In a two-digit number, the units digit is 3 more than the ten's digit. The sum of the digits is 18 less than the original number. Find the product of the digits

Options

A. 54
B. 40
C. 10
D. 28

Explanation

To solve the problem step by step, let's break it down:

### Step 1: Define the variables
Let the ten's digit of the two-digit number be \( x \). According to the problem, the unit's digit is 3 more than the ten's digit. Therefore, we can express the unit's digit as:
\[ \text{Unit's digit} = x + 3 \]

### Step 2: Write the original number
The original two-digit number can be expressed in terms of \( x \):
\[ \text{Original number} = 10x + (x + 3) = 10x + x + 3 = 11x + 3 \]

### Step 3: Set up the equation based on the sum of the digits
The problem states that the sum of the digits is 18 less than the original number. The sum of the digits is:
\[ \text{Sum of digits} = x + (x + 3) = 2x + 3 \]

According to the problem, we can set up the following equation:
\[ 2x + 3 = (11x + 3) - 18 \]

### Step 4: Simplify the equation
Now, simplify the right side of the equation:
\[ 2x + 3 = 11x + 3 - 18 \]
\[ 2x + 3 = 11x - 15 \]

### Step 5: Rearrange the equation
Next, we will rearrange the equation to isolate \( x \):
\[ 2x + 3 + 15 = 11x \]
\[ 2x + 18 = 11x \]

### Step 6: Solve for \( x \)
Now, subtract \( 2x \) from both sides:
\[ 18 = 11x - 2x \]
\[ 18 = 9x \]
\[ x = 2 \]

### Step 7: Find the unit's digit
Now that we have \( x \), we can find the unit's digit:
\[ \text{Unit's digit} = x + 3 = 2 + 3 = 5 \]

### Step 8: Find the product of the digits
Now we can find the product of the digits:
\[ \text{Product} = x \times (x + 3) = 2 \times 5 = 10 \]

### Final Answer
The product of the digits is \( 10 \).

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