it takes an HP laserjet 1300 printer 10 minute longer to complete a 600 page print job by itself than it takes an HP laserjet 2420 to complete the same job itself. Together the two printers can complete the job in 12 min. How long does it takes for each printer to do the job alone? What is the speed in pages per min of each printer?
If the 2420 does it in x minutes, then
1/12 = 1/x + 1/(x+10)
x = 20
thanks steve
To solve this problem, let's assign variables:
- Let's call the time it takes for the HP LaserJet 1300 printer to complete the job alone as 'x' minutes
- Let's call the time it takes for the HP LaserJet 2420 printer to complete the job alone as 'y' minutes
We are given that it takes the HP LaserJet 1300 printer 10 minutes longer to complete the 600-page print job than it takes the HP LaserJet 2420 printer. This can be represented as:
x = y + 10
We are also given that together, the two printers can complete the job in 12 minutes. This means that their combined work rate is equivalent to one job completed in 12 minutes. The combined work rate is the sum of the individual work rates, which is equal to the reciprocal of the time taken for each printer. This can be represented as:
1/x + 1/y = 1/12
Now, we have two equations with two variables. We can solve these equations simultaneously to find the values of 'x' and 'y', which represent the time taken for each printer to complete the job alone.
First, let's rearrange the first equation:
x - y = 10
Now, let's multiply the second equation by '12xy' to eliminate the fractions:
12y + 12x = xy
Next, let's substitute 'x' from the first equation into the second equation:
12y + 12(x - 10) = y(x)
Expanding the equation:
12y + 12x - 120 = xy
Rearranging the equation:
xy - 12y - 12x = -120
Combining like terms:
xy - 12y - 12x + 144 = 24
Now, let's rearrange the equation:
xy - 12y - 12x + 120 = 0
Using the quadratic formula:
y = (12 ± sqrt((-12)^2 - 4 * 1 * (x) * (120))) / (2 * x)
Simplifying further:
y = (12 ± sqrt(144 - 480x)) / (2x)
Now, let's find the values of 'x' and 'y' that satisfy the conditions of the problem.
By observing the problem, we can determine that the values of 'x' and 'y' are both positive. Let's consider the positive root:
y = (12 + sqrt(144 - 480x)) / (2x)
Now, we can solve this equation using trial and error, plugging in different values for 'x' and finding the corresponding values of 'y'. We need to find values of 'x' and 'y' that satisfy both equations.
For example, if we assume 'x' = 20:
y = (12 + sqrt(144 - (480 * 20))) / (2 * 20)
y = (12 + sqrt(144 - 9600)) / 40
y = (12 + sqrt(-9456)) / 40
However, we can see that this solution is not possible as the value inside the square root is negative (-9456), which would result in a complex number.
Continuing this trial and error process, we can find the values of 'x' and 'y' that satisfy both equations.