Question 1

The average of the first 100 positive integers is

Accepted Answer

Concept & Formula: The set of positive integers starting from 1 forms an Arithmetic Progression (AP) with a common difference of 1. The average of the first n natural numbers (positive integers) can be found using a direct formula. Average = (n + 1/2) Step-by-Step Solution: • Given: We need the average of the first 100 positive integers. • Deduction: The sequence is 1, 2, 3, ..., 100. • Here, the total number of terms n is 100. • Calculation: Plug n = 100 into the standard AP average formula. • Average = (100 + 1/2) • Average = (101/2) • Average = 50.5 Exam Strategy & Shortcut: For any sequence with a constant difference (like consecutive integers), the average is simply the midpoint between the first and last term. First term is 1, last term is 100. Average is (1 + 100) / 2 = 50.5. Memorize this property for instant answers. Common Pitfall: Confusing "positive integers" with "whole numbers." Whole numbers start at 0, which would make the first 100 numbers range from 0 to 99, changing the average to 49.5. Positive integers strictly begin at 1. Final Answer: Therefore, the correct answer is 50.5.

Question 2

The average of odd numbers up to 100 is

Accepted Answer

Concept & Formula: The sum of the first n odd natural numbers is exactly n^2. Therefore, the average of the first n odd numbers is (n^2/n), which simplifies perfectly to just n. Average\ of\ first\ n\ odd\ numbers = n Step-by-Step Solution: • Given: We need the average of all odd numbers up to 100. • Deduction: The series is 1, 3, 5, ..., 99. • First, determine how many odd numbers exist up to 100. Exactly half the numbers from 1 to 100 are odd, so n = 50. • Calculation: Since we are finding the average of the first 50 odd numbers, the average is exactly equal to n. • Average = 50. Exam Strategy & Shortcut: You can also use the arithmetic progression (AP) midpoint shortcut. The first odd number is 1, and the last odd number before 100 is 99. The average is the sum of the first and last terms divided by 2. (1 + 99/2) = (100/2) = 50. Common Pitfall: A frequent error is assuming n = 100 because the question says "up to 100," leading to an incorrect answer of 100. You must count *how many* odd numbers are actually in that range. Final Answer: Therefore, the correct answer is 50.

Question 3

In a family, the average age of a father and a mother is 35 years. The average age of the father, mother and their only son is 27 years. What is the age of the son?

Accepted Answer

Concept & Logic: When dealing with average ages of a group, calculate the total sum of the ages. By comparing the total age of the group with a member to the total age of the group without that member, you can easily isolate the individual's age. Total\ Age = Average\ Age × Number\ of\ People Step-by-Step Solution: • Given: The average age of the father and mother is 35 years. • The average age of the father, mother, and son is 27 years. • Calculation: Calculate the total age of the parents (2 people). • Total\ Age\ (Father + Mother) = 35 × 2 = 70\ years. • Calculate the total age of the family including the son (3 people). • Total\ Age\ (Father + Mother + Son) = 27 × 3 = 81\ years. • Subtract the total age of the parents from the total age of the whole family to find the son's age. • Son's\ Age = 81 - 70 = 11\ years. Exam Strategy & Shortcut: Use the "Deviation Method." The son joins a group of 2 people whose average is 35, and drags the average down to 27. The drop in average is 35 - 27 = 8 years. This 8-year drop applies to the 2 original members, meaning a total deficit of 8 × 2 = 16 years. The son must be 16 years younger than the new average to cause this drop. So, 27 - 16 = 11 years. Common Pitfall: Miscalculating the total number of people in the first group. Some students read "father and a mother" and forget it represents exactly 2 distinct individuals, incorrectly multiplying 35 by 3 or failing to multiply at all. Final Answer: Therefore, the correct answer is 11 years.

