To solve this, let's break down the problem into rates and proportions.

1. Calculate the rate of grading per teacher:

   • If 3 teachers grade 40 tests in 3 hours, the total grading done per hour by 3 teachers is: $\frac{40 \text{ tests}}{3 \text{ hours}} = \frac{40}{3} \approx 13.33 \text{ tests per hour}$
   • Therefore, each teacher grades: $\frac{13.33 \text{ tests}}{3} \approx 4.44 \text{ tests per hour}$

2. Calculate the time needed for 12 teachers to grade 480 tests:

   • The rate of grading by 12 teachers is: $12 \times 4.44 = 53.33 \text{ tests per hour}$
   • To find the time ($t$) it will take to grade 480 tests, use: $t = \frac{480 \text{ tests}}{53.33 \text{ tests per hour}} \approx 9 \text{ hours}$

So, it will take 9 hours for 12 teachers to grade 480 tests.

Would you like further explanation on any part of this solution?

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Related Questions:

1. What would the time be if there were only 10 teachers grading 480 tests?
2. How many tests could 15 teachers grade in 5 hours?
3. How many teachers are needed to grade 600 tests in 8 hours?
4. If the grading rate increased by 10%, how would that affect the time for 12 teachers to grade 480 tests?
5. How would the time change if only 6 teachers were grading 480 tests?

Tip:

When working with rate problems, find the unit rate per individual or item to easily scale up or down.