Problem 122 - Prealgebra

Question

Perform the indicated operation. $$6 \div \frac{3}{5}$$

Step-by-Step Solution

Answer

The result is 10.

Step 1: Understand the Division of Fractions

When dividing by a fraction, the operation can be converted to multiplication. Divide 6 by \( \frac{3}{5} \) is equivalent to multiplying 6 by the reciprocal of \( \frac{3}{5} \). The reciprocal of \( \frac{3}{5} \) is \( \frac{5}{3} \).

Step 2: Convert Division to Multiplication

Rewrite the division operation as a multiplication operation: \( 6 \div \frac{3}{5} \) becomes \( 6 \times \frac{5}{3} \).

Step 3: Multiply the Whole Number by the Fraction

Multiply 6 by \( \frac{5}{3} \). Treat 6 as a fraction by writing it as \( \frac{6}{1} \). Thus, the multiplication becomes \( \frac{6}{1} \times \frac{5}{3} \).

Step 4: Multiply Numerators and Denominators

Multiply the numerators: \( 6 \times 5 = 30 \). Multiply the denominators: \( 1 \times 3 = 3 \).

Step 5: Simplify the Resulting Fraction

The product is \( \frac{30}{3} \) which simplifies to 10 by dividing both the numerator and the denominator by 3.

Key Concepts

Reciprocal of a Fraction

When dealing with fractions, the term 'reciprocal' may sound a bit intimidating, but it’s actually a simple and useful concept.
The reciprocal of a fraction is created by flipping the fraction. This means swapping the numerator (top number) and the denominator (bottom number) of the fraction.

For example, if we take the fraction \( \frac{3}{5} \), its reciprocal is \( \frac{5}{3} \). Why do we care about reciprocals? In the context of division, finding the reciprocal of a fraction transforms a division problem into a multiplication problem, which is often easier to handle.

• To find the reciprocal, just exchange the top and bottom numbers of the fraction.
• Reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \), where \( a \) and \( b \) are non-zero numbers.
• Reciprocals are used to divide fractions by converting the operation into multiplication.

Multiplication of Fractions

The multiplication of fractions may initially sound complex, but it becomes straightforward once you know the steps.
To multiply fractions, you multiply the numerators together and then multiply the denominators together.

For instance, if you want to multiply \( \frac{6}{1} \) by \( \frac{5}{3} \), follow these steps:

• Multiply the numerators: \( 6 \times 5 = 30 \).
• Multiply the denominators: \( 1 \times 3 = 3 \).

The result of the multiplication is \( \frac{30}{3} \).
Multiplying fractions is different from adding or subtracting fractions because you don’t need a common denominator. Each operation has its own set of rules, and for multiplication, it's as simple as multiplying straight across.

Simplifying Fractions

After you have multiplied fractions, as we did when we converted a division problem into a multiplication problem, simplifying the resulting fraction is typically the final step.
Simplification means making the fraction as simple or as small as possible by dividing both the numerator and the denominator by their greatest common divisor.

In our example, the result of multiplying was \( \frac{30}{3} \).

• Divide both the numerator (30) and the denominator (3) by their greatest common divisor, which is 3 in this case.
• This simplifies \( \frac{30}{3} \) to 10 (since \( 30 \div 3 = 10 \) and \( 3 \div 3 = 1 \)).

The process of simplifying a fraction ensures that you get the smallest, most manageable form of the fraction's value, which in many practical situations helps in understanding the number quickly. Simplifying isn’t always required, but it's a good mathematical practice.