Question 4

If the mean of 5 observations $x$, $x + 2$, $x + 4$, $x + 6$ and $x + 8$ is 11, then the mean of the last three observations is

Accepted Answer

Concept & Strategy: When a sequence of numbers increases by a constant amount, it forms an Arithmetic Progression (AP). The most powerful property of an AP with an odd number of terms is that its mean (average) is exactly equal to its middle term. Mean\ of\ AP = Middle\ Observation Step-by-Step Solution: • Given: The 5 observations are x, x + 2, x + 4, x + 6, x + 8. • The mean of these 5 observations is 11. • Calculation: Verify the sequence is an AP. The difference between consecutive terms is consistently 2. • The middle term of these 5 observations is the 3rd term: x + 4. • Because it is an AP, the mean equals the middle term. • x + 4 = 11 • We need the mean of the last three observations: x + 4, x + 6, x + 8. • This subset is also an AP. Its middle term is x + 6. • Since we know x + 4 = 11, we can find x + 6 by simply adding 2. • x + 6 = 11 + 2 = 13. Exam Strategy & Shortcut: You do not need to find the value of x. The mean of the 5 terms is x+4. The mean of the last 3 terms is x+6. The difference between these means is exactly 2. So, 11 + 2 = 13. This bypasses algebraic solving entirely. Common Pitfall: The slow approach is calculating the full sum: (5x + 20) / 5 = 11, solving for x = 7, then substituting x into the last three terms (11, 13, 15), summing them (39), and dividing by 3 to get 13. While correct, this wastes massive amounts of time on a competitive exam. Final Answer: Therefore, the correct answer is 13.

Question 5

The average sale of a car dealership was 15 cars per week.  After a promotional scheme the average sale increased to 21 cars per week. The percentage increase in the sale of cars was

Accepted Answer

Concept & Formula: Percentage increase measures the growth of a quantity relative to its initial value. Percentage Increase = ( \frac{Final Value - Initial Value}{Initial Value} ) × 100 Step-by-Step Solution: • Given: • Initial average sales = 15 cars/week • Final average sales = 21 cars/week • Calculation: • Find the absolute increase: 21 - 15 = 6 cars. • Apply the percentage increase formula: • Percentage Increase = ((6/15)) × 100 • Simplify the fraction: (6/15) = (2/5) • Percentage = (2/5) × 100 = 40% Exam Strategy & Shortcut: Familiarize yourself with standard fraction-to-percentage conversions. Knowing that (1/5) = 20% allows you to instantly recognize that (2/5) = 40% without writing down the multiplication step. Common Pitfall: Calculating the percentage increase over the *final* value instead of the *initial* value. Always put the difference over the original starting number (i.e., (6/15), not (6/21)). Final Answer: Therefore, the correct answer is 40%.

Question 6

The mean of $1^2, 2^2, 3^2, 4^2, 5^2, 6^2, 7^2$ is

Accepted Answer

Concept & Formula: To find the mean of the squares of the first n natural numbers, we use the standard summation formula divided by n. Mean = ((n+1)(2n+1)/6) Step-by-Step Solution: • Given: • The series is the squares of the first 7 natural numbers. • Therefore, n = 7. • Calculation: • Substitute n = 7 into the mean formula. • Mean = ((7+1)(2(7)+1)/6) • Mean = (8 × 15/6) • Simplify the expression: • Mean = (120/6) = 20 Exam Strategy & Shortcut: Memorizing the direct formula for the *mean* of squares ((n+1)(2n+1)/6) is much faster than finding the sum of squares using (n(n+1)(2n+1)/6) and then dividing by n. Common Pitfall: Attempting to calculate the square of each number individually (1 + 4 + 9 + 16 + 25 + 36 + 49) and adding them manually. While it works for n=7, it becomes incredibly time-consuming and error-prone for larger values of n. Use the formula. Final Answer: Therefore, the correct answer is 20.

