5e+15p</=60
How many combinations of excersises (e) and problems (p)can you do in exactly one hour(60 minutes)?
I've found 5 combinations:
9e,1p
3e,3p
6e,2p
0e,4p,
12e,0p
Are there anymore? What method would you use to solve this type of problem? Thanks for the help!
graph it
e on x axis
p on y axis
e from 0 to 60/5 = 12
p from 0 to 60/15 = 4
draw line from (0,4) down to (12,0)
everything between the origin and the line in the first quadrant will do, including on the line. Just fill in the grid there.
you have them, e,p are integer solutions
when p = 0, 13 points (0,0)to (12,0)
when p = 1, 10 points (0,1) to (9,1)
when p = 3, 4 points (0,2) to (3,3)
when p = 4, 1 point (0,4)
13 + 10 + 4 + 1 = 28
Oh, sorry, I thought you wanted all integer solutions that satisfied the inequality. The ones you listed are exactly on the line.
I left out p = 2 line
(0, 2) to (6,2), seven more
28 + 7 = 35 total solutions for less than an hour if all are integer.
To find all possible combinations of exercises (e) and problems (p) that can be done in exactly one hour (60 minutes) given the inequality 5e + 15p ≤ 60, we can use a systematic approach.
First, let's analyze the given inequality equation:
5e + 15p ≤ 60
Since we know that both e and p have to be non-negative values, we can start by finding the maximum value of e that satisfies the inequality when p is 0. By substituting p = 0 into the inequality equation, we get:
5e + 15(0) ≤ 60
5e ≤ 60
e ≤ 12
So, the maximum value for e is 12.
Now, let's consider the maximum value of p that satisfies the inequality when e is 0. By substituting e = 0 into the inequality equation, we get:
5(0) + 15p ≤ 60
15p ≤ 60
p ≤ 4
So, the maximum value for p is 4.
Based on these maximum values, we can create a table to list all possible combinations of e and p within their respective ranges:
| e | p |
|----|----|
| 0 | 4 |
| 1 | 3 |
| 2 | 2 |
| 3 | 1 |
| 4 | 0 |
| 5 | NA |
| 6 | NA |
| 7 | NA |
| 8 | NA |
| 9 | NA |
| 10 | NA |
| 11 | NA |
| 12 | NA |
From the table, we can see that there are indeed more combinations:
- 5e, 15p: (1, 3), (2, 2), (3, 1), (4, 0)
- Since we are considering whole numbers, we cannot have partial exercises or problems. So values like (5, 0.5) are not considered valid in this case.
To find these combinations using a methodical approach, you can create a loop for e and p within their respective maximum ranges (e ≤ 12 and p ≤ 4) and check if the equation 5e + 15p = 60 holds true. If it does, you can add that combination to your list.
Overall, the strategy to solve this type of problem is to set up the maximum ranges for the variables involved, and then systematically test each combination within those ranges to determine which ones satisfy the given conditions or equations.