SOLUTION: John bought only cookies and Ben bought only tarts at first. John gave half of his cookies to Ben. Ben gave half of his tarts to John. John ate 14 tarts and Ben ate 16 cookies. The

Algebra ->  Percentage-and-ratio-word-problems -> SOLUTION: John bought only cookies and Ben bought only tarts at first. John gave half of his cookies to Ben. Ben gave half of his tarts to John. John ate 14 tarts and Ben ate 16 cookies. The      Log On


   

Question 1180180: John bought only cookies and Ben bought only tarts at first. John gave half of his cookies to Ben. Ben gave half of his tarts to John. John ate 14 tarts and Ben ate 16 cookies. The ratio of John's tarts to cookies became 1:7 and the ratio of Ben's tarts to cookies became 1:4.

(a) How many tarts did Ben buy?

(b) What is the ratio of the total number of tarts to the total number of cookies they had at first?
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
John bought only cookies and Ben bought only tarts at first.
John gave half of his cookies to Ben.
Ben gave half of his tarts to John.
John ate 14 tarts and Ben ate 16 cookies.
The ratio of John's tarts to cookies became 1:7 and the ratio of Ben's tarts to cookies became 1:4.
:
let c = original no. of cookies John bought
let t = no. of tarts Ben bought
:
"John gave half of his cookies to Ben.
Ben gave half of his tarts to John."
then
John has: .5c + .5t
Ben has: .5c + .5t
"John ate 14 tarts and Ben ate 16 cookies."
then
John has .5c + (.5t-14)
Ben has: (.5c - 16) + .5t
:
"The ratio of John's tarts to cookies became 1:7"
%28.5t-14%29%2F%28.5c%29 = 1%2F7
cross multiply
.5c = 7(.5t-14)
.5c = 3.5t - 98
.5c - 3.5t = - 98
multiply by -1, and write it
3.5t - .5c = 98
"and the ratio of Ben's tarts to cookies became 1:4."
%28.5t%29%2F%28.5c-16%29 = 1%2F4
cross multiply
4(.5t)= .5c - 16
2t - .5c = -16
:
use elimination with these 2 equations
3.5t - .5c = 98
2t - .5c = -16
------------------subtraction eliminates c, find t
1.5t + 0 = 114
t = 114/1.5
t = 76
(a) How many tarts did Ben buy? 76 tarts
:
(b) What is the ratio of the total number of tarts to the total number of cookies they had at first?
find c
2(76) - .5c = -16
152 - .5c = -16 - 152
-.5c = -168
c = -168/-.5
c = +336 cookies that John bought
76%2F336 = 19:84 tart to cookie ratio originally