In a pouch of notes, the number of $2, $5 and $10 are in ratio of 5:4:6. 1/5 of the $2-notes and 15 $5-notes are taken out. The remaining notes are worth $541. Find the difference between the value of $10-notes and $2-notes in the end.
$2-notes : $5-notes : $10-notes
5 : 4 : 6
real value 5n : 4n : 6n
1/5 of $2-notes are taken 5n - 1/5 * 5n = 4n
15 of $5-notes are taken 4n - 15
total remaining notes worth $541
4n * $2 + (4n - 15) * 5 + 6n * 10 = 541
8n + 20n - 75 + 60n = 541
88n = 616
n = 7
So $10-notes there are 6n = 6 (7) = 42
and $2-notes there are 4n = 4 (7) = 28
the difference of their values
= $10 * 42 - $2 * 28
= $364
Let's start by calculating the initial number of $2, $5, and $10 notes:
Let's assume the common ratio for the number of $2, $5, and $10 notes is x. This means:
Number of $2 notes = 5x
Number of $5 notes = 4x
Number of $10 notes = 6x
Now, let's calculate the total value of the remaining notes after taking out 1/5 of $2-notes and 15 $5-notes:
Value of $2 notes taken out = 1/5 * 5x * $2 = x * $2 = 2x
Value of $5 notes taken out = 15 * $5 = $75
Remaining value = Total value - Value of notes taken out
Remaining value = $541 - (2x + $75)
Remaining value = $541 - 2x - $75
Remaining value = $466 - 2x
We know that the remaining notes altogether are worth $541, so we can write the equation:
$466 - 2x = $541
We can solve this equation to find the value of x:
2x = $466 - $541
2x = -$75
x = -$75 / 2
x = -$37.5
The negative value of x doesn't make sense in the context of the problem, so the given ratios are not valid. Therefore, there is no solution to this problem, and we cannot find the difference between the value of $10-notes and $2-notes in the end.
To solve this problem, let's break it down step by step:
Step 1: Assign variables
Let's assign variables to the number of $2, $5, and $10 notes in the pouch.
Let the number of $2-notes be 5x, the number of $5-notes be 4x, and the number of $10-notes be 6x.
Step 2: Calculate the initial value
The initial value of the notes in the pouch can be calculated by multiplying the number of each type of note by its value.
The initial value is: (5x * $2) + (4x * $5) + (6x * $10) = $10x + $20x + $60x = $90x
Step 3: Remove notes
According to the problem, 1/5 of the $2-notes and 15 of the $5-notes are taken out.
The number of $2-notes taken out is (1/5) * 5x = x
The number of $5-notes taken out is 15
The remaining number of $2-notes is 5x - x = 4x
The remaining number of $5-notes is 4x - 15
Step 4: Calculate the remaining value
The remaining value of the notes can be calculated by multiplying the number of each type of note by its value.
The remaining value is: (4x * $2) + ((4x - 15) * $5) + (6x * $10) = $8x + $20x - $75 + $60x = $88x - $75
Step 5: Set up an equation
The problem states that the remaining notes are worth $541.
So, we can set up the equation: $88x - $75 = $541
Step 6: Solve the equation
Solving the equation will give us the value of x, which represents the number of $2-notes originally.
$88x - $75 = $541
$88x = $541 + $75
$88x = $616
x = $616 / $88
x = 7
Step 7: Calculate the difference in value between $10-notes and $2-notes
To find the difference between the value of $10-notes and $2-notes in the end, we need to calculate their respective values first.
The value of $10-notes is 6x * $10 = 6 * 7 * $10 = $420
The value of $2-notes is 4x * $2 = 4 * 7 * $2 = $56
The difference between the value of $10-notes and $2-notes is $420 - $56 = $364.