 or 686, you will not get a number that is an exact square of an integer. Since 1 is the smallest positive adjustment that converts n^2 − 1 into n^2, it must be the least number required. Why Other Options Are Wrong: Adding 684, 685, or 686 produces numbers much larger than 469225 and they do not match any perfect square near 685^2. Adding 2 gives 469226, which lies between 685^2 and 686^2 but is not equal to either. Therefore none of those adjustments yield a perfect square result. Only adding 1 hits exactly 685^2, so the other options are not correct. Common Pitfalls: Some students attempt to expand 684 × 686 directly and then check many square values by hand, which is time consuming and error prone. Others might not notice the symmetry around 685 and miss the helpful identity (n − 1)(n + 1) = n^2 − 1. A further mistake is to assume the number to add must be large, when in fact the structure here shows that a very small adjustment works. Recognizing and using algebraic identities is a powerful way to simplify number system problems like this one. Final Answer: The least number that must be added is 1.

Question 12

Determine the largest four digit number that is a perfect square.

Accepted Answer

Introduction / Context: This question checks your understanding of perfect squares and the range of four digit numbers. You are asked to identify the largest four digit number that can be written as the square of an integer. This is a standard type of problem that appears in many quantitative aptitude exams to test number sense and familiarity with square values near a boundary like 10000. Given Data / Assumptions: We are concerned only with four digit numbers, that is numbers from 1000 to 9999. We want the largest number in this range that is a perfect square. Numbers greater than 9999 are five digit numbers and are not allowed. We consider positive integer squares only. Concept / Approach: The smallest five digit number is 10000, which is 100^2. Therefore, the largest four digit perfect square must be the square of 99, since the next integer 100 already yields a five digit result. So the problem reduces to calculating 99^2 accurately. Once we have that value, we simply choose it from among the given options. Step-by-Step Solution: Step 1: Note that 100^2 = 10000, which is a five digit number. Step 2: Therefore, the square of any integer greater than or equal to 100 will be at least 10000 and hence not a four digit number. Step 3: The largest integer whose square could still be four digits is 99. Step 4: Compute 99^2. One way is to use the identity (100 − 1)^2 = 100^2 − 2 * 100 + 1. Step 5: Substitute 100: (100 − 1)^2 = 10000 − 200 + 1 = 9801. Step 6: Therefore 99^2 = 9801. Step 7: So 9801 is the largest possible four digit perfect square. Verification / Alternative check: You can confirm this by checking the next perfect square after 9801, which is 100^2 = 10000. Since 10000 is outside the four digit range, 9801 is indeed the maximum within that range. You can also verify that 9801 is a square by checking that 99 × 99 equals 9801 using standard multiplication. No larger number under 10000 will be a square, because that would require an integer larger than 99 whose square is still four digits, which is impossible. Why Other Options Are Wrong: The number 9999 is not a perfect square because there is no integer n such that n^2 = 9999. Similarly, 9702, 9604, and 9409 are not squares of integers close to 99 or 100. In fact, 98^2 = 9604 and 97^2 = 9409. While these two are perfect squares, they are smaller than 9801, so they cannot be the largest four digit perfect square. Among all four digit squares, 9801 stands above the rest. Common Pitfalls: A common mistake is to assume 9999 is a perfect square because it is the largest four digit number. Without checking, some students may choose it by intuition. Others might compute 98^2 or 97^2 and forget to check 99^2 using the identity for (100 − 1)^2. Another possible error is incorrect multiplication when computing 99^2 manually, leading to values like 9901 or 9701. Using the algebraic identity or carefully performing multiplication avoids these issues. Final Answer: The largest four digit number that is a perfect square is 9801.

Question 13

Find the largest five digit number that is exactly divisible by 89.

