Lakshani has $5 to spend on pens and pencils. Each pen costs $0.50 and each pencil costs $0.10. She is thinking about buying 6 pens and 20 pencils. The last pen would add five times as much to her total utility as the last pencil.
and your question is?
Sra
To find out the total cost of pens and pencils, we can multiply the number of pens by the cost of each pen and the number of pencils by the cost of each pencil.
The cost of 6 pens can be calculated as:
6 pens * $0.50 per pen = $3
The cost of 20 pencils can be calculated as:
20 pencils * $0.10 per pencil = $2
The total cost of pens and pencils is:
$3 (pens) + $2 (pencils) = $5
Now, let's define the utility of the last pen as UPen and the utility of the last pencil as UPencil. We know that the last pen adds five times as much to the total utility as the last pencil. Mathematically, this can be represented as:
UPen = 5 * UPencil --- (Equation 1)
Given that Lakshani has $5 to spend, we can also express this in terms of money:
UPen * $0.50 (cost of the last pen) + UPencil * $0.10 (cost of the last pencil) = $5
Substituting the value of UPen from Equation 1 into the above equation:
5 * UPencil * $0.50 (cost of the last pen) + UPencil * $0.10 (cost of the last pencil) = $5
Simplifying the equation:
2.5 * UPencil + 0.1 * UPencil = $5
Combining like terms:
2.6 * UPencil = $5
Now, let's solve for UPencil:
UPencil = $5 / 2.6
Calculating the value:
UPencil = $1.92
Finally, substituting this value of UPencil into Equation 1:
UPen = 5 * $1.92
Calculating the value:
UPen = $9.6
So, the utility of the last pen (UPen) is $9.6 and the utility of the last pencil (UPencil) is $1.92.
To find out how much Lakshani will spend on pens and pencils, we can calculate the total cost of the pens and the total cost of the pencils separately.
Each pen costs $0.50, and Lakshani wants to buy 6 pens. So the total cost of the pens would be:
Total cost of pens = Cost per pen * Number of pens
= $0.50 * 6
= $3.00
Similarly, each pencil costs $0.10, and Lakshani wants to buy 20 pencils. So the total cost of the pencils would be:
Total cost of pencils = Cost per pencil * Number of pencils
= $0.10 * 20
= $2.00
Now let's consider the utility. The problem states that the last pen will add five times as much utility as the last pencil. This means that the total utility of the pens should be 5 times the total utility of the pencils.
Let's assume the utility of each pencil is U, and the utility of each pen is P. The total utility of 20 pencils would be:
Total utility of pencils = Utility per pencil * Number of pencils
= U * 20
= 20U
And the total utility of 6 pens would be:
Total utility of pens = Utility per pen * Number of pens
= P * 6
= 6P
Since the problem states that the total utility of the pens should be 5 times the total utility of the pencils, we have the equation:
6P = 5 * (20U)
Simplifying this equation, we get:
6P = 100U
Now, we know that utility cannot be directly compared in monetary terms. However, for the sake of this problem, we can assume that P = $0.50 (the cost per pen) and U = $0.10 (the cost per pencil). Substituting these values into the equation, we can solve for P:
6 * 0.50 = 100 * 0.10
3 = 10
This equation is not true, which means our assumption of P = $0.50 and U = $0.10 is incorrect. Hence, there is no combination of prices for pens and pencils that satisfy the given condition of the last pen adding five times as much utility as the last pencil.
Therefore, it seems that there is an error in the given problem statement.