a coffee shop sells a cup of espresso for $2.00 and a cup of cappuccino for 2.50. on friday, rachel sold 30 more cups of cappuccino than espresso, and she sold $178.50 worth of espresso and cappuccino together. how many cups of each were sold?
x espresso and y cappuccino
for x ----- 2 x dollars
for y ----- 2.5 y dollars
so
2x + 2.5 y = 178.5
but
y = 30 + x
so
2x + 2.5(30+x) = 178.5
2e+2.5c=178.50 c=30+e
2e+2.5(30+e)=178.5
2e+75+2.5e=178.5
4.5e+75=178.5
4.5e=103.5
e=23,
c=30+e
c=30+23
c=53
Check it by plugging them back into the first equation; it works.
To solve this problem, let's use a system of equations.
Let's assume the number of cups of espresso sold is represented by 'x'.
Therefore, the number of cups of cappuccino sold would be 'x + 30' (as Rachel sold 30 more cups of cappuccino than espresso).
The total value of espresso sold would be: 2x (since the price of each espresso cup is $2.00).
The total value of cappuccino sold would be: 2.50(x + 30) (since the price of each cappuccino cup is $2.50).
And the total value of espresso and cappuccino sold combined is $178.50.
Now we can set up the equation:
2x + 2.50(x + 30) = 178.50
Let's solve for 'x' and find the number of cups of espresso sold:
2x + 2.50x + 75 = 178.50
4.50x + 75 = 178.50
4.50x = 178.50 - 75
4.50x = 103.50
x = 103.50 / 4.50
x = 23
Therefore, Rachel sold 23 cups of espresso.
To find the number of cups of cappuccino sold, substitute the value of 'x' into the expression 'x + 30':
23 + 30 = 53
Therefore, Rachel sold 53 cups of cappuccino.
In summary, Rachel sold 23 cups of espresso and 53 cups of cappuccino.