Question

Bill and Mary Ann went to the Viola bakery. Bill bought 3 danishes and 8 filled donuts for ​$14.67. Mary Ann bought 7 of each for ​$16.73. What is the price of each type of​ pastry?

Answer 1

Each donut costs $1.50.

The cost per unit of dánishes is 89 cents.

Explanation:

Givens

• Bill bought 3 dánishes and 8 donuts for $14.67.
• Mary Ann bought 7 of each for $16.73.

Let's call
`d` dánishes and
`f` donuts.

Bill's purchase can be expressed as

`3d+8f=14.67`

Mary's purchase can be expressed as

`7d+7f=16.73`

Let's isolate
`d` from the first equation

`3d+8f=14.67\\3d=14.67-8f\\d=(14.67-8f)/(3)`

Then, we substitute this equivalence in the second equation

`7d+7f=16.73\\7((14.67-8f)/(3))+7f=16.73\\(102.69-56f)/(3) +7f=16.73\\(102.69-56f+21f)/(3)=16.73\\ 102.69-35f=50.19\\-35f=50.19-102.69\\f=(-52.5)/(-35)\\ f=1.50`

Each donut costs $1.50.

Then, we use this value to find the other one.

`3d+8f=14.67\\3d+8(1.50)=14.67\\3d+12=14.67\\3d=14.67-12\\d=(2.67)/(3)\\ d=0.89`

The cost per unit of dánishes is 89 cents.

Answer 2

donuts cost $1.5

danishes cost $1.57

Explanation:

This is a typical 2-equation syestem with 2 unknown variables problem. Lets find out which are our equations and unknowns.

Bill ought 3 danishes and 8 filled donuts for ​$14.67. Lets call d the price of donuts and c the price of danishes. We can then write Bill expenditures as an equation (I will omit $ symbol for simplicity):

3 c + 8 d = 14.67 [eq 1]

Now we can do the same for Mary's expenditures, as she bought 7 of each for ​$16.73:

7 c + 7 d = 16.73 [eq. 2]

Now, lets take eq. 1 and try to get the value of one of the variables, for example c, as function of the other -in this case, d. Notice you could also do this with eq. 2.

So:

3 c + 8 d = 14.67

Subtract 8d in both sides:

3c = 14.67 - 8d

Now, divide both sides by 3:

c = (14.67 - 8d)/3

So, we have c in function of d. Now, replace this value in eq 2:

7*(14.67 - 8d)/3 + 7 d = 16.73

(7/3)*(14.67 - 8d) + 7d = 16.73

Applying distributive:

(7/3)*14.67 - (7/3)*8d + 7d = 16.73

34.23 - 18.67d + 7d = 16.73

34.23 - 11.67d = 16.73

Now, subtract 34.23 in both sides:

-11.67 d = -17.5

Dividing both sides by -11.67

d = 1.50

So, every donuts costs $1.5. If we replace this value in any of the equations of the system we get c. Lets replace in eq. 1:

3 c + 8 d = 14.67

3 c + 8*1.5 = 3c + 12 = 14.67

Subtract 12 in both sides:

3c = 4.67

Divide by 3 in both sides:

c = 1.57

Important: notice that the results may variate in some decimals or less depending on how many numbers after the coma you use. For example, d is really 1.49957155098543 but I used 1.50, if you are more precise and use 1.49957 you results may variate a little but not significantly.

So, c=1.57 and d=1.50 is the solution. A donuts cost $1.5 and a danish $1.57.