Amy had some buns and pies in the ratio 16: 9. After she sold a total of 192 buns and pies, 1/3 of the pies were left and the number of buns decreased by 3/8. Chandra bought 3/4 of Amy's remaining pies at
$3.85 each.
(a) How many buns were sold?
(b) How much money did Chandra pay for the pies?
.
Let the common ratio be $x$. Then we have $16x+9x=25x$ buns and pies to begin with. After selling $192$ buns and pies, $16x+9x-192$ buns and pies are left. We know that $\frac{1}{3}$ of the remaining $9x$ pies are left, so $\frac{2}{3}$ of the pies were sold. That means $\frac{2}{3}(9x)=6x$ pies were sold, leaving $9x-6x=3x$ pies remaining.
Also, the number of buns decreased by $\frac{3}{8}(16x)=6x$. Since we still have a ratio of $16:9$, we know that $9x$ buns were sold. Thus, we have the equation \begin{align*}
16x-6x &= 9x \\
\Rightarrow \qquad x &= 3.
\end{align*}
So, the original number of buns and pies was $16(3)=48$ and $9(3)=27$, respectively. Then, $48+27=75$ buns and pies were sold, leaving $16(3)-6(3)=30$ buns and $9(3)-3=24$ pies. Chandra bought $\frac{3}{4}(24)=18$ pies at $3.85 each, so she paid $18(3.85)=\boxed{\$69.30}$.
To solve this problem, we can break it down into several steps:
Step 1: Determine the original ratio of buns to pies.
Given that the ratio of buns to pies is 16:9, Amy initially had a total of 16 + 9 = 25 parts.
Step 2: Find the number of parts each part represents.
To find the value of each part, we divide the total number of buns and pies by the total number of parts. Since Amy sold a total of 192 buns and pies, each part represents 192 / 25 = 7.68.
Step 3: Calculate the number of buns sold.
Since the ratio of buns to pies is 16:9, the number of buns sold is 16 parts * 7.68 = 122.88, which rounds up to 123 buns.
(a) The number of buns sold is 123.
Step 4: Calculate the number of pies left.
Since 1/3 of the pies were left after Amy sold a total of 192 buns and pies, the remaining fraction of pies is 2/3. To find the number of pies left, we multiply the fraction by the total number of parts, which is 2/3 * 9 parts * 7.68 = 12.96, which rounds up to 13 pies.
Step 5: Determine the remaining number of buns.
The number of buns decreased by 3/8, which can be represented as 3/8 * 192 buns = 72 buns. Therefore, the remaining number of buns is 192 buns - 72 buns = 120 buns.
Step 6: Calculate the number of pies bought by Chandra.
Since Chandra bought 3/4 of the remaining pies, the number of pies bought is 3/4 * 13 pies = 9.75, which rounds up to 10 pies.
(b) The number of pies bought by Chandra is 10.
Step 7: Calculate the total amount Chandra paid for the pies.
Since each pie costs $3.85 and Chandra bought 10 pies, the total amount that Chandra paid is $3.85 * 10 = $38.50.
(b) Chandra paid $38.50 for the pies.