Ratio Calculation: Men and Women Increase

This problem involves calculating a new ratio after the initial numbers of men and women increase by specific percentages. We are given the initial ratio and the percentage increases for both groups.

Initial Setup

Let the initial number of men in the company be $M$ and the initial number of women be $W$.

The problem states that the ratio of men to women is 5 : 4. We can represent this as:

$$ \frac{M}{W} = \frac{5}{4} $$

To make calculations easier, let's assume the initial number of men is $5x$ and the initial number of women is $4x$, where $x$ is a common multiplier.

• Initial Men ($M$) = $5x$
• Initial Women ($W$) = $4x$

Calculating Percentage Increases

The number of men increases by 16%, and the number of women increases by 15%.

Increase in Men:

The increase is $16\%$ of $5x$.

$$ \text{Increase in Men} = 16\% \times 5x = \frac{16}{100} \times 5x = 0.16 \times 5x = 0.8x $$

The new number of men ($M'$) is the initial number plus the increase:

$$ M' = M + \text{Increase in Men} = 5x + 0.8x = 5.8x $$

Increase in Women:

The increase is $15\%$ of $4x$.

$$ \text{Increase in Women} = 15\% \times 4x = \frac{15}{100} \times 4x = 0.15 \times 4x = 0.6x $$

The new number of women ($W'$) is the initial number plus the increase:

$$ W' = W + \text{Increase in Women} = 4x + 0.6x = 4.6x $$

Determining the New Ratio

The new ratio of men to women is the ratio of the new number of men ($M'$) to the new number of women ($W'$).

$$ \text{New Ratio} = \frac{M'}{W'} = \frac{5.8x}{4.6x} $$

The variable $x$ cancels out:

$$ \text{New Ratio} = \frac{5.8}{4.6} $$

To simplify this ratio, we can multiply the numerator and denominator by 10 to remove the decimals:

$$ \text{New Ratio} = \frac{5.8 \times 10}{4.6 \times 10} = \frac{58}{46} $$

Now, we simplify the fraction $\frac{58}{46}$ by finding the greatest common divisor. Both numbers are even, so we can divide by 2:

$$ \frac{58 \div 2}{46 \div 2} = \frac{29}{23} $$

The simplified new ratio of men to women is 29 : 23.

Conclusion

After calculating the increases based on the initial ratio of 5 : 4, where men increased by 16% and women by 15%, the new ratio of men to women is determined to be 29 : 23.