To determine how many medium mirrors were broken, let's follow the steps one by one:
1. Given Information:
- Total number of mirrors: [tex]\(480\)[/tex]
- Total number of broken mirrors is derived from summing broken mirrors in small and large categories.
- Total number of not broken mirrors: [tex]\(402\)[/tex]
- Small mirrors: [tex]\(102\)[/tex] in total
- Broken small mirrors: [tex]\(4\)[/tex]
- Not broken small mirrors: [tex]\(102 - 4 = 98\)[/tex]
- Large mirrors: [tex]\(214\)[/tex] in total
- Broken large mirrors: [tex]\(6\)[/tex]
2. Calculate the remaining not broken mirrors:
- Given that the total not broken mirrors [tex]\(= 402\)[/tex]
- Already known:
- Not broken small mirrors: [tex]\(98\)[/tex]
- Not broken large mirrors (we derived that they remain not mentioned for simplicity): [tex]\(214 - 6 = 208\)[/tex]
3. Calculate the not broken medium mirrors:
- Not broken medium mirrors must cover the gap left after counted small and large not broken mirrors:
[tex]\[
\text{Total not broken mirrors} = \text{Not broken small mirrors} + \text{Not broken medium mirrors} + \text{Not broken large mirrors}
\][/tex]
[tex]\[
402 = 98 + \text{Not broken medium mirrors} + 208
\][/tex]
[tex]\[
\text{Not broken medium mirrors} = 402 - 98 - 208
\][/tex]
[tex]\[
\text{Not broken medium mirrors} = 96
\][/tex]
4. Calculate the total number of broken mirrors:
- Total mirrors: [tex]\(480\)[/tex]
- Total not broken mirrors: [tex]\(402\)[/tex]
- Therefore, total broken mirrors:
[tex]\[
\text{Total broken mirrors} = 480 - 402 = 78
\][/tex]
5. Determine the number of broken medium mirrors:
- Total determined broken mirrors found (78)
- Already known:
- Broken small mirrors: [tex]\(4\)[/tex]
- Broken large mirrors: [tex]\(6\)[/tex]
- Total already known broken mirrors:
[tex]\[
\text {Known broken mirrors} (small + large) = 4 + 6 = 10
\][/tex]
- Therefore broken mirrors that must be in the medium category:
[tex]\(
\text{Broken medium mirrors} = 78 - 10 = 68
\)[/tex]
So, the number of broken medium mirrors is [tex]\( 68 \)[/tex].
