Buying paint in large quantities is often less expensive. Five 1-gallon buckets cost 25% more than a single 5-gallon bucket. Suppose that a job requires g gallons of paint. For what values of g should a 5-gallon bucket be purchased to minimize cost? Explain.
Well, finding the perfect amount of paint for a job requires a delicate balance, or else you might end up with more leftover paint than Picasso's entire collection! Now, let's break this down.
Let's say the cost of a single 5-gallon bucket is x dollars. If five 1-gallon buckets cost 25% more, that means they would cost 1.25 * x dollars each.
So, for five 1-gallon buckets, we would be paying 5 * (1.25 * x) dollars, which simplifies to 6.25 * x dollars. On the other hand, buying a single 5-gallon bucket would cost us x dollars.
Now, assuming we need g gallons of paint, let's compare the costs for both scenarios:
If we buy g gallons using 5 1-gallon buckets, the cost would be (6.25 * x) * (g/1) dollars, which simplifies to 6.25 * g * x dollars.
If we buy g gallons using a single 5-gallon bucket, the cost would be x * (g/5) dollars, which simplifies to (g/5) * x dollars.
To minimize the cost, we need to find the values of g where buying a 5-gallon bucket is cheaper. In other words, we need to find the values of g where (g/5) * x is less than 6.25 * g * x.
Dividing both sides of the inequality by x, we get g/5 < 6.25 * g, which further simplifies to g < 31.25 * g.
Now, here's the catch. We can't have a value of g where g is less than 0, because negative paint doesn't really exist (unless you're in some sort of art experiment).
So, the values of g where a 5-gallon bucket should be purchased to minimize cost would be any positive value of g less than 31.25.
Hope that paints a clear picture for you!
To determine the values of g for which a 5-gallon bucket should be purchased to minimize cost, we can set up an equation based on the given information.
Let's assume the cost of the single 5-gallon bucket is x dollars. Since five 1-gallon buckets cost 25% more, the cost of five 1-gallon buckets will be 1.25x dollars.
Now, let's consider the amount of paint needed for the job, which is g gallons. We can compare the cost of purchasing five 1-gallon buckets versus a single 5-gallon bucket.
For the five 1-gallon buckets, the cost per gallon is (1.25x)/(5) dollars.
For the single 5-gallon bucket, the cost per gallon is x/5 dollars.
To minimize cost, we want the cost per gallon of the 5-gallon bucket to be less than the cost per gallon of the five 1-gallon buckets.
Therefore, we can set up the inequality:
x/5 < (1.25x)/(5)
To simplify, we multiply both sides of the inequality by 5:
x < 1.25x
Now, we can subtract x from both sides:
0 < 0.25x
Dividing by 0.25:
0 < x
This means that any positive value of x will satisfy the inequality.
In the context of the problem, x represents the cost of a single 5-gallon bucket. Since the cost must be positive, it means that the 5-gallon bucket should be purchased for any value of g that requires paint.
To find the values of g for which purchasing a 5-gallon bucket minimizes cost, we need to compare the cost of buying five 1-gallon buckets with the cost of buying a single 5-gallon bucket.
Let's assume the cost of a single 5-gallon bucket is C dollars. Therefore, the cost of five 1-gallon buckets, which costs 25% more than a single 5-gallon bucket, can be expressed as:
Cost of five 1-gallon buckets = C + 0.25C = 1.25C
Now, let's analyze the cost per gallon in both scenarios.
For the five 1-gallon buckets, the cost per gallon is:
Cost per gallon of five 1-gallon buckets = (Cost of five 1-gallon buckets) / (total gallons)
Since each 1-gallon bucket contains 1 gallon of paint, the total gallons in the case of five 1-gallon buckets would be 5. Therefore:
Cost per gallon of five 1-gallon buckets = (1.25C) / 5 = 0.25C
For the 5-gallon bucket, the cost per gallon is simply:
Cost per gallon of a 5-gallon bucket = C / 5
In order to minimize cost, we need to find the range of values for g (in gallons) where the cost per gallon of a 5-gallon bucket is cheaper than the cost per gallon of five 1-gallon buckets.
Therefore, we have the following inequality:
Cost per gallon of a 5-gallon bucket < Cost per gallon of five 1-gallon buckets
C / 5 < 0.25C
Now, let's solve the inequality:
C / 5 < 0.25C
Multiply both sides of the inequality by 5 to eliminate the denominator:
C < 1.25C
Now, subtract C from both sides to isolate the variable:
0 < 0.25C
Divide both sides by 0.25 to solve for C:
0 / 0.25 < C
0 < C
This tells us that the cost of a single 5-gallon bucket should be greater than zero dollars. In other words, a 5-gallon bucket should always be purchased when the cost per gallon is positive.
Since a 5-gallon bucket is more cost-effective than five 1-gallon buckets for any positive value of C, there is no specific range of values for g. A 5-gallon bucket should be purchased for any amount of gallons required to complete the job.
well, a 5-gallon bucket costs the same as 4 1-gallon cans.
So, if g <= 4, buy cans.
If 4<g<=5, but the bucket.
Naturally, if g > 5, buy enough buckets so that the remainder is less than 5. Then apply the above rule for the remainder.