Sure, let's go through the solutions step-by-step.
### 14) Nancy's Spending and Savings
Nancy spent a portion of her savings on a dress and a blouse, then we need to find her total savings and the cost of the blouse.
#### Part (a): Total Savings
1. Understanding the spending on the dress:
- Nancy spent [tex]\(\frac{3}{4}\)[/tex] of her savings on the dress.
- This leaves her with [tex]\(\frac{1}{4}\)[/tex] of her savings.
2. Understanding the spending on the blouse:
- She then spent half of the remaining [tex]\(\frac{1}{4}\)[/tex] on a blouse.
- Half of [tex]\(\frac{1}{4}\)[/tex] is [tex]\(\frac{1}{8}\)[/tex].
3. Remaining savings after purchasing the dress and blouse:
- After both purchases, she has [tex]\(\frac{1}{8}\)[/tex] of her original savings left, which amounts to [tex]$12.
4. Calculating total savings:
- If \(\frac{1}{8}\) of her savings equals $[/tex]12, then her total savings can be calculated as:
[tex]\[
\text{Total Savings} = 12 \times 8 = 96 \text{ dollars}
\][/tex]
Thus, Nancy's total savings were [tex]$96.
#### Part (b): Cost of the Blouse
1. Remaining savings after purchasing the dress:
- After spending \(\frac{3}{4}\) on the dress, remaining is \(\frac{1}{4}\) of the total savings.
- \(\frac{1}{4}\) of $[/tex]96 is [tex]\(24\)[/tex] dollars.
2. Cost of the blouse:
- Nancy spent half of the remaining [tex]\(\frac{1}{4}\)[/tex] on the blouse.
- Half of [tex]$24 is \(24 \times \frac{1}{2} = 12 \text{ dollars}\).
Thus, the cost of the blouse was $[/tex]12.
### 15) Ken's Spending
Ken spent fractions of his money on different days, ending with [tex]$120 left. We need to determine his total starting money.
1. Spending on Monday:
- Ken spent \(\frac{1}{3}\) of his total money on Monday.
- This leaves him with \(\frac{2}{3}\) of his total money.
2. Spending on Tuesday:
- Ken spent \(\frac{1}{4}\) of the remaining \(\frac{2}{3}\) on Tuesday.
- After spending \(\frac{1}{4}\) of \(\frac{2}{3}\), he is left with \(\frac{3}{4} \times \frac{2}{3} = \frac{1}{2}\) of his total money.
3. Remaining money after Tuesday:
- Ken is left with $[/tex]120, which represents [tex]\(\frac{1}{2}\)[/tex] of his total money.
4. Calculating total money:
- If [tex]\(\frac{1}{2}\)[/tex] of his total money equals [tex]$120, then his total money can be calculated as:
\[
\text{Total Money} = 120 \times 2 = 240 \text{ dollars}
\]
Thus, Ken initially had $[/tex]240.
