Problem 50 - Prealgebra

Question

Evaluate each expression. $$r^{2} \div s^{2} \text { if } r=-\frac{3}{4} \text { and } s=1 \frac{1}{3}$$

Step-by-Step Solution

Answer

The expression evaluates to \( \frac{81}{256} \).

Step 1: Substitute the Values

First, substitute the given values of \( r \) and \( s \) into the expression. We have \( r = -\frac{3}{4} \) and \( s = 1 \frac{1}{3} \). Substitute these into the expression: \( \left(-\frac{3}{4}\right)^{2} \div \left(1 \frac{1}{3}\right)^{2} \).

Step 2: Simplify \( s \) into an Improper Fraction

Convert the mixed number \( s = 1 \frac{1}{3} \) into an improper fraction. \( 1 \frac{1}{3} = \frac{3 \times 1 + 1}{3} = \frac{4}{3} \). Now the expression becomes \( \left(-\frac{3}{4}\right)^{2} \div \left(\frac{4}{3}\right)^{2} \).

Step 3: Calculate \( r^{2} \)

Find the square of \( r \). We have \( r = -\frac{3}{4} \), so \( \left(-\frac{3}{4}\right)^{2} = \frac{(-3)^{2}}{4^{2}} = \frac{9}{16} \).

Step 4: Calculate \( s^{2} \)

Find the square of \( s \). We have \( s = \frac{4}{3} \), so \( \left(\frac{4}{3}\right)^{2} = \frac{4^{2}}{3^{2}} = \frac{16}{9} \).

Step 5: Calculate the Division

Now perform the division: \( \frac{9}{16} \div \frac{16}{9} \). Dividing by a fraction involves multiplying by its reciprocal. Thus, \( \frac{9}{16} \div \frac{16}{9} = \frac{9}{16} \times \frac{9}{16} = \frac{81}{256} \).

Key Concepts

Fraction Manipulation

Fraction manipulation is a skill that helps in working with parts of a whole or representing numbers out of a particular total. This involves several sub-skills:

• Simplifying: Fractions can often be simplified by dividing the numerator and the denominator by their greatest common divisor.
• Converting: Mixed numbers need to be converted to improper fractions to simplify calculations. For instance, converting a number like \(1 \frac{1}{3}\) into \(\frac{4}{3}\) ensures uniformity in mathematical operations.
• Rewriting: Operations often require rewriting fractions. For example, converting multiplication or division into straightforward expressions makes them easier to handle.

This exercise involves changing mixed numbers to improper fractions and correctly preparing them for further calculations, such as exponentiation or division.

Exponentiation

Exponentiation is a mathematical process where a number is multiplied by itself a certain number of times. This exercise asks us to understand exponentiation involving fractions:

• Squaring: Squaring a number involves multiplying it by itself. For instance, \(r^2\), where \(r = -\frac{3}{4}\), means \((-\frac{3}{4}) \times (-\frac{3}{4}) = \frac{9}{16}\).
• Fraction Exponentiation: When exponents apply to fractions, apply the power to both the numerator and the denominator separately. For example, \(\left(\frac{4}{3}\right)^2 = \frac{4^2}{3^2} = \frac{16}{9}\).

It's key to treat the operation on both parts of the fraction equally. This ensures correctness regardless of whether dealing with integers or fractions in calculations.

Division of Fractions

Dividing fractions may sound complex, but a simple rule makes it straightforward: invert and multiply.

• Reciprocal Understanding: To divide by a fraction, multiply by its reciprocal. The reciprocal of \(\frac{16}{9}\) is \(\frac{9}{16}\).
• Applying the Rule: If our goal is to compute \(\frac{9}{16} \div \frac{16}{9}\), convert this to \(\frac{9}{16} \times \frac{9}{16}\).
• Multiplication: Multiply the numerators together and the denominators together, getting \(\frac{81}{256}\).

This technique of inverting and multiplying means that dividing by complex fractions becomes a simple multiplication task, making it easier to handle any problems involving division of fractions.