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:
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Diego starts with 20 signatures and collects 3 signatures per day, the equation for Diego's total signatures can be expressed as:
\[ D = 20 + 3d \]
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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:
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For Diego: \[ D = 20 + 3(10) = 20 + 30 = 50 \]
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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 \( d = 10 \).