Problem 91 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ -3 \frac{3}{8} \div\left(-2 \frac{1}{4}\right) $$

Step-by-Step Solution

Answer

The result is \(\frac{7}{6}\) or 1 \(\frac{1}{6}\).

Step 1: Convert Mixed Numbers to Improper Fractions

Start by converting the mixed numbers into improper fractions.For \(-3 \frac{3}{8}\):- Multiply the whole number (-3) by the denominator (8): \(-3 \times 8 = -24\)- Add the numerator (3) to the result: \(-24 + 3 = -21\)This results in the fraction \(-\frac{21}{8}\).For \(-2 \frac{1}{4}\):- Multiply the whole number (-2) by the denominator (4): \(-2 \times 4 = -8\)- Add the numerator (1) to the result: \(-8 + 1 = -9\)This results in the fraction \(-\frac{9}{4}\).

Step 2: Rewrite the Division as Multiplication

The expression involves division. Rewriting the division of fractions as multiplication makes it easier to handle.Write the problem as: \(-\frac{21}{8} \div \left(-\frac{9}{4}\right) \) This is equivalent to multiplying by the reciprocal of the second fraction.Therefore, this becomes:\(-\frac{21}{8} \times \left(-\frac{4}{9}\right)\)

Step 3: Multiply the Fractions

To multiply the fractions, multiply the numerators and the denominators separately.Numerator: \(-21 \times -4 = 84\)Denominator: \(8 \times 9 = 72\)So the product is:\(\frac{84}{72}\)

Step 4: Simplify the Fraction

Simplify \(\frac{84}{72}\) by finding the greatest common divisor (GCD) of 84 and 72.The GCD of 84 and 72 is 12.Divide both the numerator and the denominator by 12:- Numerator: \(\frac{84}{12} = 7\)- Denominator: \(\frac{72}{12} = 6\)So the simplified fraction is \(\frac{7}{6}\), or 1 \(\frac{1}{6}\) as a mixed number.

Key Concepts

Converting Mixed Numbers to Improper Fractions

To perform operations with mixed numbers, it's often best to first convert them into improper fractions. This process involves working with the whole number and the fraction separately.

• **For a mixed number like** \(-3 \frac{3}{8}\): Multiply the whole number (-3) by the denominator (8). This is \(-3 \times 8 = -24\).
• Add this result to the numerator (3). So, \(-24 + 3 = -21\).
• The improper fraction thus becomes \(-\frac{21}{8}\).

For the other mixed number, \(-2 \frac{1}{4}\), the approach is the same:

• Multiply the whole number (-2) by the denominator (4), resulting in \(-2 \times 4 = -8\).
• Add the numerator (1) to this result, giving \(-8 + 1 = -9\).
• Thus, you have the improper fraction \(-\frac{9}{4}\).

Converting mixed numbers to improper fractions simplifies the arithmetic, especially when performing division or multiplication.

Multiplication of Fractions

Once the mixed numbers are expressed as improper fractions, any division can be rewritten as a multiplication problem. This is achieved by using the reciprocal of the divisor. Let's see how this is done:

• **Divide** \(-\frac{21}{8}\) **by** \(-\frac{9}{4}\). Instead of dividing, multiply by the reciprocal of the second fraction, resulting in: \(-\frac{21}{8} \times -\frac{4}{9}\).
• **Multiply the numerators**: \(-21 \times -4 = 84\).
• **Multiply the denominators**: \(8 \times 9 = 72\).

This results in the fraction \(\frac{84}{72}\). Remember, two negative signs mean a positive product in multiplication. This step shows how closely related division and multiplication of fractions are, making complex calculations more manageable.

Simplifying Fractions

After multiplying fractions, the final step is often to simplify the result. Simplification involves finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by this number.

• The fraction \(\frac{84}{72}\) has a GCD of 12. To simplify, divide both the numerator and the denominator by 12.
• **Numerator**: \(\frac{84}{12} = 7\).
• **Denominator**: \(\frac{72}{12} = 6\).
• The simplified fraction is \(\frac{7}{6}\), which can also be written as the mixed number \(1 \frac{1}{6}\).

Simplifying fractions ensures that the result is in its simplest form. It helps in making the final result more legible and easier to interpret, a crucial step in any fraction calculation.