High School

Kate used \( \frac{1}{4} \) of a roll of ribbon to tie a present. She then gave \( \frac{3}{8} \) of the ribbon to her friend. If she had 90 cm of the ribbon left, how long was the original length of her ribbon?

Answer :

To solve this problem, we need to determine the original length of Kate's ribbon before she used or gave any parts of it away. Here's a step-by-step approach:


  1. Understand the Parts of the Ribbon Used and Given Away:


    • Kate used [tex]\frac{1}{4}[/tex] of the original length of the ribbon to tie a present.

    • Kate then gave [tex]\frac{3}{8}[/tex] of the original length of the ribbon to her friend.



  2. Formulate the Relationship Among Parts:


    • Let's denote the original length of the ribbon as [tex]x[/tex].

    • The part of the ribbon used for the present is [tex]\frac{1}{4}x[/tex].

    • The part of the ribbon given to her friend is [tex]\frac{3}{8}x[/tex].



  3. Set Up the Equation:


    • After using and giving away parts of the ribbon, Kate has 90 cm left.

    • The equation representing the total length of the ribbon used, given away, and left is:
      [tex]\left( x - \frac{1}{4}x - \frac{3}{8}x \right) = 90[/tex]



  4. Simplify the Equation:


    • First, find a common denominator for [tex]\frac{1}{4}[/tex] and [tex]\frac{3}{8}[/tex], which is 8:
      [tex]\frac{1}{4} = \frac{2}{8}[/tex]

    • Combine the fractions:
      [tex]\frac{2}{8}x + \frac{3}{8}x = \frac{5}{8}x[/tex]

    • Substitute back into the equation:
      [tex]\left( x - \frac{5}{8}x \right) = 90[/tex]

    • Simplify above:
      [tex]\frac{3}{8}x = 90[/tex]



  5. Solve for [tex]x[/tex]:


    • Multiply both sides of the equation by [tex]\frac{8}{3}[/tex] to isolate [tex]x[/tex]:
      [tex]x = 90 \times \frac{8}{3}[/tex]

    • Calculate:
      [tex]x = 240[/tex]




So, the original length of the ribbon was 240 cm.

In summary, Kate's ribbon had an initial length of 240 cm.

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