Problem 98 - Elementary Algebra

Question

Evaluate: \(\frac{3}{8} \div 6\)

Step-by-Step Solution

Answer

\( \frac{3}{8} \div 6 = \frac{1}{16} \)

Step 1 - Rewrite the Division as Multiplication

When dividing by a number, you can instead multiply by its reciprocal. Rewrite the division \( \frac{3}{8} \div 6 \) as \( \frac{3}{8} \times \frac{1}{6} \).

Step 2 - Multiply the Numerators

Multiply the numerators together: \( 3 \times 1 = 3 \).

Step 3 - Multiply the Denominators

Multiply the denominators together: \( 8 \times 6 = 48 \).

Step 4 - Simplify the Fraction

Simplify the resulting fraction \( \frac{3}{48} \) by finding the greatest common divisor (GCD) of 3 and 48, which is 3. Then divide the numerator and the denominator by the GCD: \( \frac{3 \div 3}{48 \div 3} = \frac{1}{16} \).

Key Concepts

Division

Division is the process of determining how many times one number is contained within another. When we divide fractions or a fraction by a whole number, it changes slightly. Instead of direct division, we prefer to use the concept of multiplying by the reciprocal to make the calculations more straightforward. For example, when you have \(\frac{3}{8} \div 6\), we transform it into multiplication by rewriting the division as multiplication with the reciprocal of 6.

Reciprocal

A reciprocal of a number is what you multiply that number by to get 1. To find a reciprocal, you simply flip the numerator and the denominator. For example:

• The reciprocal of \(\frac{1}{2}\) is \(\frac{2}{1}\) or just 2.
• For a whole number like 6, its reciprocal is \(\frac{1}{6}\).

When dealing with the division of fractions, we convert the problem by multiplying by the reciprocal. So, \(\frac{3}{8} \div 6\) becomes \(\frac{3}{8} \times \frac{1}{6}\). This multiplication is easier to manage, as we only need to multiply the numerators and denominations.

Fraction Simplification

Simplifying fractions means reducing them to their simplest form. This is done by dividing both the numerator and the denominator by their largest common factor. For instance:

• If you have the fraction \(\frac{3}{48}\), you find the greatest common factor of 3 and 48.
• In this case, the GCD is 3.
• Then you divide both numbers by the GCD, \(\frac{3 \div 3}{48 \div 3} = \frac{1}{16}\).

The simplified fraction is more straightforward and easier to understand and use in further calculations.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that can exactly divide two numbers. It helps in simplifying fractions. Here’s how you can find the GCD:

• List the factors of both numbers.
• Identify the largest factor that appears in both lists.

For example:

• The factors of 3 are: 1, 3.
• The factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.

The greatest common factor listed is 3. So, the GCD of 3 and 48 is 3. This helps us simplify the fraction \(\frac{3}{48}\) to \(\frac{1}{16}\).