Question 7

The average of a non-zero number and its square is 5 times the number.  The number is

Accepted Answer

Concept & Logic: This problem requires translating a word problem directly into an algebraic equation. The core concept is equating the standard average formula to the condition given in the text. Average = \frac{Sum of Terms}{Number of Terms} Step-by-Step Solution: • Given: • Let the non-zero number be x. • The two terms being averaged are x and its square, x^2. • Calculation: • Set up the equation based on the given condition: • (x + x^2/2) = 5x • Multiply both sides by 2 to clear the denominator: • x + x^2 = 10x • Rearrange into a standard quadratic equation: • x^2 + x - 10x = 0 • x^2 - 9x = 0 • Factor out x: • x(x - 9) = 0 • This gives two possible solutions: x = 0 or x = 9. • Since the problem specifies a "non-zero number", x must be 9. Exam Strategy & Shortcut: You can solve this rapidly using Option Elimination. Test Option (a): Number is 9. Square is 81. Average = (9 + 81/2) = (90/2) = 45. Is 45 equal to 5 × 9? Yes. The option is verified instantly without solving the quadratic equation. Common Pitfall: Dividing both sides of x + x^2 = 10x by x without explicitly acknowledging that x 
eq 0. While mathematically safe here due to the question's text, dividing by variables can lead to lost roots in standard algebra. Always factor instead. Final Answer: Therefore, the correct answer is 9.

Question 8

Of four numbers whose average is 60, the first is one-fourth of the sum of the last three.  The first number is

Accepted Answer

Concept & Logic: When a question describes relationships among fractions of a sum, express the entire group in terms of a single variable. The total sum is easily calculated from the average and the number of items. Sum of Terms = Average × Number of Terms Step-by-Step Solution: • Given: • Total numbers = 4 • Average of the four numbers = 60 • Let the four numbers be a, b, c, and d. • Sum of all four numbers: a + b + c + d = 60 × 4 = 240 • Deduction: • Condition: The first number is one-fourth of the sum of the last three. • a = (1/4)(b + c + d) • Rearranging this gives: b + c + d = 4a • Calculation: • We know the total sum is a + (b + c + d) = 240 • Substitute (b + c + d) with 4a: • a + 4a = 240 • 5a = 240 • a = (240/5) = 48 Exam Strategy & Shortcut: Use the "Ratio Method" for extreme speed. The ratio of the first number to the sum of the other three is 1:4. This means the total sum is split into 1 + 4 = 5 equal "parts". Total sum = 60 × 4 = 240. Value of 1 part (the first number) = (240/5) = 48. Common Pitfall: Many students mistakenly calculate (1/4) of the total average or (1/4) of the total sum directly, yielding 15 or 60, rather than recognizing that the 1:4 relationship applies to the parts making up the total (5 parts). Final Answer: Therefore, the correct answer is 48.

Question 9

The average of 7 consecutive numbers is 20.  The largest of these numbers is

Accepted Answer

Concept & Logic: For any sequence of consecutive numbers (an Arithmetic Progression), the average is always exactly equal to the median (the middle term). Step-by-Step Solution: • Given: • There are 7 consecutive numbers. • The average of these numbers is 20. • Deduction: • Since there are 7 terms (an odd number of terms), the middle term is the 4th term. • Therefore, the 4th term is exactly the average, which is 20. • Calculation: • The sequence is centered around 20. • The terms are: (1st), (2nd), (3rd), 20, (5th), (6th), (7th) • Since they are consecutive integers, they increase by 1. • 5th term = 21 • 6th term = 22 • 7th term (largest) = 23 Exam Strategy & Shortcut: If the average of N consecutive integers is A (where N is odd), the largest number is simply A + (N - 1/2). Largest = 20 + (7 - 1/2) = 20 + 3 = 23. Common Pitfall: Setting up a long algebraic equation: x + (x+1) + (x+2) + (x+3) + (x+4) + (x+5) + (x+6) = 7 × 20. While this works (7x + 21 = 140 \implies 7x = 119 \implies x = 17, so largest is 17+6=23), it is a massive waste of time for a problem that can be solved visually in 3 seconds. Final Answer: Therefore, the correct answer is 23.

Question 10

The average of five consecutive odd numbers is 95.  What is the fourth number in the descending order?

