A man's wage was reduced by 50%. Again, the wage was increased by 50%. Find his loss in percentage.

Steps to Solve

1. Calculate the wage after a 50% reduction

Start with an initial wage, let's call it $W$.

After a reduction of 50%, the new wage becomes:

$$ W_{\text{reduced}} = W - 0.5W = 0.5W $$

2. Calculate the new wage after a 50% increase

Now, we take the reduced wage $W_{\text{reduced}}$ and increase it by 50%. The new wage will be:

$$ W_{\text{increased}} = W_{\text{reduced}} + 0.5 \times W_{\text{reduced}} $$

Substituting for $W_{\text{reduced}}$:

$$ W_{\text{increased}} = 0.5W + 0.5 \times 0.5W = 0.5W + 0.25W = 0.75W $$

3. Calculate the percentage loss in wages compared to the original wage

Now, we need to find the percentage loss in wages from the original wage $W$ to the new wage $W_{\text{increased}}$. The loss can be calculated as:

$$ \text{Loss} = W - W_{\text{increased}} = W - 0.75W = 0.25W $$

To find the percentage loss:

$$ \text{Percentage Loss} = \left( \frac{\text{Loss}}{W} \right) \times 100% = \left( \frac{0.25W}{W} \right) \times 100% = 25% $$