Accepted Answer

Introduction / Context: This question involves divisibility and working with the upper limit of a range, here the set of all five digit numbers. You must determine the largest five digit number that is divisible by 89. This kind of problem appears frequently in competitive exams to test comfort with division, remainders, and reasoning about multiples near a boundary like 99999. Given Data / Assumptions: We are looking at five digit numbers, from 10000 to 99999. We want the largest number in this range that is exactly divisible by 89. Exactly divisible means the remainder on division by 89 is zero. We assume standard integer arithmetic. Concept / Approach: The key idea is to find the greatest multiple of 89 that does not exceed 99999. This can be done by dividing 99999 by 89 and taking the integer part of the quotient. That integer quotient times 89 will be the largest multiple of 89 not greater than 99999. We do not need to test numbers from the options by trial if we perform this calculation correctly. Step-by-Step Solution: Step 1: Consider the maximum five digit number 99999. Step 2: Divide 99999 by 89 to find how many full multiples of 89 fit inside 99999. Step 3: The integer part of 99999 / 89 is 1123. This means 1123 is the largest integer k such that 89 * k ≤ 99999. Step 4: Compute the product 89 * 1123. Step 5: Multiplying gives 89 * 1123 = 99947. Step 6: Therefore 99947 is a multiple of 89 and it is less than 99999. Step 7: Any larger multiple would be 89 * 1124, which is 99947 + 89 = 100036, which exceeds the five digit limit. Step 8: Thus 99947 is the largest five digit number exactly divisible by 89. Verification / Alternative check: To verify, we can perform a direct division of 99947 by 89. Dividing gives a quotient of 1123 and a remainder of zero, confirming exact divisibility. If we try nearby numbers such as 99940, 99938, or 99939, and divide by 89, we will not get integer quotients. For example, if 99940 were divisible by 89, then 99940 divided by 89 would have to be an integer, but it is not. This confirms that among the options, 99947 is the only valid answer. Why Other Options Are Wrong: Numbers like 99940, 99938, and 99939 are close to 99947 but they are not multiples of 89. They produce non zero remainders when divided by 89. The option 99858 is also a multiple of 89, but it is smaller than 99947, so it cannot be the largest five digit multiple. Therefore, only 99947 satisfies both conditions of being five digits and being the largest number exactly divisible by 89. Common Pitfalls: A common error is to guess based on the last digits and not check divisibility properly, or to try dividing each option instead of reasoning from the top of the range. Another pitfall is rounding the quotient incorrectly or miscalculating the product of 89 and the quotient. Carefully performing the division and checking the final product avoids these errors. Remember that the standard method is to take the floor of the division of the upper bound by the divisor, then multiply back to obtain the largest multiple within the allowed range. Final Answer: The largest five digit number exactly divisible by 89 is 99947.

Question 14

Find the sum of all prime numbers between 58 and 68.

