Jeff starts at 0 and counts by 3’s: 0, 3, 6, 9, 12, ... At the same time and at the same speed, Hurudaya counts backwards from 2024 by fives: 2024, 2019, 2014, ... Find the one number that both Jeff and Hurudaya count at the same time.

Let's analyze the situation for Jeff and Hurudaya:

- **Jeff** counts by 3's starting from 0. His sequence is:
  \[
  0, 3, 6, 9, 12, \dots
  \]
  This sequence can be expressed as:
  \[
  J(n) = 3n
  \]
  where \(n\) is the number of steps he takes.

- **Hurudaya** counts backwards by 5's starting from 2024. His sequence is:
  \[
  2024, 2019, 2014, 2009, \dots
  \]
  This sequence can be expressed as:
  \[
  H(n) = 2024 - 5n
  \]
  where \(n\) is the number of steps Hurudaya takes.

### Finding the Common Number
We need to find the step \(n\) at which both Jeff and Hurudaya count the same number. This means we are looking for \(n\) such that:
\[
3n = 2024 - 5n
\]
Now, let's solve for \(n\).

1. Start by adding \(5n\) to both sides:
   \[
   3n + 5n = 2024
   \]
   \[
   8n = 2024
   \]
2. Divide both sides by 8:
   \[
   n = \frac{2024}{8} = 253
   \]

### Common Number
Now that we have \(n = 253\), substitute this back into Jeff's sequence to find the common number:
\[
J(253) = 3 \times 253 = 759
\]

Thus, the number that both Jeff and Hurudaya count at the same time is **759**.

Would you like further details on this process or have any other questions?

Here are 5 related questions to expand your understanding:
1. What if Jeff counts by 4's and Hurudaya counts by 7's? How would the common number change?
2. How many steps would it take for them to count the same number if Hurudaya started at a different number?
3. How would the common number change if Jeff started at a number other than 0?
4. What if Hurudaya counted forwards instead of backwards?
5. How would the situation change if both counted at different speeds?

**Tip:** When solving for a common number between two sequences, try expressing both in terms of \(n\), then solve for \(n\) by equating the two expressions!

Explore how Jeff, counting by 3's, and Hurudaya, counting backwards by 5's from 2024, find a common number in their sequences. Learn key concepts like arithmetic sequences and linear equations with step-by-step explanations. Suitable for Grades 6-8.

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