Question

Peter takes 4 hours longer than Samuel to prepare 30 sandwiches. Working together, they can make 50 sandwiches in 150 minutes. How long would it take Peter alone to make 40 sandwiches?

Hint:

In work-rate problems, it's often easiest to define variables for the time taken to complete a specific, common amount of work (like 30 sandwiches here). This allows for direct calculation of individual rates, which can then be used in the combined work equation.

Answer 1

Step 1: Understanding the Concept:
This is a work-rate problem. The key principle is that rates are additive. If two people work together, their combined rate is the sum of their individual rates. Rate is defined as Work/Time.
Step 2: Detailed Explanation:
Let \(T_S\) be the time in hours Samuel takes to prepare 30 sandwiches.
Let \(T_P\) be the time in hours Peter takes to prepare 30 sandwiches.
From the problem statement, \(T_P = T_S + 4\).
Now, let's express their rates in sandwiches per hour. Samuel's rate, \(R_S = \frac{30}{T_S}\) sandwiches/hour.
Peter's rate, \(R_P = \frac{30}{T_P} = \frac{30}{T_S + 4}\) sandwiches/hour.
Together, they make 50 sandwiches in 150 minutes. First, convert the time to hours: \(150 \text{ minutes} = \frac{150}{60} \text{ hours} = 2.5 \text{ hours}\).
Their combined rate is \(R_{combined} = \frac{\text{Work}}{\text{Time}} = \frac{50}{2.5} = 20\) sandwiches/hour.
The combined rate is the sum of their individual rates: \[ R_S + R_P = 20 \] Substitute the expressions for their rates: \[ \frac{30}{T_S} + \frac{30}{T_S + 4} = 20 \] Divide the entire equation by 10 to simplify: \[ \frac{3}{T_S} + \frac{3}{T_S + 4} = 2 \] To solve for \(T_S\), find a common denominator: \[ \frac{3(T_S + 4) + 3T_S}{T_S(T_S + 4)} = 2 \] \[ 3T_S + 12 + 3T_S = 2(T_S^2 + 4T_S) \] \[ 6T_S + 12 = 2T_S^2 + 8T_S \] Rearrange into a standard quadratic form: \[ 2T_S^2 + 2T_S - 12 = 0 \] Divide by 2: \[ T_S^2 + T_S - 6 = 0 \] Factor the quadratic equation: \[ (T_S + 3)(T_S - 2) = 0 \] The possible values for \(T_S\) are -3 and 2. Since time cannot be negative, \(T_S = 2\) hours.
So, Samuel takes 2 hours to make 30 sandwiches.
Peter's time to make 30 sandwiches is \(T_P = T_S + 4 = 2 + 4 = 6\) hours.
Step 3: Final Answer:
The question asks for the time it would take Peter to make 40 sandwiches.
First, find Peter's rate: \[ R_P = \frac{30 \text{ sandwiches}}{6 \text{ hours}} = 5 \text{ sandwiches/hour} \] Now, calculate the time required for Peter to make 40 sandwiches: \[ \text{Time} = \frac{\text{Work}}{\text{Rate}} = \frac{40 \text{ sandwiches}}{5 \text{ sandwiches/hour}} = 8 \text{ hours} \]