SOLUTION: A bakery made some buns for charity. 3/5 of them were chicken buns and the rest were tuna buns. 7/8 of the chicken buns and 600 of the tuna buns were eaten. 7/40 of all the buns we

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Question 1200889: A bakery made some buns for charity. 3/5 of them were chicken buns and the rest were tuna buns. 7/8 of the chicken buns and 600 of the tuna buns were eaten. 7/40 of all the buns were left.
(a) How many buns were made?
(b) How many tuna buns were left?
Found 2 solutions by math_tutor2020, greenestamps:
Answer by math_tutor2020(3817) (Show Source):
You can put this solution on YOUR website!

Answers:
(a) 2000
(b) 200

Work Shown:

3/5 were chicken buns.
The rest (2/5) were tuna buns.

c = number of chicken buns
t = number of tuna buns
c+t = number of total buns
(3/5)*(total) = number of chicken buns
(3/5)*(c+t) = number of chicken buns
(3/5)*(c+t) = c
3(c+t) = 5c
3c+3t = 5c
3t = 5c-3c
3t = 2c
c = 3t/2
c = 1.5t
This will be used later in a substitution step

7/8 of the chicken buns and 600 of the tuna buns were eaten
7/40 of all the buns were left. Therefore, 33/40 of the buns were eaten
(7/8)c = number of chicken buns eaten
600 = number of tuna buns eaten
(7/8)c + 600 = number of buns eaten
(7/8)c + 600 = (33/40)*(total number of buns)
(7/8)c + 600 = (33/40)*(c+t)
(7/8)*1.5t + 600 = (33/40)*(1.5t+t) ........... substitution step
(21/16)t + 600 = (33/16)t
16*((21/16)t + 600) = 16*(33/16)t
21t + 9600 = 33t
9600 = 33t-21t
9600 = 12t
t = 9600/12
t = 800 tuna buns to start with
600 were eaten, so 800-600 = 200 tuna buns remain.

c = 1.5t
c = 1.5*800
c = 1200 chicken buns to start with
c+t = 1200+800 = 2000 total buns to start with

Check:
3/5 of 2000 = (3/5)*2000 = 1200 chicken buns to start
2/5 of 2000 = (2/5)*2000 = 800 tuna buns to start
7/8 of 1200 chicken buns = (7/8)*1200 = 1050 chicken buns eaten. Add on the 600 tuna buns eaten to get 1050+600 = 1650 total eaten.
2000-1650 = 350 buns remain
350/2000 = 7/40 of the original buns remain
This confirms the answers.

Another way to check:

Chicken Tuna Total
Start 1200 800 2000
Eaten (7/8)*1200 = 1050 600 1050+600 = 1650
End 1200-1050 = 150 800-600 = 200 150+200 = 350 (or 2000-1650 = 350)

The table mentions:
350 total buns remaining out of 2000 starting buns.
350/2000 = 7/40 of the original buns remain

Answer by greenestamps(13203) (Show Source):
You can put this solution on YOUR website!

The response from the other tutor shows a valid algebraic solution, with a typical setup.

But this problem is one in which a little time looking at the numbers in the problem can lead to a setup which makes solving the problem much easier.

The problem says that 3/5 of the buns were chicken buns and that 7/8 of the chicken buns were eaten. It also says that 7/40 of the buns were not eaten. Those two pieces of information both tell us that the total number of buns the bakery made was a multiple of 40. So....

let 40x be the number of buns the bakery made.

Then

number of chicken buns made: 3/5 of 40x, or 24x
number of tuna buns made: 2/5of 40x, or 16x

number of chicken buns eaten: 7/8 of 24x, or 21x
number of chicken buns left over: 3x

number of tuna buns eaten: 600
number of tuna buns left over: 16x-600

total number of buns left over: 7/40 of 40x = 7x

Our equation then says, very simply, that the total number of buns left over is equal to the number of chicken buns left over plus the number of tuna buns left over:

Now it is easy to answer the two questions....

(a) number of buns made: 40x = 40(50) = 2000
(b) number of tuna buns left: 16x-600 = 16(50)-600 = 800-600 = 200