One number is 2 times a first number. A third number is 100 more than the first number. If the sum of the three numbers is 312​, find the numbers.

Let's define the variables for the three numbers:

- Let the first number be \( x \).
- The second number is 2 times the first number, so it is \( 2x \).
- The third number is 100 more than the first number, so it is \( x + 100 \).

According to the problem, the sum of the three numbers is 312. Therefore, we can set up the following equation:

\[
x + 2x + (x + 100) = 312
\]

Now, let's simplify and solve for \( x \):

\[
x + 2x + x + 100 = 312
\]

\[
4x + 100 = 312
\]

Subtract 100 from both sides:

\[
4x = 212
\]

Now, divide both sides by 4:

\[
x = 53
\]

So, the first number is 53. Now, we can find the second and third numbers:

- The second number \( 2x = 2(53) = 106 \).
- The third number \( x + 100 = 53 + 100 = 153 \).

### The three numbers are:
- First number: 53
- Second number: 106
- Third number: 153

Would you like more details or have any questions?

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Here are 5 related questions to consider:

1. What would the numbers be if the third number was 200 more than the first number?
2. If the sum of the numbers was 400 instead of 312, what would the numbers be?
3. How would the solution change if the second number was 3 times the first number?
4. What would happen if the third number was 50 more than the first number?
5. Can you express the relationship between the three numbers as a function?

**Tip:** When solving word problems, always define your variables clearly and set up an equation based on the relationships given in the problem.

Learn how to solve a word problem involving three numbers with specific relationships. This problem covers basic arithmetic and equation solving techniques. Suitable for Grades 7-9 students.

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