Problem 64 - Algebra 1: Concepts and Skills

Question

Divide. Write the answer as a fraction or as a mixed number in simplest form. $$ 1 \frac{4}{5} \div 2 \frac{1}{2} $$

Step-by-Step Solution

Answer

The result of the division is \(\frac{18}{25}\).

Step 1: Convert to improper fractions

Convert both mixed fractions into improper fractions. The mixed fraction \(1 \frac{4}{5}\) can be converted into an improper fraction by multiplying the whole number by the denominator of the fractional part and adding the numerator of the fractional part. So, \(1 \frac{4}{5} = 1*5+4 = 9/5\). Similarly, convert \(2 \frac{1}{2}\) into \(2*2+1 = 5/2\). The problem now is \( \frac{9}{5}\) \( \div \) \( \frac{5}{2} \).

Step 2: Perform division

To divide fractions, we multiply the first fraction by the reciprocal (flip) of the second fraction. So, \( \frac{9}{5} \div \frac{5}{2} = \frac{9}{5} \times \frac{2}{5} \).

Step 3: Multiply fractions

Now, multiply the top numbers (numerators) to get your new numerator. And multiply the bottom numbers (denominators) to get your new denominator. \( \frac{9}{5} \times \frac{2}{5} = \frac{9*2}{5*5} = \frac{18}{25}\).

Step 4: Simplify the fraction

\(\frac{18}{25}\) is already in its simplest form because the numerator and the denominator do not have any common factors other than 1.

Key Concepts

Improper Fractions

Improper fractions are a type of fraction where the numerator is greater than or equal to the denominator. This can make them seem daunting at first, but they're crucial in performing operations like division and multiplication. To convert a mixed number, like \(1 \frac{4}{5}\), into an improper fraction, you follow these steps:

• Multiply the whole number by the denominator: \(1 \times 5 = 5\)
• Add this result to the numerator: \(5 + 4 = 9\)
• Place this sum over the original denominator: \(\frac{9}{5}\)

Converting mixed numbers to improper fractions allows for smooth calculations, especially when dealing with division as we are doing here. It's just a matter of performing some simple multiplication and addition.

Simplifying Fractions

Simplifying fractions means adjusting a fraction so that the numerator and denominator are the smallest possible integers. This is done by finding common factors between the top and bottom numbers and dividing them out.

• Check for any common factors.
• Divide both the numerator and the denominator by these factors.

In our example, \(\frac{18}{25}\), the numbers 18 and 25 share no common factors other than 1. This means that \(\frac{18}{25}\) is already in its simplest form. Always take a moment to see if simplification is possible; this makes the fraction easier to understand and work with.

Reciprocal of a Fraction

The reciprocal of a fraction is created by flipping the numerator and the denominator. This is particularly important for division. When you divide by a fraction, you multiply by its reciprocal. For instance, to divide \(\frac{9}{5}\) by \(\frac{5}{2}\), you actually multiply \(\frac{9}{5}\) by \(\frac{2}{5}\).

• Flip the second fraction to find its reciprocal.
• Multiply the fractions.

By understanding reciprocals, you turn division into multiplication, which is often more straightforward. This approach can simplify fraction division tasks and make them more manageable.