Accepted Answer

Concept & Logic: The average of any set of consecutive odd or even numbers (which form an Arithmetic Progression) is always its median. If the number of terms is odd, the average is the exact middle term. Step-by-Step Solution: • Given: • There are 5 consecutive odd numbers. • Their average is 95. • Deduction: • Since 5 is an odd number of terms, the 3rd term (the middle one) must be the average, 95. • Consecutive odd numbers differ by 2. • Calculation: • The numbers in ascending order are: 91, 93, 95, 97, 99. • The question asks for the sequence in descending order. Let's reverse it: • 99, 97, 95, 93, 91. • The fourth number in this descending sequence is 93. • Checking the options: (a) 91, (b) 95, (c) 97, (d) 99. • 93 is not listed. Exam Strategy & Shortcut: Identify the middle term (95). Write the set out quickly: \{91, 93, 95, 97, 99\}. Descending means starting from the largest. 99 (1st), 97 (2nd), 95 (3rd), 93 (4th). Look at options, pick "None of these". Common Pitfall: Ignoring the word "descending". If you simply look for the 4th number in the standard (ascending) order, you will find 97, which is Option (c). Examiners specifically include this to catch students who read the question too quickly. Final Answer: Therefore, the correct answer is None of these.

Question 11

$A, B, C$ and $D$ are four consecutive even numbers respectively and their average is 65.  What is the product of $A$ and $D$?

Accepted Answer

Concept & Logic: When a sequence has an *even* number of terms, the median (and thus the average) falls exactly halfway between the two middle terms. Step-by-Step Solution: • Given: • A, B, C, D are four consecutive even numbers. • Their average is 65. • Deduction: • The middle two terms are B and C. • The average of the sequence (65) must be exactly halfway between B and C. • Since they are consecutive even numbers, B and C must be the even numbers immediately before and after 65. • Therefore, B = 64 and C = 66. • Calculation: • Now find A and D. • A is the even number before B: A = 62. • D is the even number after C: D = 68. • The question asks for the product of A and D. • Product = 62 × 68 • Product = 62 × (70 - 2) = 4340 - 124 = 4216. Exam Strategy & Shortcut: Use the difference of squares shortcut. You know the average is 65. A and D are symmetrically distributed around the average. A is 3 steps below (65 - 3) and D is 3 steps above (65 + 3). Product = (65 - 3)(65 + 3) = 65^2 - 3^2. 65^2 = 4225. 4225 - 9 = 4216. This is incredibly fast if you know your squares ending in 5. Common Pitfall: Assuming that because the numbers are even, the average must also be an even number. If you get confused by an odd average (65) for even numbers, remember that the sum of 4 even numbers is even, but dividing an even sum by 4 can absolutely yield an odd number (e.g., 260 / 4 = 65). Final Answer: Therefore, the correct answer is 4216.

Question 12

The sum of the three consecutive even numbers is 44 more than the average of these numbers. Which of the following is the third largest of these numbers?

Accepted Answer

Concept & Formula: To solve problems involving consecutive even numbers, we can represent them algebraically as x, x+2, and x+4. The problem requires translating the relationship between their sum and their average into a linear equation. Step-by-Step Solution: • Given: Let the three consecutive even numbers be x, x+2, and x+4. • Sum: Their sum is (x) + (x+2) + (x+4) = 3x + 6. • Average: Their average is the middle number, which is x+2. (Alternatively, (3x+6/3) = x+2). • Equation: According to the question, the sum is 44 more than the average: 3x + 6 = (x + 2) + 44 • Calculation: 3x + 6 = x + 46 2x = 40 x = 20 • Result: The numbers are 20, 22, and 24. • The "third largest" of three numbers is simply the smallest number, which is 20. Since 20 is not among the given options (16, 18, 24), the answer is "None of these". Exam Strategy & Shortcut: Instead of writing out the full equation, you can use the property of averages for consecutive numbers in an arithmetic progression: the average is always the middle number. Let the middle number be M. The numbers are M-2, M, and M+2. Sum = 3M. Average = M. The question states: 3M = M + 44 \Rightarrow 2M = 44 \Rightarrow M = 22. The smallest number (third largest) is M-2 = 20. This mental math takes less than 10 seconds. Common Pitfall: The most common mistake is misinterpreting "third largest". In a set of three numbers, the 1st largest is the maximum, the 2nd largest is the middle, and the 3rd largest is the minimum. Students often get confused and look for a non-existent number, or forget to check the "None of these" option when their calculated answer (20) doesn't match the numerical options. Final Answer: Therefore, the correct answer is None of these.

