Analyzing Bank Vacancies and Cadre Requirements

The problem provides data from two tables. Table 1 lists the number of vacant posts in six different banks (A to F), and Table 2 gives the percentage breakup of required personnel across different cadres, including Agricultural Field Officers (AFO), for a bank. We are asked to find the ratio of AFO recruitment in Banks C and E combined to that in Banks D and F combined, assuming the percentage breakup is the same for all banks.

Let's first look at the data provided in the tables:

Calculating AFO Recruitment in Specific Banks

According to Table 2, the percentage of requirement for Agricultural Field Officers (AFO) is 35% of the total vacant posts in a bank. Since this percentage is assumed to be the same for all six banks, we can calculate the number of AFO posts in Banks C, E, D, and F.

• AFO posts in Bank C: This is 35% of the total vacant posts in Bank C.

  \(\text{AFO in C} = 35\% \text{ of } 600\)

  \(\text{AFO in C} = \frac{35}{100} \times 600\)

  \(\text{AFO in C} = 0.35 \times 600 = 210\)

• AFO posts in Bank E: This is 35% of the total vacant posts in Bank E.

  \(\text{AFO in E} = 35\% \text{ of } 450\)

  \(\text{AFO in E} = \frac{35}{100} \times 450\)

  \(\text{AFO in E} = 0.35 \times 450 = 157.5\)

• AFO posts in Bank D: This is 35% of the total vacant posts in Bank D.

  \(\text{AFO in D} = 35\% \text{ of } 820\)

  \(\text{AFO in D} = \frac{35}{100} \times 820\)

  \(\text{AFO in D} = 0.35 \times 820 = 287\)

• AFO posts in Bank F: This is 35% of the total vacant posts in Bank F.

  \(\text{AFO in F} = 35\% \text{ of } 560\)

  \(\text{AFO in F} = \frac{35}{100} \times 560\)

  \(\text{AFO in F} = 0.35 \times 560 = 196\)

Calculating the Ratio of AFO Recruitment

The question asks for the ratio of AFO recruitment in C and E taken together with that of D and F taken together.

• Total AFO posts in C and E together:

  \(\text{AFO (C + E)} = \text{AFO in C} + \text{AFO in E}\)

  \(\text{AFO (C + E)} = 210 + 157.5 = 367.5\)

• Total AFO posts in D and F together:

  \(\text{AFO (D + F)} = \text{AFO in D} + \text{AFO in F}\)

  \(\text{AFO (D + F)} = 287 + 196 = 483\)

Now, we need to find the ratio of AFO (C + E) to AFO (D + F).

\(\text{Ratio} = \text{AFO (C + E)} : \text{AFO (D + F)}\)

\(\text{Ratio} = 367.5 : 483\)

To simplify the ratio and remove the decimal, we can multiply both sides by 10:

\(\text{Ratio} = 3675 : 4830\)

Now, let's simplify the ratio by finding common factors. Both numbers end in 5 or 0, so they are divisible by 5.

\(3675 \div 5 = 735\)

\(4830 \div 5 = 966\)

\(\text{Ratio} = 735 : 966\)

Let's check for other common factors. The sum of digits for 735 (\(7+3+5=15\)) is divisible by 3. The sum of digits for 966 (\(9+6+6=21\)) is divisible by 3. So, both numbers are divisible by 3.

\(735 \div 3 = 245\)

\(966 \div 3 = 322\)

\(\text{Ratio} = 245 : 322\)

Now let's check for smaller prime factors. 245 ends in 5, so it is divisible by 5: \(245 = 5 \times 49\). 49 is \(7 \times 7\). So \(245 = 5 \times 7 \times 7\).

Let's check if 322 is divisible by 7:

\(322 \div 7\)

\(32 \div 7 = 4\) with remainder \(4\).

\(42 \div 7 = 6\).

So, \(322 \div 7 = 46\). Yes, 322 is divisible by 7.

Divide both parts of the ratio \(245 : 322\) by 7:

\(245 \div 7 = 35\)

\(322 \div 7 = 46\)

\(\text{Ratio} = 35 : 46\)

The simplified ratio of AFO recruitment in C and E together to that in D and F together is 35 : 46.

Revision Table: Key Calculations

Here’s a summary of the main steps and results:

Ratio (C+E) : (D+F) = \(367.5 : 483 = 35 : 46\)

Additional Information on Ratios and Percentages

A ratio is a way to compare two or more quantities. It shows the relative sizes of the quantities being compared. Ratios can be written using a colon (:) or as a fraction.

A percentage is a ratio expressed as a fraction of 100. For example, 35% means 35 out of 100, which can be written as \(\frac{35}{100}\) or 0.35 in decimal form.

To calculate a percentage of a number, you can convert the percentage to a decimal or a fraction and then multiply by the number. In this problem, we calculated 35% of the vacant posts in each bank to find the number of AFO required.

When working with ratios, it is often helpful to simplify them to their lowest terms by dividing all parts of the ratio by their greatest common divisor (GCD).