To solve the problem, we need to determine how many men were in the group given that 30 of them were women, and an equal number of men and women passed the driving test. Here's a detailed, step-by-step solution:
1. Define the variables:
- Let [tex]\( W \)[/tex] represent the total number of women. We know [tex]\( W = 30 \)[/tex].
- Let [tex]\( M \)[/tex] represent the total number of men in the group. This is what we need to find.
2. Calculate the number of women who passed the test:
- According to the problem, [tex]\(\frac{4}{5}\)[/tex] of the women passed the test.
- So, the number of women who passed the test is [tex]\(\frac{4}{5} \times W = \frac{4}{5} \times 30 = 24\)[/tex].
3. Calculate the number of men who passed the test:
- The problem states that [tex]\(\frac{3}{4}\)[/tex] of the men passed the test.
- Let [tex]\(\text{Men Passed}\)[/tex] represent this number. So, [tex]\(\text{Men Passed} = \frac{3}{4} \times M\)[/tex].
4. Set up the equation:
- We are given that the number of women who passed the test is equal to the number of men who passed the test.
- Therefore, [tex]\(\frac{4}{5} \times 30 = \frac{3}{4} \times M\)[/tex].
5. Solve the equation:
- We already calculated [tex]\(\frac{4}{5} \times 30 = 24\)[/tex].
- So, [tex]\(24 = \frac{3}{4} \times M\)[/tex].
6. Isolate [tex]\( M \)[/tex]:
- Multiply both sides of the equation by [tex]\(\frac{4}{3}\)[/tex] to solve for [tex]\( M \)[/tex]:
[tex]\[
M = 24 \times \frac{4}{3}
\][/tex]
- Compute the multiplication:
[tex]\[
M = 32
\][/tex]
Therefore, the total number of men in the group is [tex]\( 32 \)[/tex].
