The units digit of a two-digit number exceeds the tens digit by 2. Find the number if it is 4 times the sum of its digits

Let the two-digit number be represented as \( 10t + u \), where \( t \) is the tens digit and \( u \) is the units digit.

### Step 1: Setting up the first condition
The problem states that "the units digit exceeds the tens digit by 2." Therefore, we have the equation:

\[
u = t + 2
\]

### Step 2: Setting up the second condition
The problem also states that "the number is 4 times the sum of its digits." The sum of the digits is \( t + u \), and the number itself is \( 10t + u \). Therefore, the second equation is:

\[
10t + u = 4(t + u)
\]

### Step 3: Substituting the first equation into the second
Substitute \( u = t + 2 \) into the second equation:

\[
10t + (t + 2) = 4(t + (t + 2))
\]

Simplify both sides:

\[
11t + 2 = 4(2t + 2)
\]

\[
11t + 2 = 8t + 8
\]

### Step 4: Solving for \( t \)
Now, solve for \( t \) by isolating it on one side of the equation:

\[
11t - 8t = 8 - 2
\]

\[
3t = 6
\]

\[
t = 2
\]

### Step 5: Finding \( u \)
Now that we know \( t = 2 \), use the first equation to find \( u \):

\[
u = t + 2 = 2 + 2 = 4
\]

### Step 6: Verifying the solution
The number is \( 10t + u = 10(2) + 4 = 24 \).

The sum of the digits is \( 2 + 4 = 6 \), and four times the sum of the digits is \( 4 \times 6 = 24 \), which matches the number itself.

Thus, the number is \( \boxed{24} \).

---

Would you like further details or have any questions?

Here are 5 related questions you might explore:
1. What if the units digit exceeded the tens digit by 3 instead of 2?
2. How can this approach be generalized to solve similar problems with different relationships between the digits?
3. What if the number was 5 times the sum of its digits instead of 4 times?
4. How can we solve this type of problem when given a three-digit number?
5. How does this problem relate to linear equations and algebraic techniques?

**Tip:** When solving word problems, clearly translate the conditions into equations to avoid confusion and simplify the solution process.

The units digit of a two-digit number exceeds the tens digit by 2. Find the number if it is 4 times the sum of its digits.

Learn how to solve a two-digit number problem where the units digit exceeds the tens digit by 2, and the number is 4 times the sum of its digits. This problem covers key algebra concepts and is suitable for students in Grades 6-8.

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