Problem 4 - Technical Mathematics with Calculus

Question

Simplify. Leave your answers as improper fractions. $$\frac{\frac{4}{5}}{\frac{1}{5}+\frac{2}{3}}$$

Step-by-Step Solution

Answer

\(\frac{12}{13}\)

Step 1: Find a Common Denominator

To add the fractions in the denominator, find a common denominator. In this case, the lowest common denominator for \(\frac{1}{5}\) and \(\frac{2}{3}\) is 15.

Step 2: Convert and Add the Fractions

Convert each fraction in the denominator to an equivalent fraction with the common denominator of 15, then add them together. \(\frac{1}{5} = \frac{3}{15}\) and \(\frac{2}{3} = \frac{10}{15}\), so you get \(\frac{3}{15} + \frac{10}{15} = \frac{13}{15}\).

Step 3: Write Division as Multiplication

Rewrite the division of fractions as the multiplication of the first fraction by the reciprocal of the second fraction. The expression becomes \(\frac{4}{5} \times \frac{15}{13}\).

Step 4: Multiply the Fractions

Multiply the numerators and the denominators separately. \(\frac{4}{5} \times \frac{15}{13} = \frac{4 \times 15}{5 \times 13} = \frac{60}{65}\).

Step 5: Simplify the Resulting Fraction

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, 5. \(\frac{60}{65} = \frac{60 ÷ 5}{65 ÷ 5} = \frac{12}{13}\).

Key Concepts

Finding a Common Denominator

Understanding how to find a common denominator is crucial when dealing with complex fractions, especially when you need to add or subtract fractions with different denominators.

Imagine you're trying to combine two different sizes of slices from separate pizzas into a single stack of slices that are all the same size. In mathematics, this is like finding a common denominator to add fractions. It's a way of making sure that the fractions are speaking the same 'language' before you combine them.

A common denominator refers to a shared multiple of the denominators of two or more fractions. To find it, you look for the least common multiple (LCM) of the denominators. In the exercise we looked at, the LCM for 5 and 3 is 15, so we convert \(\frac{1}{5}\) to \(\frac{3}{15}\) and \(\frac{2}{3}\) to \(\frac{10}{15}\) before adding them.

Understanding the Reciprocal of a Fraction

The reciprocal of a fraction flips the fraction's numerator and denominator. For example, if you have a slice of pizza and you want to know how many slices make up a whole pizza, you're thinking about the reciprocal.

In mathematical terms, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\), assuming neither \(a\) nor \(b\) is zero, because a fraction multiplied by its reciprocal always equals 1. This flipping is particularly useful when dividing fractions, because division by a fraction is the same as multiplication by its reciprocal. In our exercise, \(\frac{4}{5}\) divided by \(\frac{13}{15}\) became \(\frac{4}{5} \times \frac{15}{13}\), simplifying the division process.

Multiplying Fractions

When it comes to multiplying fractions, it's like combining parts of parts. Think of having a quarter of a pie and then taking half of that quarter; you're multiplying the fractions of the pie you have.

To multiply fractions, you multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. So for the fractions \(\frac{a}{b}\) and \(\frac{c}{d}\), you would end up with \(\frac{a \times c}{b \times d}\). There's no need to find a common denominator when multiplying. In our earlier exercise, we multiplied \(\frac{4}{5}\) by \(\frac{15}{13}\), resulting in \(\frac{60}{65}\), which we then simplified to \(\frac{12}{13}\) by dividing both the numerator and denominator by the greatest common divisor.