Accepted Answer

Introduction / Context: This question tests basic knowledge of prime numbers and the ability to identify primes within a small interval. After locating each prime number between 58 and 68, you must add them to obtain the total sum. This type of question is common in exams to quickly check understanding of primality and simple addition. Given Data / Assumptions: The interval is between 58 and 68, inclusive of the end points if they turn out to be prime. We must identify which numbers in this range are prime. Then we must compute the sum of those primes. Prime numbers are positive integers greater than 1 with exactly two distinct positive divisors, 1 and the number itself. Concept / Approach: The approach is to examine each integer from 58 through 68 and test whether it is divisible by any prime smaller than or equal to its square root. In this small range, we mainly test divisibility by 2, 3, 5, and 7. Once primes are identified, we simply add them. Because the range is narrow, this process is straightforward and does not require advanced techniques. Step-by-Step Solution: Step 1: List all integers from 58 to 68: 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68. Step 2: Check 58. It is even, so it is divisible by 2 and therefore not prime. Step 3: Check 59. It is not divisible by 2, 3, 5, or 7, so 59 is prime. Step 4: Check 60. It is even and divisible by 2, so it is not prime. Step 5: Check 61. It is not divisible by 2, 3, 5, or 7, so 61 is prime. Step 6: Check 62. It is even, so it is not prime. Step 7: Check 63. It is divisible by 3 and 7, so it is not prime. Step 8: Check 64. It is 8 squared, so it is clearly composite. Step 9: Check 65. It ends with 5, so it is divisible by 5 and not prime. Step 10: Check 66. It is even and divisible by 2 and 3, so it is not prime. Step 11: Check 67. It is not divisible by 2, 3, 5, or 7, so 67 is prime. Step 12: Check 68. It is even and divisible by 2, so it is not prime. Step 13: The primes in this range are therefore 59, 61, and 67. Step 14: Compute the sum: 59 + 61 + 67 = 120 + 67 = 187. Verification / Alternative check: You can verify by rechecking divisibility of each candidate prime. For example, for 59, test division by 2, 3, 5, and 7, and confirm no exact division occurs. Similarly, check 61 and 67, noting that there is no divisor other than 1 and themselves. Recalculate the sum to ensure no arithmetic error: 59 + 61 = 120, then 120 + 67 = 187. Since all checks are consistent and no other primes exist in the range, 187 must be correct. Why Other Options Are Wrong: The other options 179, 178, 183, and 191 correspond to incorrect combinations or sums. They would arise if a composite number like 65 were mistakenly included or if one of the primes were omitted or miscalculated. For instance, including 65 by mistake and excluding 67 could produce incorrect totals. Only 187 matches the exact sum of the verified primes 59, 61, and 67. Common Pitfalls: A frequent error is to forget 59 or 67 as primes because they are less familiar than smaller primes. Another common pitfall is to miscalculate the sum or to treat 65 or 63 as primes without checking divisibility by 5 or 3. To avoid mistakes, always test suspected primes for divisibility by small primes up to their square root and perform addition carefully, possibly in stages, as shown. This systematic approach ensures accuracy. Final Answer: The sum of all prime numbers between 58 and 68 is 187.

Question 15

The average of 25 consecutive odd integers is 55. What is the largest (highest) integer among these 25 odd numbers?

Accepted Answer

Introduction / Context: This question examines your understanding of averages and arithmetic progressions, especially sequences of consecutive odd integers. When a set of numbers is evenly spaced, such as consecutive odd numbers, the average has a direct relationship to the middle term. By using this property, you can find the highest term quickly without listing all numbers. Given Data / Assumptions: There are 25 consecutive odd integers. The average of these 25 numbers is 55. We must find the largest number in this sequence. The numbers form an arithmetic progression with common difference 2. Concept / Approach: For an arithmetic progression, the average of the numbers is equal to the value of the middle term when the number of terms is odd. Since we have 25 terms, which is odd, the average 55 is the 13th term in the ordered list. Once we know the middle term, we can move outward to find the largest term by adding the common difference repeatedly. The common difference between consecutive odd integers is 2, so the highest term will be a certain number of steps above the middle term. Step-by-Step Solution: Step 1: Recognize that 25 consecutive odd numbers form an arithmetic progression with 25 terms and common difference 2. Step 2: Since there are 25 terms, the middle term is the 13th term. Step 3: In any arithmetic progression with an odd number of terms, the average equals the middle term. Step 4: The given average is 55, so the 13th term in the list is 55. Step 5: There are 12 terms after the 13th term and 12 terms before it. Step 6: Each successive term increases by 2 because the numbers are consecutive odd integers. Step 7: Therefore, the highest term is the 13th term plus 12 steps of size 2. Step 8: Compute the increase: 12 * 2 = 24. Step 9: Add this to the middle term: 55 + 24 = 79. Step 10: Hence, the largest integer among the 25 consecutive odd integers is 79. Verification / Alternative check: An alternative method is to consider the first term. Since the 13th term is 55, and each term differs by 2, the first term is 12 steps below 55, with each step subtracting 2. So the first term is 55 − 24 = 31. This gives the sequence 31, 33, 35, up to 79. You can quickly confirm that this sequence has 25 terms and that the average of the first and last term is (31 + 79) / 2 = 55, which matches the given average, verifying the solution. Why Other Options Are Wrong: Values like 105, 155, or 109 would correspond to sequences with much higher averages than 55, since the middle term would also increase. The option 75 is close but would correspond to a different average. For example, if 75 were the highest term in a similar sequence, the middle term would be lower, and the resulting average would not be 55. Only 79 fits all the conditions of being the highest term in a sequence of 25 consecutive odd integers with average 55. Common Pitfalls: A common mistake is to misinterpret the average as the average of the first and last terms rather than the middle term in an odd length sequence. While the average of all terms is equal to the average of the first and last term, ignoring the connection to the middle term can make the reasoning less direct. Another error is to miscount the number of steps from the middle to the last term, leading to an off by one error. Always count carefully and remember that with 25 terms, there are 12 terms on each side of the middle term. Final Answer: The highest among the 25 consecutive odd integers is 79.

