if a photocopying machine can make 40 copies in one minute, how many copies can it make in 8.4 minutes, assuming that it operates at the same rate? solve it using ratios

To solve this problem using ratios, we can set up a proportion based on the given information.

### Step 1: Set up the ratio
The photocopying machine can make 40 copies in 1 minute. We need to find out how many copies it can make in 8.4 minutes. Let's denote the number of copies as \( x \).

The ratio of time to the number of copies is the same in both cases:
\[
\frac{1 \text{ minute}}{40 \text{ copies}} = \frac{8.4 \text{ minutes}}{x \text{ copies}}
\]

### Step 2: Solve the proportion
To find \( x \), cross-multiply the terms:
\[
1 \times x = 8.4 \times 40
\]
\[
x = 8.4 \times 40
\]

Now, multiply the numbers:
\[
x = 336 \text{ copies}
\]

### Conclusion
In 8.4 minutes, the photocopying machine can make 336 copies.

Would you like further details or have any questions?

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Here are some related questions:
1. How many copies can the machine make in 15 minutes?
2. What if the machine's speed increased to 50 copies per minute?
3. How long would it take to make 500 copies?
4. If the machine had a 2-minute break every 10 minutes, how many copies would it make in 20 minutes?
5. How many copies could it make in half an hour?

**Tip:** When solving ratio problems, always ensure that the units on both sides of the proportion match for accurate results.

Learn how to calculate the number of copies made by a photocopier in 8.4 minutes using ratios. This problem involves understanding ratio and proportion concepts in mathematics. Suitable for students in Grades 6-8.

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