To find the number of bad light bulbs in a batch of 22,158 light bulbs, we can use the ratio provided: 3 out of every 268 light bulbs are bad.
First, we calculate the total number of bulbs produced (22,158) divided by the number of bulbs in the ratio (268):
\[ \frac{22158}{268} \approx 82.5 \]
Since we can't have a fraction of a batch, we take the whole number part of this division, which is 82. This means there are approximately 82 complete batches of 268 light bulbs in 22,158 light bulbs.
Next, we calculate the number of bad bulbs based on the number of batches:
\[ 82 \text{ batches} \times 3 \text{ bad bulbs per batch} = 246 \text{ bad bulbs} \]
Now, we check if there are any bad bulbs from the remaining bulbs outside of the complete batches. Since \( 82 \times 268 = 21976 \) bulbs are accounted for in complete batches, we calculate the remaining:
\[ 22158 - 21976 = 182 \text{ remaining bulbs} \]
Next, we find the proportion of bad bulbs among these remaining 182 bulbs using the same ratio (3 out of 268):
To find the expected number of bad bulbs in those 182 remaining bulbs:
\[ \text{Proportion of bad bulbs} = \frac{3}{268} \]
Now we calculate the expected number of bad bulbs in the remaining 182:
\[ \frac{3}{268} \times 182 \approx 2.042 \]
Since we can't have a fraction of a bulb, we round to the nearest whole number, which is 2.
Finally, we add the two numbers of bad bulbs together:
\[ 246 \text{ (from complete batches)} + 2 \text{ (from remaining bulbs)} = 248 \text{ bad bulbs} \]
Thus, the total number of bad light bulbs in the batch of 22,158 light bulbs is 248.