Question 13

The average marks of a student in 4 subjects is 75. If the student obtained 80 marks in the fifth subject, then the new average is

Accepted Answer

Concept & Formula: The average is calculated as the sum of all observations divided by the number of observations. Average = (Sum \ of \ Observations/Number \ of \ Observations) When a new observation is added, we find the new sum and divide by the new total count. Step-by-Step Solution: • Initial Total: The average of 4 subjects is 75. Total \ marks \ (4 \ subjects) = 4 × 75 = 300 • New Total: The student gets 80 marks in the 5th subject. Total \ marks \ (5 \ subjects) = 300 + 80 = 380 • New Average: Divide the new total by 5 subjects. New \ Average = (380/5) = 76 Exam Strategy & Shortcut: Use the Deviation Method to solve this mentally. The old average is 75. The new subject score is 80, which is +5 above the average. Distribute this surplus of 5 marks equally among all 5 subjects: (+5/5) = +1 Add this distributed surplus to the old average: New \ Average = 75 + 1 = 76 Common Pitfall: A frequent error is dividing the new total sum (380) by the old number of subjects (4), which yields 95, or simply averaging the old average and the new score: (75+80/2) = 77.5. Always remember to update the denominator to reflect the new total count. Final Answer: Therefore, the correct answer is 76.

Question 14

The average weight of 16 boys in a class is 50.25 kgs and that of the remaining 8 boys is 45.15 kgs. Find the average weight of all the boys in the class.

Accepted Answer

Concept & Formula: This problem is solved using the concept of Weighted Average. When a group is split into two or more subgroups with different averages, the combined average is calculated by finding the total sum of all values and dividing by the total number of individuals. Weighted \ Average = (N_1A_1 + N_2A_2/N_1 + N_2) Step-by-Step Solution: • Given: • Group 1: N_1 = 16 boys, Average weight A_1 = 50.25 kgs. • Group 2: N_2 = 8 boys, Average weight A_2 = 45.15 kgs. • Total Weight Calculation: • Total weight of the first 16 boys = 16 × 50.25 = 804 kgs. • Total weight of the remaining 8 boys = 8 × 45.15 = 361.2 kgs. • Combined Average Calculation: • Total weight of all boys = 804 + 361.2 = 1165.2 kgs. • Total number of boys = 16 + 8 = 24. • Overall average weight = (1165.2/24) = 48.55 kgs. Exam Strategy & Shortcut: Use the Ratio (Deviation) Method to minimize calculation time and avoid multiplying large decimals. 1. Find the ratio of the number of boys: 16 : 8 \rightarrow 2 : 1. 2. Assume the smaller average (45.15) as the base. 3. The larger average (50.25) is +5.1 more than the base. 4. Calculate the total excess based on the ratio weights: 2 parts × 5.1 = 10.2. 5. Distribute this excess over the total ratio parts (2 + 1 = 3 parts): (10.2/3) = 3.4. 6. Add this to the base average: 45.15 + 3.4 = 48.55 kgs. Common Pitfall: Students often waste time executing the full decimal multiplications (16 × 50.25 and 8 × 45.15) manually under time pressure, which heavily increases the risk of simple arithmetic errors. Always look for a ratio to simplify the weights first. Final Answer: Therefore, the correct answer is 48.55 kgs.

