Problem 89 - Prealgebra

Question

Find each quotient. (Divide) $$1 \frac{3}{4} \div 2 \frac{1}{2}$$

Step-by-Step Solution

Answer

\(1 \frac{3}{4} \div 2 \frac{1}{2} = \frac{7}{10}\).

Step 1: Convert Mixed Numbers to Improper Fractions

First, we need to convert the mixed numbers into improper fractions. The mixed number \(1 \frac{3}{4}\) can be converted by multiplying the denominator 4 by the whole number 1 and adding the numerator 3, giving us \(\frac{7}{4}\). Similarly, for \(2 \frac{1}{2}\), multiply the denominator 2 by the whole number 2 and add the numerator 1, resulting in \(\frac{5}{2}\). So: \(1 \frac{3}{4} = \frac{7}{4}\) and \(2 \frac{1}{2} = \frac{5}{2}\).

Step 2: Write the Division Problem in Fraction Format

The division problem \(1 \frac{3}{4} \div 2 \frac{1}{2}\) is now expressed as \(\frac{7}{4} \div \frac{5}{2}\).

Step 3: Convert Division to Multiplication by Reciprocal

To divide fractions, we multiply by the reciprocal of the divisor. The reciprocal of \(\frac{5}{2}\) is \(\frac{2}{5}\). Therefore, \(\frac{7}{4} \div \frac{5}{2} = \frac{7}{4} \times \frac{2}{5}\).

Step 4: Multiply the Fractions

Multiply the numerators and the denominators: \(\frac{7}{4} \times \frac{2}{5} = \frac{7 \times 2}{4 \times 5} = \frac{14}{20}\).

Step 5: Simplify the Fraction

Simplify \(\frac{14}{20}\) by dividing the numerator and the denominator by their greatest common divisor, which is 2. This results in \(\frac{7}{10}\).\(\frac{14 \div 2}{20 \div 2} = \frac{7}{10}\).

Key Concepts

Mixed Numbers

Mixed numbers combine a whole number with a fraction, making them both simple and complex at the same time. Consider a mixed number like \(1 \frac{3}{4}\), which means you have 1 whole and 3 quarters. It's helpful to keep in mind that converting mixed numbers to improper fractions simplifies calculations.

To change a mixed number to an improper fraction, follow these simple steps:

• Multiply the whole number by the denominator of the fraction part.
• Add the numerator to the result.
• Place this final result over the original denominator.

For \(1 \frac{3}{4}\), multiply 1 by 4 to get 4, then add 3, resulting in the numerator 7. Finally, place it over the denominator: \(\frac{7}{4}\). Now that it's an improper fraction, it can be used easily in operations like multiplication or division.

Improper Fractions

Improper fractions have a numerator larger than the denominator. While they might seem unwieldy, they are perfect for performing arithmetic operations such as addition, subtraction, multiplication, and division.

Consider the improper fraction \(\frac{7}{4}\). It tells us that we have more parts than make up one whole unit — in this case, more than four quarters, which is more than 1 whole.

To help visualize it, think of \(\frac{7}{4}\) as 1 full & a remaining three-quarters. When dealing with division operations, improper fractions are particularly handy as they are straightforward to work with when transforming division problems into multiplication problems. By keeping calculations consistent, they ensure results are accurate and easy to simplify.

Fraction Simplification

Simplifying fractions is a crucial step in ensuring results are as understandable as possible. When simplifying, seek the greatest common divisor (GCD) for the numerator and denominator.

Take the fraction \(\frac{14}{20}\) as an example. The GCD here is 2, as both 14 and 20 can be divided by 2.

Follow these simple steps to simplify:

• Find the largest number that divides both the numerator and denominator.
• Divide both by this number to reduce the fraction.

For \(\frac{14}{20}\), divide both numerator and denominator by 2 to simplify to \(\frac{7}{10}\).

This neatly reduced fraction is easier to interpret and use in further calculations. Remember, simplified fractions hold the same value as the original, just in a more concise form.