Understanding College Academic Performance Data

The question asks for the ratio of the total number of students in College D and College F, given a specific condition based on the academic performance data presented in the table. The key condition is that the number of male students who passed from College D and College F are equal.

Let's first look at the relevant data for Colleges D and F from the provided table:

Calculating Students in College D

Let the total number of students in College D be \(T_D\).

• The pass percentage in College D is 45%.
• Number of passed students in College D, \(P_D = 45\%\) of \(T_D = \frac{45}{100} T_D = 0.45 T_D\).
• The ratio of Male to Female among passed students in College D is 4 ∶ 1. This means out of every \(4+1=5\) passed students, 4 are male and 1 is female.
• The fraction of male students among the passed students is \(\frac{4}{4+1} = \frac{4}{5}\).
• Number of male students who passed from College D, \(M_{P,D} = \frac{4}{5} \times P_D = \frac{4}{5} \times 0.45 T_D\).
• Let's simplify this expression: \(M_{P,D} = \frac{4}{5} \times \frac{45}{100} T_D = \frac{4 \times 45}{5 \times 100} T_D = \frac{180}{500} T_D = \frac{18}{50} T_D = \frac{9}{25} T_D\).

Calculating Students in College F

Let the total number of students in College F be \(T_F\).

• The pass percentage in College F is 37.50%. Note that 37.50% is equivalent to \(\frac{37.5}{100} = \frac{375}{1000} = \frac{3 \times 125}{8 \times 125} = \frac{3}{8}\).
• Number of passed students in College F, \(P_F = 37.50\%\) of \(T_F = \frac{3}{8} T_F = 0.375 T_F\).
• The ratio of Male to Female among passed students in College F is 3 ∶ 2. This means out of every \(3+2=5\) passed students, 3 are male and 2 are female.
• The fraction of male students among the passed students is \(\frac{3}{3+2} = \frac{3}{5}\).
• Number of male students who passed from College F, \(M_{P,F} = \frac{3}{5} \times P_F = \frac{3}{5} \times 0.375 T_F\).
• Let's simplify this expression: \(M_{P,F} = \frac{3}{5} \times \frac{3}{8} T_F = \frac{3 \times 3}{5 \times 8} T_F = \frac{9}{40} T_F\).

Applying the Condition and Finding the Ratio

The problem states that the number of male students who passed from D and F are equal.

So, we have the equation: \(M_{P,D} = M_{P,F}\).

Substituting the expressions we derived:

\(\frac{9}{25} T_D = \frac{9}{40} T_F\)

We want to find the ratio of the number of students in D and F, which is \(T_D : T_F\) or \(\frac{T_D}{T_F}\).

Divide both sides of the equation by \(T_F\) (assuming \(T_F \neq 0\)) and by \(\frac{9}{25}\):

\(\frac{T_D}{T_F} = \frac{9/40}{9/25}\)

To divide by a fraction, we multiply by its reciprocal:

\(\frac{T_D}{T_F} = \frac{9}{40} \times \frac{25}{9}\)

Cancel out the '9' from the numerator and denominator:

\(\frac{T_D}{T_F} = \frac{1}{40} \times 25\)

Simplify the fraction \(\frac{25}{40}\) by dividing both numerator and denominator by their greatest common divisor, which is 5:

\(\frac{T_D}{T_F} = \frac{25 \div 5}{40 \div 5} = \frac{5}{8}\)

So, the ratio of the number of students in College D and College F is 5 ∶ 8.

Result and Conclusion

Based on the calculation, the ratio of the total number of students in College D and College F is 5 ∶ 8.

Revision Table: Key Calculations

Additional Information: Data Interpretation Tips

• Read the table headings and column labels carefully to understand what the data represents.
• Pay close attention to percentages and ratios. Remember how to convert percentages to fractions or decimals, and how to work with ratios (e.g., finding the fraction of a part within the whole).
• Break down the problem into smaller steps. Calculate intermediate values like the number of passed/failed students or the number of males/females in each category before combining them according to the question's condition.
• Set up equations based on the given information and conditions.
• Simplify calculations by converting decimals to fractions, especially when dealing with exact values like 37.5%.