To find the number of small white magnets, let's go through the problem step-by-step using the given information and forming equations.
1. Number of Magnets:
- There are 36 small magnets.
- The number of large magnets is [tex]\(\frac{1}{2}\)[/tex] the number of small magnets.
2. Calculation for Large Magnets:
[tex]\[
\text{Number of large magnets} = \frac{1}{2} \times 36 = 18
\][/tex]
3. Distribution Based on Color:
- There are 5 times as many white magnets as red magnets in total.
- There are 8 large red magnets.
4. Calculation for Large White Magnets:
[tex]\[
\text{Total large magnets} = 18
\][/tex]
[tex]\[
\text{Large white magnets} = 18 - 8 = 10
\][/tex]
5. Total Red Magnets:
- Let [tex]\(r\)[/tex] represent the total number of red magnets.
- Let [tex]\(w\)[/tex] (which is 5 times [tex]\(r\)[/tex]) represent the total number of white magnets.
6. Forming the Equation:
- From the problem: [tex]\(w = 5r\)[/tex]
- The number of red magnets:
[tex]\[
r = \text{small red magnets} + \text{large red magnets}
\][/tex]
- The number of white magnets:
[tex]\[
w = \text{small white magnets} + \text{large white magnets}
\][/tex]
- Therefore:
[tex]\[
w = \text{small white magnets} + 10
\][/tex]
7. Solving for Red and White Magnets:
[tex]\[
r = 8 + \text{small red magnets}
\][/tex]
[tex]\[
w = 5r \Rightarrow w = 5(8 + \text{small red magnets})
\][/tex]
8. Total Small Magnets:
[tex]\[
36 = \text{small red magnets} + \text{small white magnets}
\][/tex]
9. Forming Equations:
\begin{align}
r &= 8 + \text{small red magnets} \\
5r &= \text{small white magnets} + 10 \\
\end{align}
10. Expressing Variables:
[tex]\[
\text{small white magnets} = 5r - 10
\][/tex]
11. Solving the System:
[tex]\[
r = 8 + \text{small red magnets}
\][/tex]
[tex]\[
36 = \text{small red magnets} + \text{small white magnets}
\][/tex]
12. Substitute [tex]\(r\)[/tex] in [tex]\(36 = \text{small red magnets} + 5r - 10\)[/tex]:
[tex]\[
r = -10
\][/tex]
13. Calculating Small Magnets:
[tex]\(
\text{small red magnets} + \text{small white magnets} = 36
\)[/tex]
[tex]\(
\text{small white magnets} = -60
\)[/tex]
So, the number of small white magnets is [tex]\(-60\)[/tex].
Thus, following these steps given in the problem, we determine that such a scenario is numerically not possible because the number of magnets should be a positive number, indicating perhaps some assumptions or given constraints need revisiting, but based on the given data the solution stays as described above.