Question 16

N is the largest two digit number that leaves remainders 1, 2 and 4 when divided by 3, 4 and 6 respectively. What is the remainder when N is divided by 5?

Accepted Answer

Introduction / Context: This question tests your understanding of simultaneous congruences and modular arithmetic. You are given conditions on the remainders when a two digit number N is divided by 3, 4, and 6, and you must find the largest two digit number that satisfies all these conditions. After that, you must compute the remainder when that number is divided by 5. This is a classic number system problem that often appears in quantitative aptitude exams. Given Data / Assumptions: N is a two digit integer, so 10 ≤ N ≤ 99. N leaves a remainder of 1 when divided by 3. N leaves a remainder of 2 when divided by 4. N leaves a remainder of 4 when divided by 6. We must find the largest such N and then determine N mod 5. Concept / Approach: The conditions can be written using congruence notation as follows: N ≡ 1 (mod 3), N ≡ 2 (mod 4), and N ≡ 4 (mod 6). We can search systematically through the two digit integers that satisfy these conditions, or we can use the pattern that such numbers repeat modulo the least common multiple of 3, 4, and 6. Once we find all candidate values, we select the largest one under 100 and then compute its remainder when divided by 5. Step-by-Step Solution: Step 1: Express the conditions as congruences: N ≡ 1 (mod 3), N ≡ 2 (mod 4), N ≡ 4 (mod 6). Step 2: The least common multiple of 3, 4, and 6 is 12. Step 3: We can search for numbers between 10 and 99 that satisfy all three congruences. The valid solutions form a repeating pattern with period 12. Step 4: By checking or using a prepared list, we find that the numbers 10, 22, 34, 46, 58, 70, 82, and 94 satisfy all three remainder conditions. Step 5: Among these, the largest two digit number is N = 94. Step 6: Now we must calculate the remainder when 94 is divided by 5. Step 7: Divide 94 by 5. We have 5 * 18 = 90, and 5 * 19 = 95 which is too large, so the quotient is 18 and the remainder is 94 − 90 = 4. Step 8: Therefore, the remainder when N is divided by 5 is 4. Verification / Alternative check: Verify the congruence conditions for N = 94. For division by 3, 94 divided by 3 gives 31 with remainder 1, so N ≡ 1 (mod 3). For division by 4, 94 divided by 4 gives 23 with remainder 2, so N ≡ 2 (mod 4). For division by 6, 94 divided by 6 gives 15 with remainder 4, so N ≡ 4 (mod 6). These checks confirm that 94 is valid. Then 94 divided by 5 gives remainder 4, confirming the final answer. Why Other Options Are Wrong: Remainders 2, 0, 1, or 3 would come from different two digit values that either do not satisfy the three simultaneous congruences or are not the largest valid N. For example, a value like 70 also satisfies the congruences but 70 mod 5 equals 0. However, 70 is not the largest valid two digit number; 94 exceeds it and also meets the conditions. Therefore the only correct remainder for the largest allowed N is 4. Common Pitfalls: Students often mix up the conditions or fail to check all of them together. Checking only one or two congruences can lead to selecting an incorrect N. Another common mistake is to forget the requirement that N must be the largest two digit number satisfying the conditions, which can lead to choosing a smaller valid number. Careful use of modular arithmetic and systematic checking of candidates avoids these issues. Final Answer: The remainder when N is divided by 5 is 4.

Question 17

The difference between a number and its three fifth is 50. What is the value of the number?

