High School

Rebecca had a piece of ribbon. She used [tex]\(\frac{3}{8}\)[/tex] of the ribbon to tie a package. She used [tex]\(\frac{1}{5}\)[/tex] of the remainder for a bookmark. If Rebecca had 24 cm of ribbon left, what was the length of the piece of ribbon she had at first?

Answer :

Let's solve the problem step-by-step:

1. Identify the Parts Used:
- Rebecca used [tex]\(\frac{3}{8}\)[/tex] of the ribbon to tie packs.
- Then, she used [tex]\(\frac{1}{5}\)[/tex] of the remaining ribbon for a bookmark.
- She has 24 cm of ribbon left at the end.

2. Set the Equation:
- Let the original length of the ribbon be [tex]\( x \)[/tex] cm.

3. Calculate Ribbon Left After Tying Packs:
- Ribbon used for packs = [tex]\(\frac{3}{8} \times x\)[/tex].
- Remaining ribbon after tying packs = [tex]\( x - \frac{3}{8} \times x = \frac{5}{8} \times x \)[/tex].

4. Calculate Ribbon Used for Bookmark:
- Ribbon used for bookmark = [tex]\(\frac{1}{5} \times \frac{5}{8} \times x = \frac{5}{40} \times x = \frac{1}{8} \times x\)[/tex].
- Remaining ribbon after bookmark = [tex]\(\frac{5}{8} \times x - \frac{1}{8} \times x = \frac{4}{8} \times x = \frac{1}{2} \times x\)[/tex].

5. Solve for [tex]\( x \)[/tex]:
- According to the problem, the amount of ribbon left is 24 cm.
- So, [tex]\(\frac{1}{2} \times x = 24\)[/tex].
- Solving for [tex]\( x \)[/tex]:
[tex]\[
x = 24 \times 2 = 48
\][/tex]

Therefore, the original length of the ribbon Rebecca had was 48 cm.

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