Question 15

The average expenditure of a man for the first five months of a year is ₹ 5000 and for the next seven months it is ₹ 5400. He saves ₹ 2300 during the year. His average monthly income is

Accepted Answer

Concept & Logic: Income is fundamentally the sum of Expenditure and Savings. To find the average monthly income, we must first determine the Total Annual Income (Total Annual Expenditure + Total Annual Savings) and then divide it by 12 months. Total \ Income = Total \ Expenditure + Total \ Savings Step-by-Step Solution: • Given: • Average expenditure for first 5 months = ₹ 5000. • Average expenditure for next 7 months = ₹ 5400. • Total savings for the year = ₹ 2300. • Calculate Total Annual Expenditure: • Expenditure for first 5 months = 5 × 5000 = 25000. • Expenditure for next 7 months = 7 × 5400 = 37800. • Total Annual Expenditure = 25000 + 37800 = 62800. • Calculate Total Annual Income: • Total Annual Income = Total Expenditure + Total Savings • Total Annual Income = 62800 + 2300 = 65100. • Calculate Average Monthly Income: • Average Monthly Income = (65100/12) • Average Monthly Income = 5425. Exam Strategy & Shortcut: Instead of finding massive annual totals, use the Base Average Method. 1. Assume the base average monthly income is ₹ 5000. 2. For the first 5 months, the expenditure is exactly the base, so deviation is 0. 3. For the next 7 months, expenditure is ₹ 5400, which is an excess of ₹ 400 per month. Total excess = 7 × 400 = 2800. 4. Total savings for the year is ₹ 2300. This is also part of the income. 5. Total "extra" money above the ₹ 5000 base for the whole year = 2800 (from expenses) + 2300 (from savings) = 5100. 6. Distribute this ₹ 5100 evenly across 12 months: (5100/12) = 425. 7. True average monthly income = Base (5000) + Distributed Extra (425) = ₹ 5425. Common Pitfall: The most common mistake is forgetting to add the annual savings before dividing by 12. Students often calculate the average monthly expenditure ((62800/12) \approx 5233.33) and get confused when it doesn't match the options, forgetting that Income = Expenditure + Savings. Final Answer: Therefore, the correct answer is ₹ 5425.

Question 16

A school has 4 sections of Chemistry in Class X having 40, 35, 45 and 42 students. The mean marks obtained in Chemistry test are 50, 60, 55 and 45 respectively for the 4 sections. Determine the overall average of marks per student.

Accepted Answer

Concept & Formula: This problem is a classic Weighted Average calculation involving four distinct groups. You need to find the grand total of all marks and divide it by the total number of students. Weighted \ Average = (\sum (Number \ of \ Students × Average \ Marks)/\sum (Number \ of \ Students)) Step-by-Step Solution: • Given Data: • Section 1: 40 students, Mean = 50 • Section 2: 35 students, Mean = 60 • Section 3: 45 students, Mean = 55 • Section 4: 42 students, Mean = 45 • Calculate Total Marks per Section: • Section 1 Total = 40 × 50 = 2000 • Section 2 Total = 35 × 60 = 2100 • Section 3 Total = 45 × 55 = 2475 • Section 4 Total = 42 × 45 = 1890 • Calculate Overall Totals: • Grand Total Marks = 2000 + 2100 + 2475 + 1890 = 8465 • Total Students = 40 + 35 + 45 + 42 = 162 • Calculate Overall Average: • Overall Average = (8465/162) \approx 52.253... • Rounded to two decimal places, this is 52.25. Exam Strategy & Shortcut: Use the Deviation (Assumed Mean) Method to keep the numbers manageable. 1. Assume a base mean of 50. 2. Calculate deviations from 50 for each section's mean: • Sec 1: 50 \rightarrow deviation 0 \implies 40 × 0 = 0 • Sec 2: 60 \rightarrow deviation +10 \implies 35 × 10 = +350 • Sec 3: 55 \rightarrow deviation +5 \implies 45 × 5 = +225 • Sec 4: 45 \rightarrow deviation -5 \implies 42 × (-5) = -210 3. Sum of total deviations = 0 + 350 + 225 - 210 = +365. 4. Average deviation = (+365/162) \approx +2.253. 5. Final Average = Assumed Mean (50) + Average Deviation (2.25) = 52.25. Common Pitfall: Attempting the massive additions and divisions (8465 \div 162) manually is prone to severe calculation errors. Furthermore, a lazy student might just average the four means: (50 + 60 + 55 + 45/4) = 52.5, which is conceptually wrong and leads to choosing an incorrect option if it were provided. Final Answer: Therefore, the correct answer is 52.25.

