1. **Problem statement:** To finish a job in 8 days, 6 workers are needed. How many workers are required to finish the same job 2 days earlier, i.e., in 6 days? 2. **Formula used:** The amount of work done is proportional to the number of workers multiplied by the number of days. This can be expressed as: $$\text{Workers}_1 \times \text{Days}_1 = \text{Workers}_2 \times \text{Days}_2$$ This assumes all workers work at the same rate and the job size is constant. 3. **Step-by-step solution:** - Given: $6$ workers for $8$ days. - New time: $8 - 2 = 6$ days. - Let $x$ be the number of workers needed to finish in $6$ days. Using the formula: $$6 \times 8 = x \times 6$$ Simplify: $$48 = 6x$$ Divide both sides by 6: $$x = \frac{48}{6} = 8$$ 4. **Answer:** $8$ workers are needed to finish the job in 6 days. --- 1. **Problem statement:** 24 carpenters need 45 days to build a house. How many carpenters are needed to complete the job in 30 days? 2. **Formula used:** Same as above, work is constant: $$\text{Carpenters}_1 \times \text{Days}_1 = \text{Carpenters}_2 \times \text{Days}_2$$ 3. **Step-by-step solution:** - Given: $24$ carpenters for $45$ days. - New time: $30$ days. - Let $y$ be the number of carpenters needed. Using the formula: $$24 \times 45 = y \times 30$$ Simplify: $$1080 = 30y$$ Divide both sides by 30: $$y = \frac{1080}{30} = 36$$ 4. **Answer:** $36$ carpenters are needed to complete the job in 30 days.