The question asks to find which fraction Lizzy could have found that is equivalent to the fraction shown on the number line.

Steps to Solve:

1. Interpret the Number Line:

   • The number line is divided into 12 equal parts (denominator is $12$).
   • The black dot is positioned at $\frac{9}{12}$.

2. Simplify $\frac{9}{12}$:

   • Find the greatest common divisor (GCD) of $9$ and $12$, which is $3$.
   • Simplify $\frac{9}{12}$ by dividing numerator and denominator by $3$:
     $\frac{9}{12} = \frac{9 \div 3}{12 \div 3} = \frac{3}{4}.$

3. Check Answer Choices:

   • Option 1: $\frac{3}{4}$
     Calculation provided is correct:
     $9 \div 3 = 3 \quad \text{and} \quad 12 \div 3 = 4 \quad \Rightarrow \frac{3}{4}.$ This is correct.
   • Option 2: $\frac{4}{10}$
     Calculation involves incorrect subtraction: $\frac{9 - 5}{12 - 2} = \frac{4}{10}.$
     $\frac{4}{10}$ is not equivalent to $\frac{9}{12}$.
     Incorrect.
   • Option 3: $\frac{3}{8}$
     Incorrect division of $\frac{9}{16}$:
     $9 \div 3 \neq 3$ when denominator is $16$.
     Incorrect.
   • Option 4: $\frac{1}{3}$
     Division is incorrect: $9 \div 9 = 1 \quad \text{and} \quad 12 \div 4 = 3$, but $\frac{1}{3}$ is not equivalent to $\frac{9}{12}$.
     Incorrect.

Final Answer:

The correct fraction Lizzy found is $\frac{3}{4}$.

Would you like a detailed explanation of any step?

Relative Questions:

1. How do you simplify fractions step by step?
2. Why is $\frac{3}{4}$ equivalent to $\frac{9}{12}$?
3. How do you find the GCD of two numbers?
4. Can subtraction ever be used to find equivalent fractions?
5. How do you verify if two fractions are equivalent?

Tip:

Always simplify a fraction by finding the GCD of the numerator and denominator—it ensures the fraction is in its simplest form.