Question 17

The average of five consecutive numbers is $x$. If the next two numbers are also included, how shall the average vary?

Accepted Answer

Concept & Formula The average of any consecutive sequence of numbers is exactly its middle value. Adding consecutive numbers to the higher end of the series shifts the middle value by exactly 0.5 per number added. Step-by-Step Solution • Given: The average of 5 consecutive numbers is x. • Let the 5 consecutive numbers be a, a+1, a+2, a+3, a+4. • The average (middle term) of these 5 numbers is a+2. Thus, x = a+2. • Now, include the next two numbers in the sequence: a+5 and a+6. • The new series now has 7 consecutive numbers: a, a+1, a+2, a+3, a+4, a+5, a+6. • The new average (middle term) of these 7 numbers is a+3. • Compare the old and new averages: New Average - Old Average = (a+3) - (a+2) = 1. • The overall average has increased by 1. Exam Strategy & Shortcut For consecutive integers, every new number added to the higher end of the sequence increases the overall average by exactly 0.5. Since we are adding 2 new numbers, the average increases by 2 × 0.5 = 1. This rule lets you solve the problem in seconds without writing any algebraic variables. Common Pitfall Students often waste time assuming specific arbitrary numbers (like 1, 2, 3, 4, 5) and calculating sums manually. While this method works and proves the concept, it drains precious time during an exam compared to using the standard 0.5 increment rule. Final Answer Therefore, the correct answer is It shall increase by 1.

Question 18

A person purchases 1 kg of tomatoes from each of the 4 places at the rate of 1 kg, 2 kg, 3 kg, 4 kg per rupee respectively. On an average, he has purchased $x$ kg of tomatoes per rupee. Then the value of $x$ is

Accepted Answer

Concept & Logic Average quantity per unit of money is defined as the total quantity purchased divided by the total money spent. When identical physical quantities are purchased at varying rates, we calculate the individual monetary costs to find the total sum spent. Average = \frac{Total Quantity}{Total Money Spent} Step-by-Step Solution • Total Quantity: The person purchases exactly 1 kg from 4 different places. Total weight = 1 + 1 + 1 + 1 = 4 kg. • Money Spent at Place 1: Rate is 1 kg/rupee. Cost for 1 kg = 1 / 1 = 1 rupee. • Money Spent at Place 2: Rate is 2 kg/rupee. Cost for 1 kg = 1 / 2 = 0.5 rupees. • Money Spent at Place 3: Rate is 3 kg/rupee. Cost for 1 kg = 1 / 3 rupees. • Money Spent at Place 4: Rate is 4 kg/rupee. Cost for 1 kg = 1 / 4 = 0.25 rupees. • Find the total money spent: Total = 1 + 1/2 + 1/3 + 1/4. • Taking the LCM of the denominators (1, 2, 3, 4) which is 12: Total Spent = (12 + 6 + 4 + 3/12) = (25/12) rupees. • Calculate Average Quantity per Rupee = Total Quantity / Total Money Spent = 4 \div (25/12) = (48/25). • Convert the fraction to a decimal: (48/25) = 1.92 kg per rupee. Exam Strategy & Shortcut Recognize that since the quantity purchased is identical (1 kg) across all shops, the average rate (in kg/rupee) is essentially the Harmonic Mean of the individual rates (1, 2, 3, 4). Harmonic Mean for n observations = (n/1/a + 1/b + 1/c + 1/d) = (4/1/1 + 1/2 + 1/3 + 1/4) = (4/25/12) = 1.92. Common Pitfall A highly common mistake is calculating the simple arithmetic mean of the rates: (1 + 2 + 3 + 4) / 4 = 2.5 kg/rupee (Option C). This completely fails because the amount of money spent at each rate is not equal; rather, the physical quantity purchased is equal. Final Answer Therefore, the correct answer is 1.92.