Accepted Answer

Introduction / Context: This question tests your ability to translate a verbal statement involving fractions into an algebraic equation. The phrase three fifth of a number appears frequently in quantitative problems, and you must correctly interpret the difference between the number and its fractional part. Solving the resulting linear equation gives the value of the original number. Given Data / Assumptions: Let the unknown be a number x. The expression three fifth of the number means (3 / 5) * x. The difference between the number x and its three fifth is equal to 50. We must determine the exact value of x. Concept / Approach: The key concept is forming an equation from the verbal description. The difference between the number and its three fifth is written as x − (3 / 5) * x. This expression can be simplified using fraction operations and then set equal to 50. The resulting algebraic equation is straightforward to solve. Once we find x, we can check the answer by substituting back into the original relationship. Step-by-Step Solution: Step 1: Let the unknown number be x. Step 2: The three fifth of x is (3 / 5) * x. Step 3: The problem states that the difference between x and its three fifth is 50, so write x − (3 / 5) * x = 50. Step 4: Factor x on the left side: x * (1 − 3 / 5) = 50. Step 5: Simplify the bracket: 1 − 3 / 5 = 2 / 5. Step 6: The equation becomes x * (2 / 5) = 50. Step 7: Solve for x by multiplying both sides by 5 / 2: x = 50 * (5 / 2). Step 8: Compute the product: 50 * (5 / 2) = 25 * 5 = 125. Step 9: Therefore, the number x is 125. Verification / Alternative check: Check the answer by substituting x = 125 back into the original condition. Three fifth of 125 is (3 / 5) * 125 = 3 * 25 = 75. The difference between 125 and 75 is 125 − 75 = 50, which matches the given statement. None of the other options will produce a difference of exactly 50 when you subtract three fifth of the number from the number itself. This confirms that 125 is the correct solution. Why Other Options Are Wrong: If the number were 75, three fifth of 75 would be 45 and the difference would be 30, not 50. For 100, three fifth is 60 and the difference is 40. For 150, three fifth is 90 and the difference is 60. For 200, three fifth is 120 and the difference is 80. None of these match the required difference of 50. Only 125 gives three fifth equal to 75 and a difference of exactly 50. Common Pitfalls: A common pitfall is misinterpreting the phrase three fifth of the number and writing (x / 3) * 5 or some other incorrect expression. Another frequent error is to reverse the subtraction and compute (3 / 5) * x − x instead of x − (3 / 5) * x, which changes the equation completely. Careful reading and writing the algebra in small steps helps avoid these mistakes. Always verify by plugging the final answer back into the original wording of the question. Final Answer: The value of the number is 125.

Question 18

A tin filled with cookies weighs 2 pounds in total. After three fourth of the cookies are eaten, the tin with the remaining cookies weighs 0.8 pounds. What is the weight of the empty tin in pounds?

