Understanding the Data and the Problem

The question provides a table showing the percentage distribution of students who qualified an Entrance Exam from seven different schools (A through G) over two years, 2020 and 2021. We are also given the actual number of students who qualified specifically from School G in both years.

The goal is to calculate the percent rise in the number of students qualifying from School D from 2020 to 2021. To do this, we first need to find the actual number of students who qualified from School D in each year.

The key to solving this problem is to use the information about School G to find the total number of students who qualified the exam in each year. Once we have the total numbers, we can use the percentages for School D to find the actual number of qualified students from School D in 2020 and 2021.

Calculating Total Qualified Students in 2020 and 2021

The table shows that in 2020, 12% of the total qualified students were from School G. We are told that 90 students from School G qualified in 2020. We can use this to find the total number of qualified students in 2020.

Let \( T_{2020} \) be the total number of students who qualified in 2020.

According to the data:

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

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

To find \( T_{2020} \), we divide 90 by 0.12:

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

So, the total number of students who qualified the Entrance Exam in 2020 was 750.

Similarly, in 2021, 15% of the total qualified students were from School G. We are told that 135 students from School G qualified in 2021.

Let \( T_{2021} \) be the total number of students who qualified in 2021.

According to the data:

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

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

To find \( T_{2021} \), we divide 135 by 0.15:

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

So, the total number of students who qualified the Entrance Exam in 2021 was 900.

Determining Qualified Students from School D

Now that we know the total number of qualified students in each year, we can find the actual number of qualified students from School D using its percentage distribution from the table.

From the table, in 2020, 10% of the total qualified students were from School D.

Number of students from School D in 2020 = \( 10\% \text{ of } T_{2020} \)

Number of students from School D in 2020 = \( 0.10 \times 750 = 75 \)

So, 75 students from School D qualified in 2020.

From the table, in 2021, 14% of the total qualified students were from School D.

Number of students from School D in 2021 = \( 14\% \text{ of } T_{2021} \)

Number of students from School D in 2021 = \( 0.14 \times 900 = 126 \)

So, 126 students from School D qualified in 2021.

Calculating Percent Rise for School D

We need to find the percent rise in the number of students qualifying from School D from 2020 (75 students) to 2021 (126 students). The formula for percent rise is:

\( \text{Percent Rise} = \frac{\text{Change in Number}}{\text{Original Number}} \times 100 \)

Here, the Original Number is the number of qualified students from School D in 2020, which is 75.

The Change in Number is the difference between the number of qualified students in 2021 and 2020:

\( \text{Change} = 126 - 75 = 51 \)

Now, we can calculate the percent rise:

\( \text{Percent Rise} = \frac{51}{75} \times 100 \)

\( \text{Percent Rise} = \frac{51}{75} \times 100 = \frac{17 \times 3}{25 \times 3} \times 100 = \frac{17}{25} \times 100 \)

\( \text{Percent Rise} = 17 \times \frac{100}{25} = 17 \times 4 = 68 \)

The percent rise in the number of students qualifying from School D from 2020 to 2021 is 68%.

Summary of Calculations

Percent Rise for School D = \( \frac{126 - 75}{75} \times 100 = \frac{51}{75} \times 100 = 68\% \)

Conclusion

The percent rise in the number of students qualifying from School D from 2020 to 2021 is 68%.

Revision Table: Key Concepts Revisited

• Data Interpretation from Tables: Understanding how to read and extract relevant information.
• Percentage Calculations: Using percentages to find total values or specific component values.
• Percent Rise/Change Formula: Applying the formula \( \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100 \) to find percentage increase.
• Problem Solving Steps: Breaking down a complex problem into smaller, manageable steps (find total, find specific value, calculate change).

Additional Information: Percentages and Ratios

Percentages are a way of expressing a proportion out of 100. In this problem, the percentage distribution tells us the share of qualified students from each school relative to the total number of qualified students in that year.

For example, saying 10% of students from School D qualified in 2020 means that for every 100 students who qualified overall in 2020, 10 were from School D.

The information about School G provides a link between the percentage distribution and the actual numbers. Since we know both the percentage and the actual count for School G, we can determine the base value, which is the total number of qualified students for that year. This total number is crucial because the percentages for all other schools (including School D) are based on this total.

Understanding percent rise is also important. A percent rise indicates how much a value has increased relative to its starting value. A 68% rise means the number of students from School D qualifying in 2021 was 68% higher than the number in 2020.