What is the next fraction in this sequence? Simplify your answer.



9

10

 **,**  

3

4

 **,**  

3

5

 **,**  

9

20

Let's analyze the given sequence of fractions: \( \frac{9}{10}, \frac{3}{4}, \frac{3}{5}, \frac{9}{20} \).

### Step 1: Observe the Numerators
The numerators are: 9, 3, 3, 9.
- The numerators alternate between 9 and 3.

### Step 2: Observe the Denominators
The denominators are: 10, 4, 5, 20.
- The denominators seem to follow a pattern related to multiplication and division:
  - \( 10 \) is reduced by a factor of 2 to get \( 5 \) after passing through \( 4 \) (which is \( 2 \times 2 \)), and then the denominator \( 20 \) is \( 5 \times 4 \).

### Step 3: Predict the Next Numerator and Denominator
- **Numerator**: The sequence alternates between 9 and 3. Since the last numerator is 9, the next numerator should be 3.
  
- **Denominator**: The denominators have gone from \( 10 \) to \( 4 \) to \( 5 \) to \( 20 \). Observing this, it seems to be multiplying by small integers and returning back to a base figure.
    - After \( 20 \) (which is \( 5 \times 4 \)), one possible pattern could be returning to 10 (which is \( 20 \div 2 \)), or \( 4 \times 2 \), which would be 8.

Let's select the denominator as 8 (following the earlier multiplication pattern).

### Next Fraction Prediction:
The next fraction is \( \frac{3}{8} \).

This fraction is already in its simplest form, so the answer is:

\[
\boxed{\frac{3}{8}}
\]

Would you like more details or have any questions?

Here are five related questions for further exploration:
1. What would happen if the numerator didn't alternate but followed a different pattern?
2. How would you predict the next term if the denominators were powers of 2?
3. What if the numerators and denominators formed arithmetic sequences independently?
4. How does the pattern change if the sequence is defined by a recursive formula?
5. Can you create your own sequence where the next fraction follows a rule based on the previous two fractions?

**Tip:** In sequences involving fractions, check if the numerators and denominators follow their own patterns independently before combining them.

Explore how to predict the next fraction in the sequence 9/10, 3/4, 3/5, 9/20 with step-by-step analysis. Understand the alternating numerator pattern and the changing denominator pattern involving multiplication and division. Suitable for students in Grades 6-8.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation