Calculating Machine Output Over Time

This question asks us to determine how many tools a machine can cut in 25 hours, given its performance over a shorter period. This is a typical problem involving direct proportion, assuming the machine works at a constant rate.

In a direct proportion scenario, as the time increases, the number of tools cut also increases proportionally.

First, let's find the rate at which the machine cuts tools. The rate is the number of tools cut per unit of time (in this case, per hour).

• Given: Machine cuts 240 tools in 10 hours.
• Rate of cutting = $\frac{\text{Total number of tools cut}}{\text{Total time taken}}$

Using the given information:

Rate = $\frac{240 \text{ tools}}{10 \text{ hours}}$

Rate = 24 tools per hour

This means the machine cuts 24 tools every hour.

Now we need to find out how many tools the machine will cut in 25 hours. We use the rate we just calculated:

• Time = 25 hours
• Rate = 24 tools per hour
• Number of tools cut = Rate $\times$ Time

Plugging in the values:

Number of tools cut = $24 \text{ tools/hour} \times 25 \text{ hours}$

Let's perform the multiplication:

$24 \times 25$

$24 \times 25 = 600$

So, the machine will cut 600 tools in 25 hours.

Let's summarize the calculation steps:

Alternatively, we can set up a direct proportion equation. Let $T_1 = 10$ h, $N_1 = 240$ tools, $T_2 = 25$ h, and $N_2$ be the unknown number of tools. The direct proportion relationship is:

$\frac{N_1}{T_1} = \frac{N_2}{T_2}$

Substitute the known values into the equation:

$\frac{240}{10} = \frac{N_2}{25}$

Simplify the left side:

$24 = \frac{N_2}{25}$

Now, solve for $N_2$ by multiplying both sides by 25:

$N_2 = 24 \times 25$

$N_2 = 600$

Both methods confirm that the machine will cut 600 tools in 25 hours.

Revision Table: Understanding Direct Proportion

Additional Information: Proportional Reasoning

Proportional reasoning is a fundamental mathematical skill used to solve problems involving ratios and rates. It helps us understand how quantities change in relation to each other. In problems like the one we solved, identifying whether the relationship is directly proportional or inversely proportional is the first key step.

• In direct proportion, the quantities change in the same direction (both increase or both decrease).
• In inverse proportion, the quantities change in opposite directions (one increases while the other decreases). An example is speed and time for a fixed distance; increasing speed decreases the time taken.

For the machine cutting tools, assuming the machine's efficiency doesn't change, more time always means more tools cut, making it a direct proportion.