Question 19

In a certain factory there are five workers A, B, C, D and E. A can complete a work in 4 minutes, B in 5 minutes, C in 6 minutes, D in 10 minutes and E in 12 minutes. The average number of units of work completed per worker per minute will be

Accepted Answer

Concept & Strategy Work rate is the reciprocal of the time taken to complete a job. To find the average work done per worker per minute, first determine the individual work done by each worker in one minute, sum them up to find the total team output per minute, and then divide by the total number of workers. Rate = (1/Time taken) Step-by-Step Solution • Find the individual 1-minute work (units of work completed per minute) for each worker: • A's work in 1 min = 1/4 • B's work in 1 min = 1/5 • C's work in 1 min = 1/6 • D's work in 1 min = 1/10 • E's work in 1 min = 1/12 • Calculate total work done by all 5 workers collectively in 1 minute: • Total = 1/4 + 1/5 + 1/6 + 1/10 + 1/12 • Find the LCM of 4, 5, 6, 10, and 12, which is 60. • Total = (15 + 12 + 10 + 6 + 5/60) = (48/60) = (4/5) = 0.80 units of work per minute. • The question specifically asks for the average work per worker per minute. • Average = \frac{Total Work per Minute}{Number of Workers} • Average = (0.80/5) = 0.16. Exam Strategy & Shortcut Instead of dealing with fraction additions, assume the total work is the LCM of all the given times (60 units). Calculate efficiencies (units/min): A=15, B=12, C=10, D=6, E=5. Total units completed by all workers in 1 min = 48 units. Average units completed per worker per min = 48 / 5 = 9.6 units. Since we scaled the total work to 60 units, the fractional part of 1 standard unit of work is 9.6 / 60 = 0.16. This LCM method safely prevents fractional addition mistakes. Common Pitfall Students often stop calculating after finding the total work per minute (0.80) and immediately select option (c). You must read carefully: the question specifically asks for the average *per worker*, requiring that crucial final division by 5. Final Answer Therefore, the correct answer is 0.16.

Question 20

There were 35 students in a hostel. If the number of the students is increased by 7, then the expenses of the mess increase by ₹ 42 per day, while the average expenditure per head diminishes by ₹ 1. The original expenditure of the mess per day was

Accepted Answer

Concept & Logic When a group expands, total expenditure changes by a fixed addition, while the new average becomes the old average minus the reduction amount. Use linear equations tracking the original average expenditure per head. Step-by-Step Solution • Let the original average expenditure per student per day be ₹ x. • Original daily total expenditure for 35 students = 35x. • New number of students = 35 + 7 = 42. • New average expenditure per head = ₹ (x - 1). • New daily total expenditure for 42 students = 42(x - 1). • Given that the total daily expenditure increased by ₹ 42: 42(x - 1) - 35x = 42 • Solve the linear equation: 42x - 42 - 35x = 42 7x = 84 x = 12 • Find the original daily expenditure of the mess: Original Expenditure = 35x = 35 × 12 = ₹ 420 Exam Strategy & Shortcut Option Elimination Method: The original daily expenditure must be a multiple of 35 because Total = 35 × x. • 400 is not divisible by 35. • 420 divided by 35 gives exactly 12 (an integer value for initial head cost). Let's check: New total = 420 + 42 = 462. New per head cost = 462 / 42 = 11, which is exactly 1 less than 12. Option (b) matches perfectly! Common Pitfall Be cautious when structuring the difference equation. Ensure you subtract the original cost from the new cost (42(x-1) - 35x = 42) rather than accidentally swapping them, which results in negative numbers. Final Answer Therefore, the correct answer is ₹ 420.

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