Last month Joe the painter painted many rooms. He used 3 coats of paint on one third of the rooms he painted. On two fifths of the remaining rooms he used 2 coats of paint, and he only used 1 coat of paint on the remaining 24 rooms. What was the total number of coats of paint Joe painted last month?

Answer

To find the total number of coats of paint Joe painted last month, we can break down the problem step by step.

### Step 1: Define the total number of rooms
Let the total number of rooms Joe painted be \( X \).

### Step 2: Calculate the number of rooms painted with 3 coats
Joe used 3 coats of paint on one-third of the rooms:
\[ \text{Rooms with 3 coats} = \frac{X}{3} \]
The total coats of paint used on these rooms is:
\[ \text{Total coats for these rooms} = 3 \times \frac{X}{3} = X \]

### Step 3: Calculate the remaining rooms
After painting one-third of the rooms, the remaining rooms are:
\[ \text{Remaining rooms} = X - \frac{X}{3} = \frac{2X}{3} \]

### Step 4: Calculate the number of rooms painted with 2 coats
Joe used 2 coats of paint on two-fifths of the remaining rooms:
\[ \text{Rooms with 2 coats} = \frac{2}{5} \times \frac{2X}{3} = \frac{4X}{15} \]
The total coats of paint used on these rooms is:
\[ \text{Total coats for these rooms} = 2 \times \frac{4X}{15} = \frac{8X}{15} \]

### Step 5: Calculate the remaining rooms after the previous steps
After painting the rooms with 3 coats and 2 coats, the remaining rooms are:
\[ \text{Remaining rooms} = \frac{2X}{3} - \frac{4X}{15} \]
To simplify this, we need a common denominator (which is 15):
\[ \frac{2X}{3} = \frac{10X}{15} \]
Thus,
\[ \text{Remaining rooms} = \frac{10X}{15} - \frac{4X}{15} = \frac{6X}{15} = \frac{2X}{5} \]

### Step 6: Set the remaining rooms to 24
It is given that there are 24 rooms remaining where he used 1 coat of paint:
\[ \frac{2X}{5} = 24 \]

### Step 7: Solve for \( X \)
To find \( X \), we multiply both sides by 5:
\[ 2X = 120 \implies X = 60 \]

### Step 8: Calculate the total number of coats of paint
Now that we know \( X = 60 \), we can calculate the total number of coats:
1. Coats from rooms with 3 coats:
\[ \text{Total coats} = 60 \]
2. Coats from rooms with 2 coats:
\[ \text{Rooms with 2 coats} = \frac{4 \times 60}{15} = 16 \implies \text{Total coats} = 16 \times 2 = 32 \]
3. Coats from the remaining 24 rooms:
\[ \text{Total coats} = 24 \times 1 = 24 \]

### Step 9: Sum the total coats
Now we sum all the coats:
\[ \text{Total coats} = 60 + 32 + 24 = 116 \]

Thus, the total number of coats of paint Joe painted last month is **116**.

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