Let's solve the problem step-by-step, given the information:
### Part (a): How much did Sarah spend on the vase?
1. Each plate cost [tex]$\$[/tex]18.60[tex]$. Sarah bought 15 plates, so the total cost of the plates is:
\[
\text{Plates cost} = 15 \times 18.60 = \$[/tex]279.00
\]
2. The pan costs [tex]$\frac{1}{9}$[/tex] of the total cost of the plates. Therefore, the cost of the pan is:
[tex]\[
\text{Pan cost} = \frac{279.00}{9} = \$31.00
\][/tex]
3. To find how much Sarah had initially, let's denote her initial amount of money as [tex]\( M \)[/tex]. According to the problem:
[tex]\[
M = 6 \times (\text{Plates cost} + \text{Pan cost})
\][/tex]
Substituting the values found:
[tex]\[
M = 6 \times (279.00 + 31.00) = 6 \times 310.00 = \$1860.00
\][/tex]
4. After buying the toaster with [tex]$\frac{1}{6}$[/tex] of her initial money, Sarah had:
[tex]\[
M - \frac{1}{6}M = \frac{5}{6}M
\][/tex]
The amount left after buying the toaster:
[tex]\[
\frac{5}{6} \times 1860 = \$1550.00
\][/tex]
5. Sarah then bought the pan and plates, costing a total of:
[tex]\[
279.00 + 31.00 = \$310.00
\][/tex]
Subtracting this from the amount left after buying the toaster:
[tex]\[
1550.00 - 310.00 = \$1240.00
\][/tex]
6. Sarah used [tex]$\frac{1}{3}$[/tex] of her remaining money on a vase. Since she had [tex]$\$[/tex]1240.00[tex]$ remaining at this point, she spent:
\[
\text{Vase cost} = \frac{1}{3} \times 1240.00 = \$[/tex]413.33
\]
### Conclusion for Part (a):
- Sarah spent [tex]\(\$413.33\)[/tex] on the vase.
---
### Part (b): How much did Sarah have at first?
1. We already determined that Sarah's initial amount of money, [tex]\( M \)[/tex], is calculated as:
[tex]\[
M = 6 \times (\text{Plates cost} + \text{Pan cost})
\][/tex]
2. Substituting the values:
[tex]\[
M = 6 \times (279.00 + 31.00) = 6 \times 310.00 = \$1860.00
\][/tex]
### Conclusion for Part (b):
- Sarah initially had [tex]\(\$1860.00\)[/tex].
In summary:
(a) Sarah spent [tex]\(\$413.33\)[/tex] on the vase.
(b) Sarah initially had [tex]\(\$1860.00\)[/tex].
