Vanessa bought a set of military medals through an antique dealer for $120 .she gave two of the medals to her father and resold the rest,charging $4 more per medal than she paid .if vanessa sold the medals for a total of $156.how much did she charge for each medal?.

$12

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To find out how much Vanessa charged for each medal, we need to set up an equation based on the given information.

Let's assume that Vanessa bought x medals initially.

The cost of the x medals she bought is $120, which means the cost price of each medal is $120/x.

She gave two medals to her father, so she has x - 2 medals left to sell.

Vanessa resold the rest of the medals, charging $4 more per medal than she paid.
This means she charged $120/x + $4 for each medal she sold.

The total amount she earned from selling the medals is $156, which means the quantity of medals she sold multiplied by the price per medal equals $156.
So, (x - 2) * ($120/x + $4) = $156.

Now, we can solve this equation to find the value of x, which is the number of medals she bought initially.

(x - 2) * ($120/x + $4) = $156
120(x - 2)/x + 4(x - 2) = 156
120(x - 2) + 4x(x - 2) = 156x
120x - 240 + 4x^2 - 8x = 156x
4x^2 - 284x - 240 = 0

To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula. If we factorize the equation, it becomes:

4(x + 5)(x - 12) = 0

From this, we get two possible solutions:
x + 5 = 0 (which yields x = -5, but since we can't have a negative number of medals, we discard this solution)
or
x - 12 = 0 (which yields x = 12)

Therefore, Vanessa initially bought 12 medals.

Now, we can calculate how much she charged for each medal by substituting x = 12 into the equation:

$120/12 + $4 = $10 + $4 = $14

Therefore, Vanessa charged $14 for each medal she sold.