Question 235712: After a sale, a store had taken in $105 more on its $ 59.50 dresses than on its $39.50 dresses. The total sales were $2870. How many dresses were sold at each price? write a system and solve it.
Answer by MathPro(14)   (Show Source):
You can put this solution on YOUR website! Good Afternoon,
Your problem is:
After a sale, a store had taken in $105 more on its $ 59.50 dresses than on its $39.50 dresses. The total sales were $2870. How many dresses were sold at each price? write a system and solve it.
I’ll digress and state that if you bought two items at a $1.00 each the total cost would be 2($1.00) = $2.00.
Back to the problem. Call the number of dresses sold that cost $59.50, x, and the number of dresses sold that cost $39.50, y. Then the total sale amount of the dresses of type x is $59.50x and the total sale amount of the dresses of type y is $39.50y.
The total amount sold is the sum of the total sales of dresses of type x and the total sales of dresses of type y. We are given that the total amount of the sales is $2870.00. We can write this as
$59.50x + $39.50y = $2870.00 equation (1)
The first sentence of the problem tells us that the total sales of dresses of type x is larger than the total sales of dresses of type y by $105.00. This can be written as:
$59.50x - $39.50y = $105.00 equation (2).
Now we have two simultaneous equations:
$59.50x + $39.50y = $2870.00 equation (1)
$59.50x - $39.50y = $105.00 equation (2)
We can substitute for x from equation (2) into equation (1) by doing the following:
Add $39.50y to each side of equation (2): $59.50x - $39.50y + $39.50y = $105.00 + $39.50y
$59.50x = $105.00 + $39.50y equation(3)
We can divide by $59.50 in equation (3) to get x = $105.00/$59.50 + $39.50y/$59.50y and then substitute for x in equation (1) to get $59.50($105.00/$59.50 + $39.50y/$59.50y) + $39.50y = $2870.00 ect ..
But we can save a lot of time by just leaving the numerical number multiplied by x in equation (3) as follows:
$59.50x = $105.00 + $39.50y equation(3)
If we now look at equation (1) we can substitute for $59.50x to give:
$105.00 + $39.50y + $39.50y = $2870.00 equation (4)
Subtract $105.00 from both sides:
$(2)39.50y = $ 2870.00 - $105.00 = $2765.00
Divide both sides by 2:
$39.50y = $2765.00/2 = 1382.50
Substitute for $39.50y in equation (2)
$59.50x - $1382.50 = $105.00
Add $1382.50 to both sides:
$59.50x = $105.00 + $1382.50 = $1487.5
Now we have $59.50x = $1487.5 equation (5)
and $39.50y= 1382.50 equation (6)
Now lets solve for x by dividing both sides of equation (5) by $59.50
x = $1487.5/$59.50 = 25
Similarly, lets solve for y in equation (6):
y = $1382.50/$39.50 = 35
Our answers are x = 25 and y = 35
Lets check our results:
Lets substitute for x and y into equation (1):
$59.50(25) + $39.50(35) = $2870.00 equation (1)
$1487.5 + $ 1382.5 = $2870.00
The left side is equal to the right side so our answer is correct.
Good Luck!
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