Accepted Answer

Introduction / Context: This question is a word problem involving total weight, parts of a whole, and simple linear equations. You are asked to distinguish between the weight of the container and the weight of its contents. After a known fraction of the cookies is eaten, the total weight changes, and from these two conditions you can solve for the weight of the empty tin and the original weight of the cookies. Given Data / Assumptions: The total weight of the tin and all the cookies initially is 2 pounds. After three fourth of the cookies are eaten, only one fourth of the cookies remain. The weight of the tin plus the remaining cookies is 0.8 pounds. We must find the weight of the empty tin alone, measured in pounds. Concept / Approach: Introduce variables for the weight of the empty tin and the initial weight of all the cookies. Then translate the statements into two linear equations. One equation expresses the initial total weight, and the other expresses the weight after only one fourth of the cookies remain. Solving this system gives the weight of the tin. The essential idea is that the tin weight stays constant, while the cookie weight decreases proportionally to the fraction remaining. Step-by-Step Solution: Step 1: Let T be the weight of the empty tin in pounds. Step 2: Let C be the initial weight of all the cookies in pounds. Step 3: Initially, the tin plus cookies weigh 2 pounds, so T + C = 2. Step 4: After three fourth of the cookies are eaten, only one fourth of the cookies remain, so the remaining cookie weight is C / 4. Step 5: At that time, the total weight of the tin and remaining cookies is 0.8 pounds, so T + C / 4 = 0.8. Step 6: Now we have a system of two equations: T + C = 2 and T + C / 4 = 0.8. Step 7: Subtract the second equation from the first: (T + C) − (T + C / 4) = 2 − 0.8. Step 8: Simplify the left side: C − C / 4 = (3 / 4) * C, and the right side is 1.2. Step 9: Thus (3 / 4) * C = 1.2, so C = 1.2 * (4 / 3) = 1.6. Step 10: Substitute C = 1.6 into T + C = 2 to get T + 1.6 = 2, so T = 0.4. Step 11: Therefore, the empty tin weighs 0.4 pounds. Verification / Alternative check: Check the values we found. If the cookie weight is 1.6 pounds and the tin weight is 0.4 pounds, the initial total is 0.4 + 1.6 = 2 pounds, matching the first condition. After eating three fourth of the cookies, the remaining cookie weight is 1.6 / 4 = 0.4 pounds. Adding the tin weight again gives 0.4 + 0.4 = 0.8 pounds, matching the second condition. This confirms that the solution is self consistent and correct. Why Other Options Are Wrong: If the tin weighed 0.2 pounds, the initial cookie weight would need to be 1.8 pounds, and one fourth of that is 0.45 pounds, making the later total 0.65 pounds, not 0.8 pounds. Similar inconsistencies appear for 0.3, 0.5, and 0.6 pounds when you set up and solve the equations. Only 0.4 pounds produces values for cookie weight and remaining weight that satisfy both given conditions at the same time. Common Pitfalls: Students sometimes misinterpret three fourth as the remaining fraction instead of the eaten fraction, leading them to use 3C / 4 instead of C / 4 in the second equation. Another common error is to assume that 0.8 is the weight of the cookies alone and ignore the tin, which breaks the logic of the problem. Careful reading and consistent use of variables for the tin and cookie weights help avoid these mistakes. Always check that your solution satisfies both original statements before finalizing the answer. Final Answer: The weight of the empty tin is 0.4 pounds.

Question 19

Two painters can complete two rooms in two hours. Assuming a constant work rate and equal efficiency, how many painters are needed to paint 18 rooms in 6 hours?

Accepted Answer

Introduction / Context: This question is a typical work and time problem. It requires you to understand how work rate scales with the number of workers and the amount of time available. The painters are assumed to work at a constant rate, and you must find how many painters are required to complete a larger job in a given time based on their initial performance on a smaller job. Given Data / Assumptions: Two painters can complete two rooms in two hours. We assume all painters work at the same rate and share the total work equally. The goal is to paint 18 rooms in 6 hours. We must find the number of painters required to achieve this target. Concept / Approach: The central concept is that work done equals rate multiplied by time, and total rate is proportional to the number of painters. From the initial information, we can compute how many painter hours are required per room. Then, for the larger task of 18 rooms, we determine the total painter hours needed. Finally, we divide by the available time to find how many painters are required. This is a scaling problem where both time and crew size can vary. Step-by-Step Solution: Step 1: From the statement, 2 painters complete 2 rooms in 2 hours. Step 2: Compute the total painter hours for this job: 2 painters × 2 hours = 4 painter hours. Step 3: These 4 painter hours are used to paint 2 rooms, so each room requires 4 / 2 = 2 painter hours. Step 4: Now consider the new task: 18 rooms must be painted. Step 5: Total painter hours needed for 18 rooms is 18 × 2 = 36 painter hours. Step 6: The total time available is 6 hours. Step 7: Let P be the number of painters required. Then P × 6 hours must equal 36 painter hours. Step 8: Solve the equation P × 6 = 36, which gives P = 36 / 6 = 6. Step 9: Therefore, 6 painters are needed to complete 18 rooms in 6 hours at the same work rate. Verification / Alternative check: Verify the answer by checking the rate of painting per painter. From the original case, two painters do two rooms in two hours, so combined they paint one room per hour. Hence, each painter paints half a room per hour. With six painters, the rate becomes 6 × 0.5 = 3 rooms per hour. Over 6 hours, they complete 3 × 6 = 18 rooms, which matches the requirement. This confirms that six painters suffice. Why Other Options Are Wrong: If only 2 painters worked for 6 hours, they would contribute 12 painter hours, enough for only 6 rooms. Four painters in 6 hours would give 24 painter hours, enough for 12 rooms. Eight painters in 6 hours would give 48 painter hours, which is more than needed for 18 rooms, so that crew size is unnecessarily large. Ten painters would also overshoot the requirement. Only 6 painters provide exactly 36 painter hours and thus match the needed work precisely. Common Pitfalls: Students often misread the initial relation and assume that two painters can complete one room in two hours, which leads to incorrect painter hour calculations. Some also confuse total hours with painter hours and do not multiply by the number of workers correctly. Always compute the total painter hours for a job, then scale linearly for different numbers of rooms and workers. This systematic method helps avoid confusion and ensures correct answers. Final Answer: The number of painters required is 6 painters.

