Seven times a two -digit number is equal to four times the number obtained by reversing the order of its digits. If the difference between the digits is 3, find the number.

Options

A. 30
B. 32
C. 34
D. 36

Explanation

To solve the problem, we will follow these steps:

### Step 1: Define the two-digit number
Let the two-digit number be represented as \(10A + B\), where \(A\) is the tens digit and \(B\) is the units digit.

**Hint:** The two-digit number can be expressed in terms of its digits.

### Step 2: Write the equation for reversing the digits
When the digits are reversed, the number becomes \(10B + A\). According to the problem, seven times the original number is equal to four times the reversed number. Therefore, we can write the equation:

\[ 7(10A + B) = 4(10B + A) \]

**Hint:** Set up the equation based on the relationship given in the problem.

### Step 3: Expand the equation
Expanding both sides of the equation gives:

\[ 70A + 7B = 40B + 4A \]

**Hint:** Distribute the multiplication across the terms.

### Step 4: Rearrange the equation
Now, let's rearrange the equation to isolate terms involving \(A\) and \(B\):

\[ 70A - 4A = 40B - 7B \]
\[ 66A = 33B \]

**Hint:** Move all terms involving \(A\) to one side and those involving \(B\) to the other.

### Step 5: Simplify the equation
Dividing both sides by 33 gives us:

\[ 2A = B \]

**Hint:** Simplify the equation to express one variable in terms of the other.

### Step 6: Use the second condition
We also know from the problem that the difference between the digits is 3. This gives us the second equation:

\[ B - A = 3 \]

**Hint:** Write down the second equation based on the information provided.

### Step 7: Substitute \(B\) in the second equation
Now substitute \(B\) from the first equation into the second equation:

\[ (2A) - A = 3 \]
\[ A = 3 \]

**Hint:** Replace \(B\) with \(2A\) in the difference equation.

### Step 8: Find \(B\)
Now that we have \(A\), we can find \(B\):

\[ B = 2A = 2(3) = 6 \]

**Hint:** Use the value of \(A\) to calculate \(B\).

### Step 9: Form the two-digit number
The two-digit number is:

\[ 10A + B = 10(3) + 6 = 36 \]

**Hint:** Combine \(A\) and \(B\) to form the final two-digit number.

### Conclusion
Thus, the two-digit number is **36**.

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