Three consecutive whole numbers are such that if they be divided by `5, 3` and `4` respectively, the sum of the quotients is `40`. Find the numbers.

Options

A. `53, 54` and `55`
B. `37, 38` and `39`
C. `50, 51` and `52`
D. `55, 56` and `57`

Answer

To solve the problem of finding three consecutive whole numbers such that when they are divided by 5, 3, and 4 respectively, the sum of the quotients is 40, we can follow these steps:

### Step 1: Define the consecutive numbers
Let the three consecutive whole numbers be:
- First number: \( x \)
- Second number: \( x + 1 \)
- Third number: \( x + 2 \)

### Step 2: Set up the equation based on the problem statement
According to the problem, when these numbers are divided by 5, 3, and 4 respectively, the sum of the quotients is 40. We can express this mathematically as:
\[ \frac{x}{5} + \frac{x + 1}{3} + \frac{x + 2}{4} = 40 \]

### Step 3: Find the least common multiple (LCM)
To eliminate the fractions, we need to find the least common multiple (LCM) of the denominators (5, 3, and 4). The LCM of these numbers is:
\[ \text{LCM}(5, 3, 4) = 60 \]

### Step 4: Multiply through by the LCM
Now, multiply the entire equation by 60 to eliminate the fractions:
\[ 60 \left( \frac{x}{5} \right) + 60 \left( \frac{x + 1}{3} \right) + 60 \left( \frac{x + 2}{4} \right) = 60 \times 40 \]
This simplifies to:
\[ 12x + 20(x + 1) + 15(x + 2) = 2400 \]

### Step 5: Expand and simplify the equation
Now, expand the equation:
\[ 12x + 20x + 20 + 15x + 30 = 2400 \]
Combine like terms:
\[ (12x + 20x + 15x) + (20 + 30) = 2400 \]
This gives us:
\[ 47x + 50 = 2400 \]

### Step 6: Isolate the variable
Now, isolate \( x \) by subtracting 50 from both sides:
\[ 47x = 2400 - 50 \]
This simplifies to:
\[ 47x = 2350 \]

### Step 7: Solve for \( x \)
Now, divide both sides by 47:
\[ x = \frac{2350}{47} \]
Calculating this gives:
\[ x = 50 \]

### Step 8: Find the three consecutive numbers
Now that we have \( x \), we can find the three consecutive numbers:
- First number: \( x = 50 \)
- Second number: \( x + 1 = 51 \)
- Third number: \( x + 2 = 52 \)

### Final Answer
The three consecutive whole numbers are **50, 51, and 52**.

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