Problem 47 - Elementary Algebra

Question

A printer prints 30 pages in 1 minute. How long will it take to print a 720 -page booklet?

Step-by-Step Solution

Answer

It will take 24 minutes to print.

Step 1 - Determine the Printing Rate

First, identify that the printer can print 30 pages per minute.

Step 2 - Set Up the Equation

To find out how long it will take to print 720 pages, set up the equation where the number of pages (720) is equal to the rate of printing (30 pages per minute) times the time in minutes.This gives the equation: \[ 720 = 30 \times t \]

Step 3 - Solve for Time

Solve the equation for the time \( t \). Divide both sides of the equation by 30 to isolate \( t \):\[ t = \frac{720}{30} \]

Step 4 - Calculate the Time

Calculate \( \frac{720}{30} \) to find the time.\( \frac{720}{30} = 24 \).Therefore, it will take 24 minutes to print the booklet.

Key Concepts

Understanding Rate of Work

When we talk about rate of work, we are looking at how quickly something can be done. In the case of a printer, this means how many pages it can print in a specific amount of time, like a minute. Let's say a printer can produce 30 pages every minute. This is the printer's work rate. Rates are helpful because they give us a standard measurement that can be used to calculate outputs, like how long a task will take if you know how fast the work is being done. If you ever use a different machine, this same understanding can help you determine how quickly it works based on its rate of output.

• Rate of work = Output per unit of time
• In this problem: 30 pages per minute

To find out how long it will take to do a different amount of work (like printing a 720-page booklet), you can multiply the rate of work by the amount of time to cover the total output needed.

Solving Equations for Time

To find out how much time is needed to complete a task given a rate of work, you can set up an equation. The basic form of this equation is:

1. Total amount of work = Rate of work × Time
2. In our example: 720 pages = 30 pages per minute × Time (in minutes)

Now, to solve for time, you need to find the value of the variable representing time, which we'll call \( t \). Our equation is:\[ 720 = 30 \times t \]To isolate the time variable, we perform the inverse operation of multiplication on both sides of the equation. That means dividing both sides by the rate (30 pages per minute):\[ t = \frac{720}{30} \]Solving this equation tells us the amount of time the printer will need to finish the 720-page booklet.

Applying Division to Find Solutions

Division is a mathematical operation that is very useful when it comes to solving equations like the one we encountered. When the rate and the quantity of work are known, division helps find the unknown variable, which in this case is time.Let's break down the division used in our problem:

• Total pages we need to print: 720
• Pages printer can print per minute: 30

So each division operation tells us how many units of time it will take to complete a task. To find out how many 30-page batches fit into 720 pages, we perform the division \( \frac{720}{30} \).Calculating this, we find \( 24 \), meaning the printer will take 24 minutes to print the entire booklet. This understanding of division to find an unknown variable is a cornerstone of algebra and solving unit rate problems.