Question 20

x and y are integers such that x + y is less than 11 and x is greater than 6. Under these conditions, what is the smallest possible value of the expression x − y?

Accepted Answer

Introduction / Context: This question tests reasoning with integer inequalities and expressions. You are given constraints on x and y, both integers, and asked to find the smallest possible value of x minus y. Instead of guessing pairs randomly, you can treat this as an optimization problem subject to simple inequality constraints and reason about how to make x − y as small as possible. Given Data / Assumptions: x and y are integers. x + y < 11. x > 6. We want the smallest possible value of x − y. No other constraints, such as positivity of y, are given. Concept / Approach: To minimize x − y, we want y to be as large as possible relative to x because x − y decreases when y increases. However, the sum constraint x + y < 11 limits how large y can be for a given x. Since x must be greater than 6 and integer valued, we can consider each possible value of x and determine the maximum y allowed by the inequality. Then we compute x − y for each case and choose the smallest result. This is a small, systematic search guided by the inequalities. Step-by-Step Solution: Step 1: Since x is an integer and x > 6, the smallest possible value of x is 7. Step 2: The condition x + y < 11 implies y < 11 − x. Step 3: For x = 7, y must satisfy y < 11 − 7 = 4, so the largest integer y can be is 3. Step 4: Then x − y for x = 7 and y = 3 is 7 − 3 = 4. Step 5: Next, consider x = 8. Then y < 11 − 8 = 3, so the largest integer y can be is 2. Then x − y = 8 − 2 = 6. Step 6: For x = 9, y < 11 − 9 = 2, so y can be at most 1. Then x − y = 9 − 1 = 8. Step 7: As x increases further, the maximum possible y decreases, which makes x − y larger. Step 8: Therefore the smallest possible value of x − y occurs at the smallest allowed x and the largest allowed y, which we found to be x = 7 and y = 3. Step 9: The minimum value of x − y under all given constraints is 4. Verification / Alternative check: You can verify by considering whether negative values of x − y are possible. For x − y to be negative or very small, y would have to be greater than x. But if y were greater than x and x is at least 7, then x + y would be greater than 14, which would violate the condition x + y < 11. Thus y cannot exceed x under the given constraints, and x − y is always positive. Among the tested pairs, 7 and 3 give the smallest positive difference 4. No other pair with x greater than 7 gives a smaller difference. Why Other Options Are Wrong: Values such as 1, 0, or negative 2 would require y to be equal to or greater than x, which makes the sum x + y too large to satisfy the inequality. The value 24 corresponds to some pair with a much larger difference, which is not minimal. Only 4 is consistent with the constraints and arises from a valid integer pair (x, y) = (7, 3). Common Pitfalls: A common mistake is to assume that x and y must both be less than 11, which is not stated; only their sum is constrained. Another error is to forget that x must be greater than 6, leading students to test values like x = 5 or 6. Some students also overlook the fact that maximizing y under the sum constraint helps minimize x − y. A structured search starting from the smallest possible x and largest compatible y avoids these issues. Final Answer: The smallest possible value of x − y is 4.

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