Problem 5 - Elementary Algebra

Question

Perform the indicated multiplications and divisions and express your answers in simplest form. $$\left(\frac{17}{9}\right) \div\left(-\frac{19}{9}\right)$$

Step-by-Step Solution

Answer

\( \frac{-17}{19} \)

Step 1: Rewrite the Division as Multiplication

To solve the expression \( \frac{17}{9} \div -\frac{19}{9} \), rewrite the division problem as a multiplication problem by taking the reciprocal of the second fraction. This becomes \( \frac{17}{9} \times -\frac{9}{19} \).

Step 2: Multiply the Numerators

Multiply the numerators of the fractions: \( 17 \times -9 = -153 \).

Step 3: Multiply the Denominators

Multiply the denominators of the fractions: \( 9 \times 19 = 171 \).

Step 4: Simplify the Fraction

The multiplication gives the fraction \( \frac{-153}{171} \). Find the greatest common divisor (GCD) of 153 and 171, which is 9, and simplify the fraction: \( \frac{-153 \div 9}{171 \div 9} = \frac{-17}{19} \).

Step 5: Confirm Simplest Form

Ensure the fraction \( \frac{-17}{19} \) is in its simplest form by confirming that 17 and 19 are co-prime numbers, meaning they have no common factors other than 1.

Key Concepts

Fraction Division

Dividing fractions can seem tricky, but it's simpler when you know the key rule: you must "invert and multiply." This means that when you divide one fraction by another, you flip the second fraction—the divisor—and multiply.
So, if you have a problem like \( \frac{17}{9} \div -\frac{19}{9} \), you'll change it to \( \frac{17}{9} \times -\frac{9}{19} \). This changes a difficult problem into a manageable one.
Remember, the sign of the fraction must also be considered. In the example, the divisor \(-\frac{19}{9}\) becomes \(-\frac{9}{19}\) after you take the reciprocal, which keeps the negative sign in place.

• Flip the second fraction
• Change the division to multiplication
• Keep the sign of the fraction in mind

Practice converting division problems into multiplication ones, and you'll quickly become comfortable with fraction division.

Multiplication of Fractions

Once you've set up a multiplication problem, the next step is straightforward: multiply the numerators and then the denominators.
For instance, consider the problem \( \frac{17}{9} \times -\frac{9}{19} \). First, multiply the numerators, \(17\) and \(-9\), which gives you \(-153\).
Next, multiply the denominators, \(9\) and \(19\), which results in \(171\). The result is \( \frac{-153}{171} \).

• Multiply the numerators to get the new numerator
• Multiply the denominators to get the new denominator

After doing this, the next step is to simplify the resulting fraction, but always remember these two multiplication steps as your foundation.

Simplifying Fractions

Simplifying fractions makes them easier to handle. You do this by dividing the numerator and denominator by their greatest common divisor (GCD).
In our example, the fraction \( \frac{-153}{171} \) can be simplified. First, find the GCD of \(153\) and \(171\), which is 9. Then divide both the numerator and the denominator by 9:

• \(-153 \div 9 = -17\)
• \(171 \div 9 = 19\)

This gives you the simplified fraction of \( \frac{-17}{19} \).
Lastly, verify if the fraction is in its simplest form by checking that the numerator and denominator are co-prime, meaning their only common factor is 1. Since 17 and 19 have no common divisors other than 1, \( \frac{-17}{19} \) is indeed in simplest form.
Simplifying is crucial to keep fractions neat and easy to work with.