Which of the following decimals terminate? Which non - terminating decimals repeat, and which do not ?

Options

A. `(11)/(250)`
B. `(393)/(7)`
C. `(1,283)/(741)`
D. `(sqrt3)/(sqrt2)`

Answer

To determine which of the given decimals terminate, which non-terminating decimals repeat, and which do not, we can follow these steps:

### Step 1: Identify the fractions
We have four fractions to analyze:
1. \( \frac{11}{250} \)
2. \( \frac{393}{7} \)
3. \( \frac{1283}{741} \)
4. \( \frac{\sqrt{3}}{\sqrt{3}} \)

### Step 2: Determine if the decimal terminates
A decimal terminates if its denominator (in simplest form) can be expressed as \( 2^m \times 5^n \), where \( m \) and \( n \) are non-negative integers.

- **For \( \frac{11}{250} \)**:
- The denominator \( 250 = 2^1 \times 5^3 \).
- Since it can be expressed in the required form, this decimal **terminates**.

- **For \( \frac{393}{7} \)**:
- The denominator \( 7 \) cannot be expressed as \( 2^m \times 5^n \).
- Therefore, this decimal **does not terminate**.

- **For \( \frac{1283}{741} \)**:
- The denominator \( 741 \) cannot be expressed as \( 2^m \times 5^n \).
- Therefore, this decimal **does not terminate**.

- **For \( \frac{\sqrt{3}}{\sqrt{3}} \)**:
- This simplifies to \( 1 \), which is a terminating decimal.

### Conclusion for Terminating Decimals:
- **Terminating**: \( \frac{11}{250} \) and \( \frac{\sqrt{3}}{\sqrt{3}} \)

### Step 3: Determine non-terminating decimals
Next, we need to check which of the non-terminating decimals are repeating and which are non-repeating.

- **For \( \frac{393}{7} \)**:
- Performing long division gives \( 56.142857142857... \) (the digits "142857" repeat).
- Therefore, this decimal is **non-terminating and repeating**.

- **For \( \frac{1283}{741} \)**:
- Performing long division gives \( 1.73143946018933873... \) (the digits "731439" repeat).
- Therefore, this decimal is **non-terminating and repeating**.

- **For \( \frac{\sqrt{3}}{\sqrt{3}} \)**:
- As mentioned earlier, this simplifies to \( 1 \), which is terminating, so it does not fall into the non-terminating category.

### Conclusion for Non-Terminating Decimals:
- **Non-terminating and repeating**: \( \frac{393}{7} \) and \( \frac{1283}{741} \)
- **Non-terminating and non-repeating**: None in this case.

### Summary of Results:
1. **Terminating**: \( \frac{11}{250} \) and \( \frac{\sqrt{3}}{\sqrt{3}} \)
2. **Non-terminating and repeating**: \( \frac{393}{7} \) and \( \frac{1283}{741} \)
3. **Non-terminating and non-repeating**: None