Sean, Tim, Fred and Mark paid a total of $132 for a souvenir. Mark paid 3/8 of what the rest paid. Sean paid 20% of what the rest paid. Tim and Fred paid an equal amount of money. How much did Fred pay?
Fred's amount --- x
Tim's amount ---- x
Mark's amount --- y
Sean's amount --- z
Mark paid 3/8 of what the rest paid ---> y = (3/8)(132 - 2x - z)
8y = 396 - 6x - 3z
6x + 8y + 3z = 396 , #1
Sean paid 20% of what the rest paid ---> z = (1/5)(132-2x-y)
5z = 132 - 2x - y
2x + y + 5z = 132 , #2
2x + y + z = 132 , #3
I get x = 66 , y = 0, z = 0
Perhaps I misinterpreted the "what the rest paid" part of the question
Fred-- 66
Tim --- 66
which adds up to 132, and the rest paid 0
3/8 of 0 is 0
1/5 of 0 is 0
so 66 + 66 + 0 + 0 = 132 , my answer meets the conditions
x = 1/5 (2y + 2)
5x - 2y - 2 = 0 —> (1)
z = 3/8 (2y + x)
8z = 8y + 3x —> (2)
x + 2y + z = 132 —> (3)
we get 3
40x - 11y - 82 = 0
3x + 6y - 82 = 0
— — 4
————————-
37x - 22y = 0
x = 22/37y
y = 37
8x + 16y + 8z = 10
3x + 16y - 8 = 0
+ + -
————————
11x + 22y = 1056
x = 22 z = 36
Fred paid $37
Let's break down the information given step-by-step to find out how much Fred paid.
1. Mark paid 3/8 of what the rest paid.
Let's assume the total amount paid by the rest is x dollars.
Therefore, Mark paid (3/8) * x dollars.
2. Sean paid 20% of what the rest paid.
The amount paid by the rest is still x dollars.
Therefore, Sean paid (20/100) * x dollars, which simplifies to (1/5) * x dollars.
3. Tim and Fred paid an equal amount of money.
Let's assume the amount paid by Tim and Fred is y dollars each.
Now, let's write an equation to solve for y.
Mark's payment + Sean's payment + Tim's payment + Fred's payment = Total payment
((3/8) * x) + ((1/5) * x) + 2y = $132
To simplify, we can find a common denominator for the fractions:
((15/40) * x) + ((8/40) * x) + 2y = $132
((23/40) * x) + 2y = $132
Since Tim and Fred paid an equal amount, we can represent it as 2y.
Substituting the value of 2y into the equation:
(23/40)x + 2y = $132
(23/40)x + (23/40)x = $132
(46/40)x = $132
x = (40/46) * $132
x = $114
Now that we know the value of x, we can find the amount Tim and Fred paid (y):
2y = (3/8) * x
2y = (3/8) * $114
2y = $42.75
y = $42.75 / 2
y = $21.38
Therefore, Fred paid $21.38.
To find out how much Fred paid, we need to determine the amounts paid by each person.
Let's start by calculating the amount Mark paid. We know that Mark paid 3/8 of what the rest paid. So the rest of the group paid 1 - 3/8 = 5/8 of the total amount.
To find the amount the rest of the group paid, we can subtract Mark's payment from the total:
Amount paid by the rest = Total amount - Mark's payment
Amount paid by the rest = $132 - Mark's payment
Next, we need to find the amount Sean paid, which is 20% of what the rest paid. We can calculate this by multiplying the amount paid by the rest by 20% (or 0.2):
Sean's payment = (Amount paid by the rest) * 0.2
Since Tim and Fred paid an equal amount, let's call that amount X. So, Tim's payment would also be X.
Now, we can set up an equation to find the amount paid by the rest:
(Amount paid by the rest) = Sean's payment + Tim's payment + Fred's payment
(Amount paid by the rest) = Sean's payment + X + X
(Amount paid by the rest) = Sean's payment + 2X
Substituting all the values in the equation, we have:
$132 - Mark's payment = Sean's payment + 2X
Now, let's substitute the values we calculated earlier. We have:
$132 - Mark's payment = (Amount paid by the rest) * 0.2 + 2X
Since we know that (Amount paid by the rest) = $132 - Mark's payment, we can substitute this value into the equation:
$132 - Mark's payment = ($132 - Mark's payment) * 0.2 + 2X
Simplifying the equation, we have:
$132 - Mark's payment = ($132 * 0.2 - Mark's payment * 0.2) + 2X
Expanding the terms, we get:
$132 - Mark's payment = $26.4 - 0.2 * Mark's payment + 2X
Moving the terms around, we have:
- Mark's payment + 0.2 * Mark's payment = $26.4 - $132 + 2X
Combining like terms, we get:
- 0.8 * Mark's payment = $26.4 - $132 + 2X
Simplifying further, we have:
- 0.8 * Mark's payment = -$105 + 2X
Now, we need to express Mark's payment in terms of X. To do this, we can isolate Mark's payment on one side of the equation:
Mark's payment = ($105 - 2X) / 0.8
Finally, we want to find the amount Fred paid, which is equal to X. So, Fred paid:
Fred's payment = X = ($105 - 2X) / 0.8
Simplifying this equation, we have:
0.8 * X = $105 - 2X
Now, we can isolate X on one side of the equation:
0.8 * X + 2X = $105
Combining like terms, we have:
2.8 * X = $105
Dividing both sides by 2.8, we find:
X = $105 / 2.8
Evaluating this expression, we get:
X ≈ $37.50
Therefore, Fred paid approximately $37.50.