2 numbers added together equals 180 one number is 1/3rd of the other. what are the numbers?

Let's solve this problem step by step.

Let the two numbers be \( x \) and \( y \).

1. We are given that the sum of the two numbers is 180:
   \[
   x + y = 180
   \]

2. We are also told that one number is \( \frac{1}{3} \) of the other. Without loss of generality, let's assume \( x \) is \( \frac{1}{3} \) of \( y \):
   \[
   x = \frac{1}{3}y
   \]

3. Substitute \( x = \frac{1}{3}y \) into the first equation:
   \[
   \frac{1}{3}y + y = 180
   \]

4. Combine like terms:
   \[
   \frac{1}{3}y + \frac{3}{3}y = \frac{4}{3}y = 180
   \]

5. To solve for \( y \), multiply both sides by 3 to get rid of the fraction:
   \[
   4y = 540
   \]

6. Now, divide both sides by 4:
   \[
   y = 135
   \]

7. Substitute \( y = 135 \) back into the equation \( x = \frac{1}{3}y \) to find \( x \):
   \[
   x = \frac{1}{3} \times 135 = 45
   \]

So, the two numbers are \( 45 \) and \( 135 \).

Would you like further details or have any questions?

Here are 5 related questions to explore further:

1. What would the numbers be if the sum was 240 instead of 180?
2. How would the solution change if one number was 1/4th of the other?
3. What is the relationship between the numbers if the sum is negative?
4. How can you express the smaller number as a fraction of the larger number for different sums?
5. What if both numbers were integers and one was 2/3rd of the other?

**Tip:** When solving problems with fractions, always consider multiplying through by the denominator to simplify the equation.

Learn how to solve the problem where two numbers add up to 180 and one number is one-third of the other. Detailed step-by-step solution provided. Suitable for Grades 7-9.

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