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### 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.
### 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.
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