Understanding the Entrance Exam Data

The problem provides a table showing the percentage distribution of students who qualified for an Entrance Exam from seven different schools (A through G) over two years, 2020 and 2021. Crucially, it gives the actual number of students who qualified from School G in both years: 90 in 2020 and 135 in 2021.

We are asked to find the difference, Q - P, where P is the average number of students qualifying from Schools B, C, and D in 2020, and Q is the average number of students qualifying from Schools E, F, and G in 2021.

Analyzing the Provided Data Table

Here is the distribution data:

We also know:

• Number of students qualifying from School G in 2020 = 90
• Number of students qualifying from School G in 2021 = 135

Calculating Total Qualified Students in Each Year

The percentage distribution for School G is given for both years, along with the actual number of students. This allows us to find the total number of students who qualified from all schools in each year.

For 2020:

School G represents 12% of the total qualified students. Let \(T_{2020}\) be the total number of qualified students in 2020.

\(12\% \text{ of } T_{2020} = 90\)

\(0.12 \times T_{2020} = 90\)

\(T_{2020} = \frac{90}{0.12} = \frac{90}{\frac{12}{100}} = 90 \times \frac{100}{12} = \frac{9000}{12} = 750\)

So, the total number of students qualifying in 2020 was 750.

For 2021:

School G represents 15% of the total qualified students. Let \(T_{2021}\) be the total number of qualified students in 2021.

\(15\% \text{ of } T_{2021} = 135\)

\(0.15 \times T_{2021} = 135\)

\(T_{2021} = \frac{135}{0.15} = \frac{135}{\frac{15}{100}} = 135 \times \frac{100}{15} = \frac{13500}{15} = 900\)

So, the total number of students qualifying in 2021 was 900.

Calculating P: Average for Schools B, C, D in 2020

P is the average number of students qualifying from Schools B, C, and D in 2020. We need to find the actual number of students from each of these schools using the total for 2020 (\(T_{2020} = 750\)) and their respective percentages.

• Students from School B in 2020: 14% of 750 = \(0.14 \times 750 = 105\)
• Students from School C in 2020: 16% of 750 = \(0.16 \times 750 = 120\)
• Students from School D in 2020: 10% of 750 = \(0.10 \times 750 = 75\)

Now, calculate the average P:

\(P = \frac{\text{Students from B} + \text{Students from C} + \text{Students from D}}{3}\)

\(P = \frac{105 + 120 + 75}{3} = \frac{300}{3} = 100\)

The average number of students qualifying from Schools B, C, and D in 2020 (P) is 100.

Calculating Q: Average for Schools E, F, G in 2021

Q is the average number of students qualifying from Schools E, F, and G in 2021. We need to find the actual number of students from each of these schools using the total for 2021 (\(T_{2021} = 900\)) and their respective percentages.

• Students from School E in 2021: 16% of 900 = \(0.16 \times 900 = 144\)
• Students from School F in 2021: 13% of 900 = \(0.13 \times 900 = 117\)
• Students from School G in 2021: 15% of 900 = \(0.15 \times 900 = 135\) (This matches the given number for School G in 2021, confirming our total calculation)

Now, calculate the average Q:

\(Q = \frac{\text{Students from E} + \text{Students from F} + \text{Students from G}}{3}\)

\(Q = \frac{144 + 117 + 135}{3} = \frac{396}{3} = 132\)

The average number of students qualifying from Schools E, F, and G in 2021 (Q) is 132.

Finding the Difference Q - P

The final step is to calculate the difference between Q and P.

\(Q - P = 132 - 100 = 32\)

The difference Q - P is 32.

Summary of Calculation Steps

1. Used School G's actual numbers and percentages to find the total qualified students in 2020 and 2021.
2. Calculated the actual number of students qualifying from Schools B, C, and D in 2020.
3. Calculated the average P from the numbers in step 2.
4. Calculated the actual number of students qualifying from Schools E, F, and G in 2021.
5. Calculated the average Q from the numbers in step 4.
6. Subtracted P from Q to find the difference.

Revision Table: Entrance Exam Data Analysis

Additional Information: Percentage and Average Concepts

This problem combines two fundamental quantitative concepts: percentages and averages.

Percentage: A percentage represents a part of a whole, expressed as a fraction of 100. If 'x' is a percentage of a total 'T', the corresponding value is \((x/100) \times T\). Conversely, if a value 'V' corresponds to 'x%' of a total, the total is \(T = V / (x/100) = V \times (100/x)\). In this problem, we used the latter formula to find the total number of qualified students in each year using the known value and percentage for School G.

Average: The average (or mean) of a set of numbers is the sum of the numbers divided by the count of the numbers. For a set of values \(v_1, v_2, ..., v_n\), the average is given by \(\frac{v_1 + v_2 + ... + v_n}{n}\). We used this formula to calculate the average number of qualified students for specified schools in specified years.

Understanding how to work with percentages to find totals and how to calculate averages is crucial for solving data interpretation problems like this one.