Problem 87 - Prealgebra and Introductory Algebra

Question

Perform the indicated operation. $$-\frac{5}{16} \div\left(-\frac{3}{8}\right)$$

Step-by-Step Solution

Answer

The solution to \( -\frac{5}{16} \div -\frac{3}{8} \) is \( \frac{5}{6} \).

Step 1: Substitute Division with Multiplication

The rule for division of fractions is that division of a fraction can be replaced with multiplication by its reciprocal. So, the problem \( -\frac{5}{16} \div -\frac{3}{8} \) becomes \( -\frac{5}{16} * \frac{8}{3} \).

Step 2: Simplify the Negative Signs

The rule for division of negative numbers is that a negative number divided by another negative number gives a positive result. So -5/16 divided by -3/8 is the same as 5/16 divided by 3/8.

Step 3: Multiply the Fractions

To multiply the fractions, simply multiply the numerators (the top part of the fraction) together and the denominators (the bottom part) together independently. So the result is \( \frac{5*8}{16*3} = \frac{40}{48} \).

Step 4: Simplify the Fraction

To simplify a fraction, find the largest number that can divide both the numerator and the denominator. For 40 and 48, that number is 8. So, \( \frac{40}{48} \) simplifies to \( \frac{5}{6} \).

Key Concepts

Multiplication of Fractions

Multiplying fractions is a straightforward process, but it's essential to understand it clearly to perform operations correctly. When multiplying two fractions, like \( \frac{a}{b} \cdot \frac{c}{d} \), you follow a simple rule:

• Multiply the numerators: \( a \cdot c \)
• Multiply the denominators: \( b \cdot d \)
• Combine them: the result is \( \frac{a \cdot c}{b \cdot d} \)

For example, in the problem \( -\frac{5}{16} \cdot \frac{8}{3} \), you multiply the numerators \(-5 \cdot 8\) to get \(-40\), and the denominators \(16 \cdot 3\) to get \(48\), resulting in the fraction \(\frac{-40}{48}\). The process doesn't stop there, though, as simplification is key to arriving at the final answer.

Simplifying Fractions

Once you've multiplied fractions, you often get a result that can be simplified. Simplifying a fraction means reducing it to its simplest form, where the numerator and the denominator have no common factors other than one.

To simplify \( \frac{40}{48} \):

• Find the greatest common divisor (GCD) of the numerator and the denominator. In this case, it's 8.
• Divide both the numerator and the denominator by this number: \( \frac{40 \div 8}{48 \div 8} = \frac{5}{6} \)

Simplification not only makes a fraction easier to understand but also ensures it represents the same quantity in the most concise way possible. Remember, always check for a greatest common divisor to reduce the fraction completely.

Division of Fractions

Dividing fractions might seem tricky at first, but once you understand the concept, it's not too hard. The division operation can be converted into multiplication using the reciprocal of the divisor. The reciprocal of a fraction \( \frac{a}{b} \) is simply \( \frac{b}{a} \) - you flip the numerator and the denominator.

In our exercise, dividing \( -\frac{5}{16} \) by \( -\frac{3}{8} \) involves multiplying \( -\frac{5}{16} \) by the reciprocal of \( -\frac{3}{8} \), which is \( \frac{8}{3} \). The rule of thumb is:

• Change division to multiplication
• Use the reciprocal of the fraction you're dividing by (the second fraction)

This transformation simplifies the operation, and as shown, once the signs are addressed, you can proceed with multiplication. It's essentially a two-step process: flip the second fraction and then multiply, following the earlier outlined method. If negative signs are present on both the numbers involved in division, they cancel out, resulting in a positive outcome.