To determine how long it will take Sophie and Simon to peel a pile of potatoes together, follow these steps:
1. Determine Individual Rates:
- Sophie can peel the entire pile in 45 minutes. Therefore, her rate of peeling is:
[tex]\[
\text{Rate of Sophie} = \frac{1}{45} \text{ piles per minute}
\][/tex]
- Simon can peel the entire pile in 30 minutes. Therefore, his rate of peeling is:
[tex]\[
\text{Rate of Simon} = \frac{1}{30} \text{ piles per minute}
\][/tex]
2. Combine Their Rates:
- When Sophie and Simon work together, their rates add up. Therefore, the combined rate is:
[tex]\[
\text{Combined Rate} = \frac{1}{45} + \frac{1}{30}
\][/tex]
3. Find a Common Denominator:
- To add the fractions, find a common denominator. The least common multiple of 45 and 30 is 90. Transform the rates accordingly:
[tex]\[
\frac{1}{45} = \frac{2}{90}
\][/tex]
[tex]\[
\frac{1}{30} = \frac{3}{90}
\][/tex]
- Add the fractions:
[tex]\[
\text{Combined Rate} = \frac{2}{90} + \frac{3}{90} = \frac{5}{90} = \frac{1}{18} \text{ piles per minute}
\][/tex]
4. Calculate the Time Taken Together:
- The combined rate means they can peel 1 pile in 18 minutes. Therefore, the time taken to peel all the potatoes together is:
[tex]\[
\text{Time together} = \frac{1}{\text{Combined Rate}} = \frac{1}{\frac{1}{18}} = 18 \text{ minutes}
\][/tex]
So, if Sophie and Simon work together, it will take them 18 minutes to peel the entire pile of potatoes.
