Respuesta :
We need to write and solve a system of equations to see how many washcloths she gets.
We will see that the solution is 17.
The information given is:
First, she buys $91 worth of sheets in a day where each element costs $1.60 less.
So, if she buys S sheets and the price of each sheet is P₁, we have that:
S*(P₁ - $1.60) = $91
Then she returns the sheets and the credit that she gets ignores the discount, thus, she gets:
S*P₁
And with this, she purchases 16 more articles than what she got originally. If she gets T towels and W washcloths we can write
T + W = S + 16
If we define the price of the towel as P₂ and the price of the washcloth as P₃ we will have that:
T*P₂ + W*P₃ = S*P₁
We also know that for the price of one sheet, she can get a towel and a washcloth
P₂ + P₃ = P₁
Finally, we also know that:
Washcloths cost $2.70
She takes six more washcloths than towels.
We can write these two as:
P₃ = $2.70
W = T + 6
Now we can write all our equations:
- S*(P₁ - $1.60) = $91
- T + W = S + 16
- T*P₂ + W*P₃ = S*P₁
- P₃ = $2.70
- W = T + 6
- P₂ + P₃ = P₁
We want to solve this system for W, the amount of washcloths that she got.
To solve the system we need to isolate variables in one given equation and replace them in another equations.
For example, we can see that in fourth equation we have P₃ isolated, then we can replace it in the other equations:
- S*(P₁ - $1.60) = $91
- T + W = S + 16
- T*P₂ + W*$2.70 = S*P₁
- W = T + 6
- P₂ + $2.70 = P₁
Now we can isolate T in the second equation to get:
T = S + 16 - W
And replace it in the other equations with the variable "T".
- S*(P₁ - $1.60) = $91
- (S + 16 - W)*P₂ + W*$2.70 = S*P₁
- W = S + 16 - W + 6
- P₂ + $2.70 = P₁
Now P₁ is isolated in the last equation, so we can replace that in the other ones:
- S*(P₂ + $2.70 - $1.60) = $91
- (S + 16 - W)*P₂ + W*$2.70 = S*(P₂ + $2.70)
- W = S + 16 - W + 6
You can see how we reduced the number of equations and variables. Now we must keep doing this, remember to simplify the equations so they are easier to read:
- S*(P₂ + $1.10) = $91
- (S + 16 - W)*P₂ + W*$2.70 = S*(P₂ + $2.70)
- 2*W = S + 22
Now we can isolate S in the third equation:
S = 2*W - 22
And replace it in the other two:
- (2*W - 22)*(P₂ + $1.10) = $91
- (2*W - 22 + 16 - W)*P₂ + W*$2.70 = (2*W - 22)*(P₂ + $2.70)
Finally, we need to isolate P₂ in one of the equations, let's do it in the first one.
P₂ = $91/(2*W - 22) - $1.10
Now we replace this in the last equation, and we solve it all for W:
(2*W - 22 + 16 - W)*P₂ + W*$2.70 = (2*W - 22)*(P₂ + $2.70)
(W - 6)*P₂ + W*$2.70 = (2*W - 22)*P₂ + (2*W - 22)*$2.70
(W - 6)*P₂ - (2*W - 22)*P₂ = (2*W - 22)*$2.70 - W*$2.70
(W - 6 - 2*W + 22)*P₂ = (W - 22)*$2.70
Now we replace P₂ = $91/(2*W - 22) - $1.10:
(- W + 16)*( $91/(2*W - 22) - $1.10) = (W - 22)*$2.70
(- W + 16)*$91/(2*W - 22) - (- W + 16)*$1.10 = (2*W - 22)*(W - 22)*$2.70
(- W + 16)*$91 - (2*W - 22)*(- W + 16)*$1.10 = (2*W - 22)*(W - 22)*$2.70
W*(-$150.40) + ($1,843.20) - ($2.20)*W^2 = $5.40*W^2 - $178.20*W + $1,306.80
Now we have a quadratic equation, let's simplify this to:
W^2*($2.20 - $5.40) + W*(-$150.40 + $178.20) + ($1,843.20 - $1,306.80) = 0
W^2*(-$3.20) + W*($27.80) + $536.40 = $0
Now the solutions will be given by the Bhaskara's formula:
[tex]W = \frac{-27.80 \pm \sqrt{(27.80)^2 - 4*(-3.20)*536.40} }{2*(-3.20)} \\\\W = 17.31[/tex]
Where I only selected the positive solution, as the negative one does not have a sense (she can't have a negative number of washcloths).
Here we have a problem, as we did not get a whole number, this is due to the amount of operations that we performed and some rounding error, that is magnified.
Thus, we need to round W to the next whole number, which is 17, this means that she gets 17 washcloths.
If you want to learn more, you can read:
https://brainly.com/question/12895249
