Calculating Candidate Ratios for Ph.D. Entrance Exam

The question asks for the ratio of candidates who passed from Institute B to the candidates enrolled in the same institute, based on the provided data table.

First, let's understand the given data:

• Total number of candidates enrolled for the Ph.D. entrance exam = 17100
• Total number of candidates who passed the exam = 11400
• The table shows the percentage distribution of enrolled and passed candidates across seven different Institutes (A - G).

Step 1: Calculate the number of candidates enrolled in Institute B

The table shows that 22% of the total enrolled candidates were from Institute B. The total number of enrolled candidates is 17100.

Number of enrolled candidates in Institute B = 22% of 17100

Using the percentage formula:

Number of enrolled candidates in B = $\frac{22}{100} \times 17100$

Number of enrolled candidates in B = $0.22 \times 17100$

Number of enrolled candidates in B = $3762$

So, $3762$ candidates were enrolled from Institute B.

Step 2: Calculate the number of candidates who passed from Institute B

The table shows that 18% of the total passed candidates were from Institute B. The total number of passed candidates is 11400.

Number of passed candidates from Institute B = 18% of 11400

Using the percentage formula:

Number of passed candidates from B = $\frac{18}{100} \times 11400$

Number of passed candidates from B = $0.18 \times 11400$

Number of passed candidates from B = $2052$

So, $2052$ candidates passed the exam from Institute B.

Step 3: Calculate the ratio of passed candidates to enrolled candidates for Institute B

We need to find the ratio of (candidates who passed from Institute B) to (candidates enrolled in Institute B).

Ratio = $\frac{\text{Number of passed candidates from B}}{\text{Number of enrolled candidates in B}}$

Ratio = $\frac{2052}{3762}$

Now, we simplify the fraction. We can find common factors. Both numbers are even, so they are divisible by 2.

$\frac{2052 \div 2}{3762 \div 2} = \frac{1026}{1881}$

The sum of digits of 1026 is $1+0+2+6=9$ (divisible by 3). The sum of digits of 1881 is $1+8+8+1=18$ (divisible by 3). So, both numbers are divisible by 3.

$\frac{1026 \div 3}{1881 \div 3} = \frac{342}{627}$

Again, check divisibility by 3. Sum of digits of 342 is $3+4+2=9$ (divisible by 3). Sum of digits of 627 is $6+2+7=15$ (divisible by 3). So, both numbers are divisible by 3.

$\frac{342 \div 3}{627 \div 3} = \frac{114}{209}$

Now, let's look for other common factors. Let's try prime numbers. 114 is divisible by 2 ($114 = 2 \times 57$) and by 3 ($114 = 3 \times 38$). It's also divisible by 19 ($114 = 6 \times 19$). Let's check if 209 is divisible by 19. $209 \div 19$. $19 \times 10 = 190$. $209 - 190 = 19$. So, $209 = 190 + 19 = 19 \times 10 + 19 \times 1 = 19 \times (10+1) = 19 \times 11$. So, both 114 and 209 are divisible by 19.

$\frac{114 \div 19}{209 \div 19} = \frac{6}{11}$

The simplified ratio is $\frac{6}{11}$, which can be written as $6 \ratio 11$.

Conclusion on Ratio Calculation

The ratio of candidates who passed from Institute B to the candidates enrolled in the same Institute is $6 \ratio 11$. This calculation uses the total number of enrolled and passed candidates along with the percentage distributions from the table.

Comparing this with the given options, the ratio $6 \ratio 11$ matches one of the choices.

Revision Table: Ph.D. Exam Data Analysis

Additional Information: Data Interpretation & Ratios

Data interpretation questions often require extracting specific pieces of information from tables, graphs, or charts and performing calculations like percentages, averages, or ratios. Understanding how to convert percentages to actual numbers and simplify ratios is crucial for solving such problems accurately.

• Percentage Calculation: To find a percentage of a total number, convert the percentage to a decimal (divide by 100) and multiply by the total number. For example, $22\% = 0.22$, so $22\%$ of 17100 is $0.22 \times 17100$.
• Ratio Simplification: A ratio represents a relationship between two numbers. To simplify a ratio expressed as a fraction ($\frac{a}{b}$), divide both the numerator ($a$) and the denominator ($b$) by their greatest common divisor (GCD). This gives the ratio in its simplest form. In the example, we simplified $\frac{2052}{3762}$ by repeatedly dividing by common factors until no more common factors existed other than 1.
• Context is Key: Always pay close attention to what the percentages are based on. In this problem, the enrolled percentage is out of total enrolled, while the passed percentage is out of total passed. These are different bases, so you cannot directly compare or ratio the percentages themselves; you must calculate the actual numbers first.

Practicing various types of data interpretation problems with different formats (tables, bar graphs, pie charts, line graphs) helps improve speed and accuracy for exams like Ph.D. entrance tests.