Susan has 1/3 as many hair clips as Betty. When Betty gave away 55 of her hair clips to Betty the number of hair clips Susan has is twice the number of hair clips Betty has left. How many hair clips does Betty have at first?

Let the number of hair clips Betty initially has be denoted as \( x \). According to the problem:

1. Susan has \( \frac{1}{3} \) as many hair clips as Betty, so Susan initially has \( \frac{x}{3} \) hair clips.
2. Betty gave away 55 hair clips, so she is left with \( x - 55 \) hair clips.
3. After Betty gave away 55 hair clips, Susan now has twice as many hair clips as Betty has left, meaning:
   \[
   \frac{x}{3} = 2(x - 55)
   \]

### Solving the equation:
1. Start by solving for \( x \):
   \[
   \frac{x}{3} = 2(x - 55)
   \]
2. Multiply both sides of the equation by 3 to eliminate the fraction:
   \[
   x = 6(x - 55)
   \]
3. Distribute the 6 on the right side:
   \[
   x = 6x - 330
   \]
4. Subtract \( 6x \) from both sides:
   \[
   -5x = -330
   \]
5. Divide both sides by -5:
   \[
   x = 66
   \]

### Answer:
Betty initially has **66 hair clips**.

Would you like a detailed explanation or further clarification on any step?

Here are 5 related questions to deepen understanding:
1. What if Susan had 1/2 as many hair clips as Betty?
2. What happens if Betty gives away 30 hair clips instead of 55?
3. If both Betty and Susan received 20 additional hair clips, how would that affect the equation?
4. How would the solution change if Betty gave Susan 55 hair clips directly?
5. What if Susan originally had 1/4 the number of clips as Betty?

**Tip:** Always set up equations carefully by translating word problems into mathematical expressions.

Learn how to solve a real-world word problem where Susan has 1/3 as many hair clips as Betty, and Betty gives away 55 clips. After this, Susan has twice the number of clips Betty has left. This problem uses algebra and linear equations to find how many hair clips Betty originally had. Suitable for Grades 6-8.

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