Math  /  Algebra

QuestionA school office has an ink-jet printer and a laser printer. The ink-jet printer takes 10 seconds to start and then prints 20 pages per minute. The laser printer takes 5 seconds to start and then prints 30 pages per minute.
A teacher prints a document that takes 70 seconds to print on the ink-jet printer. How long would it take to print on the laser printer? \square seconds

Studdy Solution

STEP 1

What is this asking? If a document takes 70 seconds to print on a slow printer, how long will it take on a fast printer? Watch out! Don't forget about the start-up times for each printer!

STEP 2

1. Ink-jet printing time
2. Number of pages
3. Laser printing time

STEP 3

We know the **total printing time** on the ink-jet printer is **70 seconds**.
The printer takes **10 seconds** to warm up.

STEP 4

So, the **actual printing time** (without warm-up) is 7010=60\textbf{70} - \textbf{10} = \textbf{60} **seconds**.

STEP 5

The ink-jet printer prints **20 pages** per minute, which is **60 seconds**.
Let's figure out how many pages were printed in those **60 seconds**.

STEP 6

Since the **printing rate** is **20 pages per 60 seconds**, that means it prints 20 pages60 seconds=13\frac{\textbf{20} \text{ pages}}{\textbf{60} \text{ seconds}} = \frac{1}{3} **page per second**.

STEP 7

The document printed for **60 seconds**, so the **total number of pages** is 60 seconds13pagessecond=20\textbf{60} \text{ seconds} \cdot \frac{1}{3} \frac{\text{pages}}{\text{second}} = \textbf{20} **pages**!

STEP 8

Now, let's see how long the laser printer takes to print those same **20 pages**.
The laser printer prints **30 pages** per minute, which is also **60 seconds**.

STEP 9

That means the laser printer prints at a rate of 30 pages60 seconds=12\frac{\textbf{30} \text{ pages}}{\textbf{60} \text{ seconds}} = \frac{1}{2} **page per second**.

STEP 10

To print **20 pages**, the laser printer needs 20 pages12pagessecond=20 pages2secondspage=40\frac{\textbf{20} \text{ pages}}{\frac{1}{2} \frac{\text{pages}}{\text{second}}} = \textbf{20} \text{ pages} \cdot 2 \frac{\text{seconds}}{\text{page}} = \textbf{40} **seconds**.

STEP 11

Don't forget, the laser printer also has a **start-up time** of **5 seconds**.
So, the **total printing time** on the laser printer is 40+5=45\textbf{40} + \textbf{5} = \textbf{45} **seconds**.

STEP 12

It would take **45 seconds** to print the document on the laser printer.

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