The usual price of a purse was 4/5 of the usual price of a wallet. During a sale, Jerry bought the two items at $20 discount each. The discounted price of the purse was 3/5 of the discounted price of the wallet. What was the percentage discount given for the purse?
Let's call the usual price of the wallet "W" and the usual price of the purse "P".
We know that:
P = (4/5)W
During the sale, both items had a $20 discount:
P' = P - 20
W' = W - 20
We also know that:
P' = (3/5)W'
We can use substitution to solve for P:
P' = (3/5)(W - 20)
P - 20 = (3/5)(W - 20)
P = (3/5)W + 4
Now we can set up an equation to solve for the percentage discount given for the purse:
Discounted price = (1 - percentage discount)(original price)
P' = (1 - D)(P)
Substituting in the values we know:
P - 20 = (1 - D)(P)
(3/5)W + 4 - 20 = (1 - D)[(4/5)W]
Simplifying:
(3/5)W - 16 = (4/5)W - (4/5)WD
(1/5)W = (4/5)WD + 16
W = (20/4D + 80)
Now we can plug this value for W back into one of the earlier equations to solve for P:
P = (4/5)W
P = (4/5)(20/4D + 80)
P = 16/D + 64
Substituting this value for P back into the equation we set up earlier:
P' = (1 - D)(P)
(3/5)(16/D + 64 - 20) = (1 - D)(16/D + 64)
Simplifying:
9.6/D + 24 = (1 - D)(16/D + 64)
9.6/D + 24 = 16/D + 64 - 64D
Combining like terms and solving for D:
73.6/D = 40 - 64D
73.6 = 40D - 64D^2
64D^2 - 40D + 73.6 = 0
Using the quadratic formula:
D = (-(-40) ± sqrt((-40)^2 - 4(64)(73.6))) / 2(64)
D = (40 ± sqrt(800)) / 128
D = 0.062 or D = 0.85
The percentage discount given for the purse is 6.2% if we use the smaller value for D, or 85% if we use the larger value for D.
However, the value of D = 85% doesn't make sense in this context, since it would mean the discounted price of the purse was negative. Therefore, we can conclude that the percentage discount given for the purse was 6.2%.
To find the answer, let's break down the problem and solve it step by step.
Let's assume the usual price of the wallet is W dollars.
According to the problem, the usual price of the purse is 4/5 of the usual price of the wallet, so the usual price of the purse is (4/5)W dollars.
During the sale, Jerry bought both items at a $20 discount each. Therefore, the discounted price of the wallet is W - 20 dollars, and the discounted price of the purse is (4/5)W - 20 dollars.
According to the problem, the discounted price of the purse is 3/5 of the discounted price of the wallet. Mathematically, we can express this as:
(4/5)W - 20 = (3/5)(W - 20)
Now, let's solve this equation to find the value of W.
Multiply through by 5 to get rid of the fractions:
4W - 100 = 3(W - 20)
Expand and simplify:
4W - 100 = 3W - 60
Subtract 3W from both sides:
W - 100 = -60
Add 100 to both sides:
W = 40
Now we have the value of W, which is the usual price of the wallet.
To find the discounted price of the purse, we substitute W = 40 into the expression (4/5)W - 20:
(4/5)(40) - 20 = 32 - 20 = 12 dollars
The discounted price of the purse is 12 dollars.
To calculate the percentage discount given for the purse, we need to determine the difference between the usual price and the discounted price of the purse, and then express it as a percentage of the usual price.
The usual price of the purse is (4/5)W = (4/5)(40) = 32 dollars.
The difference between the usual price and the discounted price is 32 - 12 = 20 dollars.
To find the percentage discount, we divide the difference by the usual price and multiply by 100:
(20/32) * 100 = 62.5%
Therefore, the percentage discount given for the purse is 62.5%.
The discounted price of the purse is actually 3/5 of the discounted price of the wallet, not the usual price. So, we need to adjust our calculations accordingly.
Let's start again:
Let's assume the usual price of the wallet is W dollars.
According to the problem, the usual price of the purse is 4/5 of the usual price of the wallet, so the usual price of the purse is (4/5)W dollars.
During the sale, Jerry bought both items at a $20 discount each. Therefore, the discounted price of the wallet is W - 20 dollars, and the discounted price of the purse is 3/5 of (W - 20) dollars.
According to the problem, the discounted price of the purse is 3/5 of the discounted price of the wallet. Mathematically, we can express this as:
(3/5)(W - 20) = (4/5)W - 20
Now, let's solve this equation to find the value of W.
Multiply through by 5 to get rid of the fractions:
3W - 60 = 4W - 100
Subtract 3W from both sides:
-W - 60 = -100
Add 60 to both sides:
-W = -40
Multiply by -1 to get W by itself:
W = 40
Now we have the value of W, which is the usual price of the wallet.
To find the discounted price of the purse, we substitute W = 40 into the expression 3/5 of (W - 20):
3/5 * (40 - 20) = 3/5 * 20 = 12 dollars
The discounted price of the purse is 12 dollars.
To calculate the percentage discount given for the purse, we need to determine the difference between the usual price and the discounted price of the purse, and then express it as a percentage of the usual price.
The usual price of the purse is (4/5)W = (4/5)(40) = 32 dollars.
The difference between the usual price and the discounted price is 32 - 12 = 20 dollars.
To find the percentage discount, we divide the difference by the usual price and multiply by 100:
(20/32) * 100 = 62.5%
Therefore, the percentage discount given for the purse is 62.5%.
Let's start by assigning some variables to the given information:
Let P be the usual price of the purse.
Let W be the usual price of the wallet.
According to the problem, the usual price of the purse was 4/5 of the usual price of the wallet, so we can write the equation: P = (4/5)W.
During the sale, Jerry bought the two items at a $20 discount each. So, the discounted price of the purse will be P - $20, and the discounted price of the wallet will be W - $20.
The discounted price of the purse was 3/5 of the discounted price of the wallet. We can write this as an equation: P - $20 = (3/5)(W - $20).
To determine the percentage discount given for the purse, we need to compare the usual price and the discounted price of the purse. The percentage discount can be calculated using the formula: (P - (P - $20)) / P * 100.
Now, let's solve the equations step by step:
First, substitute P = (4/5)W into the equation P - $20 = (3/5)(W - $20):
(4/5)W - $20 = (3/5)(W - $20).
Next, simplify the equation:
4W/5 - $20 = 3W/5 - 3($20/5).
4W/5 - $20 = 3W/5 - 3($4).
4W/5 - $20 = 3W/5 - $12.
4W/5 - $20 = 3W/5 - $12.
Then, simplify further:
(4W - 5($20)) / 5 = (3W - 5($12)) / 5.
(4W - $100) / 5 = (3W - $60) / 5.
Now, eliminate the denominators by multiplying both sides by 5:
5[(4W - $100) / 5] = 5[(3W - $60) / 5].
4W - $100 = 3W - $60.
Rearrange the equation:
4W - 3W = $100 - $60.
W = $40.
Substitute W = $40 into the equation P = (4/5)W:
P = (4/5)($40).
P = $32.
Now, calculate the percentage discount given for the purse:
Percentage discount = (P - (P - $20)) / P * 100.
Percentage discount = ($32 - ($32 - $20)) / $32 * 100.
Percentage discount = ($32 - $12) / $32 * 100.
Percentage discount = $20 / $32 * 100.
Percentage discount = 20/32 * 100.
Percentage discount = 62.5%.
Therefore, the percentage discount given for the purse is 62.5%.