Understanding the Employee Ratio Problem

This problem involves tracking the number of male and female employees in a company as it changes due to new hires. We are given initial and final ratios of male to female employees, as well as the ratio in which new employees joined and the final total number of employees. The goal is to find the initial total number of employees.

Setting Up Equations for Employee Ratios

Let's break down the information given and represent the unknown quantities using variables:

• Initially, the ratio of male to female employees was 5:4. We can represent the initial number of male employees as $5x$ and the initial number of female employees as $4x$, where $x$ is a common multiplier. The initial total number of employees was $5x + 4x = 9x$.
• Some new male and female employees joined in the ratio 2:3. We can represent the number of new male employees as $2y$ and the number of new female employees as $3y$, where $y$ is another common multiplier. The total number of new employees was $2y + 3y = 5y$.
• After the new employees joined, the total number of employees became 1620. This gives us our first equation:
  $$ \text{Initial total} + \text{New total} = \text{Final total} $$ $$ 9x + 5y = 1620 \quad \cdots (1) $$
• The ratio of male to female employees after joining became 68:67.
  • Total male employees after joining = Initial males + New males = $5x + 2y$.
  • Total female employees after joining = Initial females + New females = $4x + 3y$.
  This gives us our second equation based on the final ratio:
  $$ \frac{5x + 2y}{4x + 3y} = \frac{68}{67} \quad \cdots (2) $$

Solving for Initial Employee Count using Simultaneous Equations

Now we need to solve the system of equations to find the value of $x$, which will help us determine the initial total number of employees ($9x$).

From equation (2), we can cross-multiply:

$$ 67(5x + 2y) = 68(4x + 3y) $$

Distribute the numbers:

$$ 335x + 134y = 272x + 204y $$

Gather like terms (move all $x$ terms to one side and $y$ terms to the other):

$$ 335x - 272x = 204y - 134y $$

$$ 63x = 70y $$

We can simplify this equation by dividing both sides by the greatest common divisor of 63 and 70, which is 7:

$$ \frac{63x}{7} = \frac{70y}{7} $$

$$ 9x = 10y $$

This equation gives a relationship between $x$ and $y$. We need to find the initial total, which is $9x$. Notice that the equation directly relates $9x$ and $10y$. Let's use equation (1) along with this relationship.

From $9x = 10y$, we can express $10y$ as $9x$. Equation (1) is $9x + 5y = 1620$. We need to substitute $y$ in terms of $x$ (or $x$ in terms of $y$), or use the direct relationship. Let's express $y$ in terms of $x$: $y = \frac{9x}{10}$.

Substitute this expression for $y$ into equation (1):

$$ 9x + 5\left(\frac{9x}{10}\right) = 1620 $$

Simplify the term with $y$:

$$ 9x + \frac{45x}{10} = 1620 $$

$$ 9x + \frac{9x}{2} = 1620 $$

To eliminate the fraction, multiply the entire equation by 2:

$$ 2(9x) + 2\left(\frac{9x}{2}\right) = 2(1620) $$

$$ 18x + 9x = 3240 $$

Combine the $x$ terms:

$$ 27x = 3240 $$

Now, solve for $x$:

$$ x = \frac{3240}{27} $$

Performing the division:

$$ x = 120 $$

The question asks for the initial total number of employees, which is $9x$.

Initial total employees = $9 \times x = 9 \times 120$

$$ \text{Initial total employees} = 1080 $$

Verification of the Solution

Let's check if our values satisfy the conditions:

• $x = 120$. Initial males = $5 \times 120 = 600$. Initial females = $4 \times 120 = 480$. Initial total = $600 + 480 = 1080$.
• From $9x = 10y$, we have $1080 = 10y$, so $y = 108$.
• New males = $2y = 2 \times 108 = 216$. New females = $3y = 3 \times 108 = 324$. Total new = $216 + 324 = 540$.
• Total employees after joining = Initial total + Total new = $1080 + 540 = 1620$. (Matches the given total).
• Total males after joining = $600 + 216 = 816$.
• Total females after joining = $480 + 324 = 804$.
• Ratio of males to females after joining = $816 : 804$. Divide both by 12: $816/12 = 68$, $804/12 = 67$. The ratio is $68:67$. (Matches the given final ratio).

The calculation is consistent with all given conditions. The initial total number of employees was 1080.

Initial Employee Count Conclusion

Based on the initial ratio of male and female employees (5:4), the ratio of new employees joining (2:3), the final total number of employees (1620), and the final ratio of males to females (68:67), the initial total number of employees in the company was 1080.

Revision Table: Key Concepts for Ratio Problems

Additional Information: Ratios and Proportions

A ratio is a comparison of two or more quantities of the same kind, measured in the same unit. It shows how much of one quantity there is compared to another. Ratios can be written as $a:b$, $a/b$, or "a to b". Proportions are statements that two ratios are equal, like $\frac{a}{b} = \frac{c}{d}$. Solving ratio and proportion problems often involves setting up equations based on the given relationships.

In problems like this, where quantities change, it's important to distinguish between the initial amounts, the change amounts, and the final amounts. Representing these with different variables or expressions derived from the initial variables helps in setting up accurate equations.

When dealing with ratios of sums or differences of quantities (like the ratio of total males to total females after some joined), setting up a proportion and cross-multiplying is a standard method to derive a linear equation relating the variables.