Problem 91 - Fundamentals of Mathematics

Question

Perform each multiplication and division. $$\frac{5}{8} \div 1 \frac{1}{4}$$

Step-by-Step Solution

Answer

\(\frac{1}{2}\)

Step 1: Convert Mixed Number to Improper Fraction

We start by converting the mixed number \(1 \frac{1}{4}\) into an improper fraction. Multiply the whole number (1) by the denominator (4) and add the numerator (1): \(1 \times 4 + 1 = 5\). Therefore, \(1 \frac{1}{4} = \frac{5}{4}\).

Step 2: Division Using Multiplication of Reciprocals

To divide by a fraction, multiply by its reciprocal. The reciprocal of \(\frac{5}{4}\) is \(\frac{4}{5}\). Thus, \(\frac{5}{8} \div \frac{5}{4} = \frac{5}{8} \times \frac{4}{5}\).

Step 3: Multiply Fractions

Multiply the numerators and the denominators: \(5 \times 4 = 20\) and \(8 \times 5 = 40\). So, \(\frac{5}{8} \times \frac{4}{5} = \frac{20}{40}\).

Step 4: Simplify the Fraction

Simplify \(\frac{20}{40}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 20: \(\frac{20 \div 20}{40 \div 20} = \frac{1}{2}\).

Key Concepts

Mixed Number Conversion

Fraction division often involves mixed numbers, which need to be converted to improper fractions. For example, the mixed number \(1 \frac{1}{4}\) must be changed to a more manageable form. This process involves multiplying the whole number by the denominator, then adding the numerator. So, for \(1 \frac{1}{4}\), you'd multiply 1 by 4 and add 1, resulting in \(5\), which becomes \(\frac{5}{4}\).
- The whole number tells us how many full sets we have.- Multiply the whole number by the denominator to convert it into parts.- Add this number to the numerator to complete the conversion.
This conversion step is crucial because it allows us to deal with just numerators and denominators, simplifying the arithmetic.

Reciprocal Multiplication

In division involving fractions, the concept of reciprocals is key. Dividing by a fraction is the same as multiplying by its reciprocal. A reciprocal is simply swapping the numerator and the denominator of the original fraction. For instance, the reciprocal of \(\frac{5}{4}\) is \(\frac{4}{5}\). This turns a division problem into a multiplication one:
- Understand that reciprocals inverting a fraction makes calculations easier.- Multiplication with reciprocals can simplify expressions significantly.
So, \(\frac{5}{8} \div \frac{5}{4}\) becomes \(\frac{5}{8} \times \frac{4}{5}\). This eliminates the complexity of division and employs a straightforward multiplication instead.

Fraction Simplification

Once fractions are multiplied, simplifying the result is often necessary. Simplification makes fractions easier to interpret and work with. In this case, multiplying gave us \(\frac{20}{40}\). To simplify, divide both the numerator and the denominator by their greatest common divisor (GCD), which in this instance is 20:

• Find the GCD of 20 and 40 — which is 20.
• Divide the numerator by 20: \(20 \div 20 = 1\).
• Divide the denominator by 20: \(40 \div 20 = 2\).

The simplified form is \(\frac{1}{2}\).
Simplifying fractions is crucial for expressing the result in the most reduced form, making it more intuitive and easy to use in further calculations.