Problem 73 - Beginning and Intermediate Algebra

Question

Perform the following operations. Write answers in lowest terms. $$ \frac{7}{8} \div 3 \frac{1}{4} $$

Step-by-Step Solution

Answer

The answer is \(\frac{7}{26}\).

Step 1: Convert Mixed Number to Improper Fraction

First, convert the mixed number \(3 \frac{1}{4}\) into an improper fraction. To do this, multiply the denominator by the whole number and add the numerator. \[3 \frac{1}{4} = \frac{3 \times 4 + 1}{4} = \frac{12 + 1}{4} = \frac{13}{4}\]

Step 2: Rewrite the Division as Multiplication

Convert the division operation into a multiplication by taking the reciprocal of the second fraction. \[\frac{7}{8} \div \frac{13}{4} = \frac{7}{8} \times \frac{4}{13}\]

Step 3: Perform the Multiplication

Multiply the numerators and the denominators:\[\frac{7 \times 4}{8 \times 13} = \frac{28}{104}\]

Step 4: Simplify the Fraction

Simplify the fraction \(\frac{28}{104}\) by finding the greatest common divisor (GCD) of 28 and 104, which is 4. Divide both numerator and denominator by 4:\[\frac{28 \div 4}{104 \div 4} = \frac{7}{26}\]

Step 5: Check the Simplification

Verify that \(\frac{7}{26}\) is fully simplified. Since 7 is a prime number and does not divide 26, the fraction is in its simplest form.

Key Concepts

Improper Fractions

Understanding improper fractions is essential when working with division and simplification problems in fractions. An improper fraction is a type of fraction where the numerator (top number) is larger than the denominator (bottom number). This contrasts with a proper fraction, where the numerator is smaller than the denominator.
To convert a mixed number (like the example 3 \(\frac{1}{4}\)) into an improper fraction, follow these simple steps:

• Multiply the whole number by the denominator of the fractional part.
• Add the result to the numerator of the fractional part.
• Write this sum as the numerator over the original denominator.

For example, converting 3 \(\frac{1}{4}\) becomes \(\frac{13}{4}\) because 3 multiplied by 4 is 12, and 12 plus 1 equals 13.
Knowing how to transform mixed numbers into improper fractions enables smoother calculations in more complex fraction operations.

Division of Fractions

Dividing fractions often intimidates students, but it's quite straightforward with the reciprocal method. The key is to change division into multiplication. Here's how you can do it:

• Take the reciprocal of the divisor (the fraction you are dividing by). This means you flip its numerator and denominator.
• Convert the division into a multiplication using the reciprocal.
• Multiply the fractions as you would normally do.

In the case of dividing \(\frac{7}{8} \div 3 \frac{1}{4}\), first convert 3 \(\frac{1}{4}\) into \(\frac{13}{4}\), then flip it to \(\frac{4}{13}\). Now, multiply \(\frac{7}{8} \times \frac{4}{13}\), resulting in \(\frac{28}{104}\).
Mastering this method is very useful when encountering division with fractions, making it easier to solve fraction problems confidently.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, with the smallest possible numerator and denominator possible that retain the same value. To achieve this, you find the greatest common divisor (GCD) of both the numerator and denominator, and divide each by the GCD.

• Identify a number that divides both numerator and denominator evenly.
• Divide both by the GCD to simplify.

For example, when simplifying \(\frac{28}{104}\), the GCD is 4 because both numbers can be evenly divided by 4. Dividing each by their GCD results in \(\frac{7}{26}\).
It's important to check that the simplified fraction cannot be reduced further, often confirmed if the numerator is a prime number not factorable by the denominator. Simplification ensures fractions are easy to interpret and compare, critical for accurate arithmetic and real-life applications.