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?

jiejxu May 11, 2021

#1**+1 **

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?

Let $c$ be the number of cookies John bought

Let $t$ be the number of tarts Ben bought

We have that $\frac{\frac{1}{2}t - 14}{\frac{1}{2}c} = \frac{1}{7} (1)$

We also have that $\frac{\frac{1}{2}c - 16}{\frac{1}{2}t} = \frac{1}{4} (2)$

$(1) \Rightarrow \frac{1}{2}t - 14 = \frac{1}{14}c \Rightarrow 7t - 196 = c (3)$

$(2) \Rightarrow \frac{1}{2}c - 6 = \frac{1}{8}t \Rightarrow 4c - 48 = t (4).$

Can you take it from here?

MathProblemSolver101 May 11, 2021