Sophie and Simon are peeling a pile of potatoes for lunch in the cafeteria. Sophie can peel all the potatoes by herself in 45 minutes, while it would take Simon 30 minutes to do the job working alone. If Sophie and Simon work together to peel the potatoes, how long will it take them?

A. 15 minutes
B. 18 minutes
C. 38 minutes



Answer :

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.