Photo Printing 471255

1. **State the problem:** We need to find the time it takes for a printer to print a given number of photos, knowing that printing $6$ photos takes $2$ minutes. 2. **Identify the formula:** The time taken is proportional to the number of photos printed. We can write the formula as: $$\text{Time} = k \times \text{Number of photos}$$ where $k$ is the constant of proportionality. 3. **Find the constant $k$:** Given $6$ photos take $2$ minutes, $$2 = k \times 6$$ Dividing both sides by $6$: $$k = \frac{2}{6} = \frac{\cancel{2}}{\cancel{6}} = \frac{1}{3}$$ So, $k = \frac{1}{3}$ minutes per photo. 4. **Calculate the time for each number of photos:** - For $6$ photos: $$\text{Time} = \frac{1}{3} \times 6 = 2 \text{ minutes}$$ - For $4$ photos: $$\text{Time} = \frac{1}{3} \times 4 = \frac{4}{3} = 1.33 \text{ minutes}$$ - For $15$ photos: $$\text{Time} = \frac{1}{3} \times 15 = 5 \text{ minutes}$$ - For $24$ photos: $$\text{Time} = \frac{1}{3} \times 24 = 8 \text{ minutes}$$ - For $27$ photos: $$\text{Time} = \frac{1}{3} \times 27 = 9 \text{ minutes}$$ 5. **Compare with given times:** - $6$ photos: $2$ minutes (matches given) - $4$ photos: calculated $1.33$ minutes, given $4$ minutes (given value seems incorrect based on proportionality) - $15$ photos: $5$ minutes (matches given) - $24$ photos: $8$ minutes (matches given) - $27$ photos: $9$ minutes (matches given) 6. **Conclusion:** The time for $4$ photos should be approximately $1.33$ minutes, not $4$ minutes, if the printer speed is consistent. The other values match the proportional relationship. **Final table with corrected time for $4$ photos:** | Number of photos | 6 | 4 | 15 | 24 | 27 | |------------------|---|---|----|----|----| | Time (minutes) | 2 | 1.33 | 5 | 8 | 9 |