Problem 48 - Elementary Algebra

Question

For exercises \(23-74\), evaluate. $$ \frac{9}{10} \div \frac{3}{5} $$

Step-by-Step Solution

Answer

The result of \( \frac{9}{10} \div \frac{3}{5} \) is \( \frac{3}{2} \).

Step 1: Understand the Problem

You need to divide two fractions: \[ \frac{9}{10} \div \frac{3}{5} \]

Step 2: Apply the Division Rule for Fractions

To divide fractions, multiply the first fraction by the reciprocal of the second fraction. The reciprocal of \( \frac{3}{5} \) is \( \frac{5}{3} \). So, \[ \frac{9}{10} \div \frac{3}{5} = \frac{9}{10} \times \frac{5}{3} \]

Step 3: Multiply the Fractions

Multiply the numerators together and the denominators together: \[ \frac{9 \times 5}{10 \times 3} = \frac{45}{30} \]

Step 4: Simplify the Fraction

Divide both the numerator and the denominator by their greatest common divisor, which is 15: \[ \frac{45 \div 15}{30 \div 15} = \frac{3}{2} \]

Key Concepts

fraction multiplication

Multiplying fractions might seem tricky at first, but it's actually quite simple. When multiplying two fractions, you follow a straightforward rule: multiply the numerators (the top numbers) together and multiply the denominators (the bottom numbers) together. For example, if you have the fractions \(\frac{a}{b}\) and \(\frac{c}{d}\), the product will be \[\frac{a\cdot c}{b\cdot d}\]. This rule makes solving problems like our example, \(\frac{9}{10}\cdot\frac{5}{3}\), much easier. You just multiply 9 by 5 for the numerators and 10 by 3 for the denominators, resulting in \(\frac{45}{30}\).

fraction simplification

Once you have your fraction result from a multiplication (or any operation), you often need to simplify it. Simplifying fractions makes them easier to work with and understand. The process of simplifying a fraction involves dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD is the largest number that evenly divides both the numerator and the denominator. In our example of \(\frac{45}{30}\), the GCD is 15. Dividing both 45 and 30 by 15 gives you the simplified fraction, \(\frac{3}{2}\). Simplifying fractions isn't just about making numbers smaller; it helps clearly show the relationship between the numerator and the denominator.

reciprocal

The concept of the reciprocal is vital in division with fractions. The reciprocal of a fraction simply swaps the numerator and the denominator. For instance, the reciprocal of \(\frac{3}{5}\) is \(\frac{5}{3}\). To divide fractions, you multiply by the reciprocal of the fraction you're dividing by. In our example problem, to divide \(\frac{9}{10}\) by \(\frac{3}{5}\), you change the division into multiplication by the reciprocal of \(\frac{3}{5}\). This turns the expression into \(\frac{9}{10}\times\frac{5}{3}\), which you can then solve by fraction multiplication, as already discussed.