A man earns $40 for a period of 5 hours that he works per day. Thereafter , for every additional hour , he earns $5 per hour. If the man earns $P in a day working for t hours, where ` t gt 5` and is an integer , and ` P le 60`, which of the following sets of value of P and t satisfy the above conditions ?

Options

A. P = 60 , t = 10
B. P = 55, t = 10
C. P = 50 , t = 9
D. P = 50, t = 7

Answer

To solve the problem step-by-step, we need to establish the earnings of the man based on the hours he works.

### Step 1: Establish the Earnings Formula
The man earns $40 for the first 5 hours. For every additional hour beyond 5 hours, he earns $5 per hour. Therefore, if he works for \( t \) hours (where \( t > 5 \)), his total earnings \( P \) can be expressed as:

\[ P = 40 + 5(t - 5) \]

### Step 2: Simplify the Earnings Formula
We can simplify the formula for \( P \):

\[ P = 40 + 5(t - 5) = 40 + 5t - 25 = 5t + 15 \]

### Step 3: Set Up the Conditions
We know from the problem that:
1. \( t > 5 \) (t is an integer).
2. \( P \leq 60 \).

From the earnings formula \( P = 5t + 15 \), we can set up the inequality:

\[ 5t + 15 \leq 60 \]

### Step 4: Solve the Inequality
Now, we can solve for \( t \):

\[ 5t \leq 60 - 15 \]
\[ 5t \leq 45 \]
\[ t \leq 9 \]

### Step 5: Determine Possible Values of \( t \)
Since \( t \) must be an integer greater than 5 and less than or equal to 9, the possible integer values for \( t \) are:

\[ t = 6, 7, 8, 9 \]

### Step 6: Calculate Corresponding Values of \( P \)
Now we can calculate the corresponding values of \( P \) for each possible \( t \):

- For \( t = 6 \):
\[ P = 5(6) + 15 = 30 + 15 = 45 \]

- For \( t = 7 \):
\[ P = 5(7) + 15 = 35 + 15 = 50 \]

- For \( t = 8 \):
\[ P = 5(8) + 15 = 40 + 15 = 55 \]

- For \( t = 9 \):
\[ P = 5(9) + 15 = 45 + 15 = 60 \]

### Step 7: List the Valid (P, t) Pairs
The valid pairs of \( (P, t) \) that satisfy the conditions \( t > 5 \) and \( P \leq 60 \) are:

1. \( (45, 6) \)
2. \( (50, 7) \)
3. \( (55, 8) \)
4. \( (60, 9) \)

### Conclusion
The sets of values of \( P \) and \( t \) that satisfy the conditions are:
- \( (45, 6) \)
- \( (50, 7) \)
- \( (55, 8) \)
- \( (60, 9) \)