Express `frac(84)(-147)` as a rational number with denominator
(i) 7,
(ii) -49.

Answer

To express the rational number \( \frac{84}{-147} \) with the specified denominators, we will follow these steps:

### Step 1: Simplify the fraction \( \frac{84}{-147} \)

First, we need to simplify the fraction \( \frac{84}{-147} \) by finding the greatest common divisor (GCD) of the numerator and denominator.

1. The GCD of 84 and 147 is 21.
2. Now, divide both the numerator and the denominator by their GCD:

\[ \frac{84 \div 21}{-147 \div 21} = \frac{4}{-7} \]

So, \( \frac{84}{-147} = \frac{4}{-7} \).

### Step 2: Express with denominator 7

To express \( \frac{4}{-7} \) with a positive denominator of 7, we can multiply both the numerator and the denominator by -1:

\[ \frac{4 \times -1}{-7 \times -1} = \frac{-4}{7} \]

Thus, the rational number with a denominator of 7 is \( \frac{-4}{7} \).

### Step 3: Express with denominator -49

Now, we need to express \( \frac{84}{-147} \) with a denominator of -49.

From our simplified fraction \( \frac{4}{-7} \), we can convert it to have a denominator of -49 by multiplying the denominator by 7:

\[ \frac{4 \times 7}{-7 \times 7} = \frac{28}{-49} \]

Thus, the rational number with a denominator of -49 is \( \frac{28}{-49} \).

### Final Answers

1. With denominator 7: \( \frac{-4}{7} \)
2. With denominator -49: \( \frac{28}{-49} \)

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