Problem 39 - Algebra A Combined Function

Question

Multiply or divide as indicated. See Examples 11 through 14 and \(21 .\) $$ \frac{3}{4} \div \frac{1}{20} $$

Step-by-Step Solution

Answer

The result is 15.

Step 1: Understand Division of Fractions

Dividing by a fraction is equivalent to multiplying by its reciprocal. This means we convert the division into multiplication by flipping the second fraction.

Step 2: Find the Reciprocal

The reciprocal of a fraction is obtained by swapping its numerator and denominator. For \(\frac{1}{20}\), the reciprocal is \(\frac{20}{1}\).

Step 3: Convert Division to Multiplication

Replace the division operation with multiplication by the reciprocal. Thus, \(\frac{3}{4} \div \frac{1}{20}\) becomes \(\frac{3}{4} \times \frac{20}{1}\).

Step 4: Multiply the Fractions

To multiply fractions, multiply the numerators together and the denominators together. The expression is \(\frac{3 \times 20}{4 \times 1}\). This simplifies to \(\frac{60}{4}\).

Step 5: Simplify the Result

Divide both the numerator and the denominator by their greatest common factor. Here, both \(60\) and \(4\) can be divided by \(4\), yielding \(\frac{15}{1}\). So, \(\frac{15}{1} = 15\).

Key Concepts

Reciprocal

Understanding the reciprocal is a crucial step in solving fraction division problems. The reciprocal of a fraction is simply found by swapping its numerator and denominator.
This means if you have a fraction like \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\).
The reason we use the reciprocal in division is that dividing by a number is the same as multiplying by its reciprocal.
This helps convert the division problem into a multiplication problem, which is often easier to handle.

• For example, the reciprocal of \(\frac{1}{20}\) is \(\frac{20}{1}\).
• This switch from division to multiplication is fundamental for solving division of fractions.

You end up multiplying the first fraction by the reciprocal of the second, invert-and-multiply technique.
This simplifies many fraction problems and is essential for quick computation.

Multiplication of Fractions

Once the division has been converted into multiplication by using the reciprocal, the next step involves multiplying fractions together.
The process of multiplying fractions is straightforward: you multiply the numerators together and then multiply the denominators together.

• For example, with \(\frac{3}{4}\) and \(\frac{20}{1}\), the operation becomes: \(\frac{3 \times 20}{4 \times 1}\).
• This results in \(\frac{60}{4}\).

By breaking down the multiplication into these simpler steps, you ensure accuracy in your calculations, and often, broader simplifications can be applied at this stage.
By simplifying beforehand, you may find yourself needing to do less work later on.

Simplifying Fractions

After multiplying fractions, it is crucial to simplify the resulting fraction to its simplest form.
This means reducing the fraction so that the numerator and denominator are as small as possible while still retaining the same value.Simplifying involves dividing the numerator and the denominator by their greatest common factor (GCF).
In our example with \(\frac{60}{4}\), the greatest common factor is \(4\).

• Divide the numerator \(60\) by \(4\), and get \(15\).
• Divide the denominator \(4\) by \(4\), and get \(1\).

Thus, the simplified form is \(\frac{15}{1}\), which equals \(15\).
Always check for simplification opportunities after performing arithmetic operations to make sure your answer is clear and correct.