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Let's break down the problem step by step to find out how many cats Juanita had at the start.

1. Let \( x \) be the number of cats Juanita had initially.

2. She adopted more cats so that the total number became twice the original number:
   \[
   2x \quad \text{(the new number of cats after adoption)}.
   \]

3. She gave away 18 cats:
   \[
   2x - 18 \quad \text{(after giving away 18 cats)}.
   \]

4. Then she sold one-third of the herd:
   \[
   \frac{2x - 18}{3} \quad \text{(one-third of the remaining herd)}.
   \]
   After selling one-third, she would have two-thirds of the herd left:
   \[
   \frac{2}{3}(2x - 18) \quad \text{(remaining after selling one-third)}.
   \]

5. She was given 2 mother cats, each with 6 kittens (2 mothers + 12 kittens = 14 new cats):
   \[
   \frac{2}{3}(2x - 18) + 14 \quad \text{(after adding the 14 cats)}.
   \]

6. We are told that at this point, Juanita had 34 cats left:
   \[
   \frac{2}{3}(2x - 18) + 14 = 34.
   \]

Now let's solve this equation step by step:

1. Subtract 14 from both sides:
   \[
   \frac{2}{3}(2x - 18) = 20.
   \]

2. Multiply both sides by 3 to eliminate the fraction:
   \[
   2(2x - 18) = 60.
   \]

3. Expand the left side:
   \[
   4x - 36 = 60.
   \]

4. Add 36 to both sides:
   \[
   4x = 96.
   \]

5. Divide by 4:
   \[
   x = 24.
   \]

Thus, Juanita originally had **24 cats** at the start.

Would you like further details or have any questions? Here are 5 related questions to deepen your understanding:

1. How would the answer change if she had adopted three times her original number of cats instead of twice?
2.

Juanita had a herd of cats on the Furball Ranch. Then she adopted more cats so that her total number was twice the original number. She gave away 18 cats to loving families, she then sold one-third of the herd, and finally was given 2 mother cats each with 6 kittens apiece. At this time, Juanita had 34 cats left. How many cats did she have at the start?

Learn how to solve this algebraic word problem where Juanita adopts, gives away, and sells cats on her Furball Ranch. This problem involves basic algebraic equations and solving for an unknown. Suitable for Grades 7-8.

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