Question
Diego and Stefan each want to run for president of their school's student body council. In order to do so, they must collect a certain number of signatures and get a nomination. So far, Diego has 20 signatures, and Stefan has 10. Diego is collecting signatures at an average rate of 3 per day, whereas Stefan is averaging 4 signatures every day. Assuming that their rate of collection stays the same, eventually the two will have collected the same number of signatures. How long will that take?
Write a system of equations, graph them, and type the solution.
Write a system of equations, graph them, and type the solution.
Answers
GPT-4o mini
To approach the problem, we'll define two variables to represent the number of days they collect signatures:
- Let \( d \) be the number of days until they have the same number of signatures.
- Let \( D \) be the total number of signatures Diego has after \( d \) days.
- Let \( S \) be the total number of signatures Stefan has after \( d \) days.
**Step 1: Set up the equations**
Given that:
- Diego starts with 20 signatures and collects 3 signatures per day, the equation for Diego's total signatures can be expressed as:
\[
D = 20 + 3d
\]
- Stefan starts with 10 signatures and collects 4 signatures per day, the equation for Stefan's total signatures is:
\[
S = 10 + 4d
\]
**Step 2: Set the equations equal to each other**
To find when they will have the same number of signatures, we set \( D \) equal to \( S \):
\[
20 + 3d = 10 + 4d
\]
**Step 3: Solve for \( d \)**
Rearranging the equation gives:
\[
20 - 10 = 4d - 3d
\]
\[
10 = d
\]
So, \( d = 10 \). This means it will take 10 days for Diego and Stefan to have the same number of signatures.
**Step 4: Check the solution**
Now let's calculate the number of signatures each will have after 10 days:
- For Diego:
\[
D = 20 + 3(10) = 20 + 30 = 50
\]
- For Stefan:
\[
S = 10 + 4(10) = 10 + 40 = 50
\]
Both Diego and Stefan will have 50 signatures after 10 days.
**Conclusion**
It will take **10 days** for Diego and Stefan to have the same number of signatures.
**Summary of the system of equations:**
\[
\begin{align*}
D & = 20 + 3d \\
S & = 10 + 4d \\
\end{align*}
\]
With the solution \( d = 10 \).
- Let \( d \) be the number of days until they have the same number of signatures.
- Let \( D \) be the total number of signatures Diego has after \( d \) days.
- Let \( S \) be the total number of signatures Stefan has after \( d \) days.
**Step 1: Set up the equations**
Given that:
- Diego starts with 20 signatures and collects 3 signatures per day, the equation for Diego's total signatures can be expressed as:
\[
D = 20 + 3d
\]
- Stefan starts with 10 signatures and collects 4 signatures per day, the equation for Stefan's total signatures is:
\[
S = 10 + 4d
\]
**Step 2: Set the equations equal to each other**
To find when they will have the same number of signatures, we set \( D \) equal to \( S \):
\[
20 + 3d = 10 + 4d
\]
**Step 3: Solve for \( d \)**
Rearranging the equation gives:
\[
20 - 10 = 4d - 3d
\]
\[
10 = d
\]
So, \( d = 10 \). This means it will take 10 days for Diego and Stefan to have the same number of signatures.
**Step 4: Check the solution**
Now let's calculate the number of signatures each will have after 10 days:
- For Diego:
\[
D = 20 + 3(10) = 20 + 30 = 50
\]
- For Stefan:
\[
S = 10 + 4(10) = 10 + 40 = 50
\]
Both Diego and Stefan will have 50 signatures after 10 days.
**Conclusion**
It will take **10 days** for Diego and Stefan to have the same number of signatures.
**Summary of the system of equations:**
\[
\begin{align*}
D & = 20 + 3d \\
S & = 10 + 4d \\
\end{align*}
\]
With the solution \( d = 10 \).