Problem 56 - Prealgebra

Question

Multiply and divide as indicated. $$13 \div \frac{1}{3}$$

Step-by-Step Solution

Answer

The result is 39.

Step 1: Understand the Problem

We need to divide 13 by the fraction \( \frac{1}{3} \). This means finding how many times \( \frac{1}{3} \) fits into 13.

Step 2: Reciprocal of the Fraction

The reciprocal of a fraction \( \frac{a}{b} \) is \( \frac{b}{a} \). Therefore, the reciprocal of \( \frac{1}{3} \) is 3.

Step 3: Convert Division into Multiplication

To divide by a fraction, multiply by its reciprocal. Thus, the division \( 13 \div \frac{1}{3} \) becomes \( 13 \times 3 \).

Step 4: Multiply the Whole Number and Reciprocal

Multiply 13 by 3: \( 13 \times 3 = 39 \).

Step 5: Conclusion

The result of \( 13 \div \frac{1}{3} \) is 39.

Key Concepts

Reciprocals

In mathematics, reciprocals play a vital role, especially when dealing with fractions.
Reciprocals are essentially the 'flip' of a fraction.
For any fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \).
This concept helps us transform division into multiplication.

To better understand, consider the fraction \( \frac{1}{3} \).
Its reciprocal is simply 3 (or \( \frac{3}{1} \)).
The switch happens because multiplying a fraction by its reciprocal equals 1:

• \( \frac{1}{3} \times 3 = 1 \)

This property makes reciprocals particularly useful in division, offering a straightforward mechanism to simplify complex arithmetic operations.

Multiplication and Division

Division using reciprocals is a unique mathematical technique that leverages the power of multiplication. Instead of directly dividing by a fraction, we multiply by its reciprocal.
This approach simplifies the calculation and ensures accuracy.

When dividing a number by a fraction:

• Find the reciprocal of the fraction.
• Convert the division into a multiplication problem with the reciprocal.
• Proceed by multiplying the numbers.

For example, given the problem \( 13 \div \frac{1}{3} \), follow the steps to simplify this into \( 13 \times 3 \), which can be calculated directly to yield 39.
This turns the division process into something more familiar and manageable.

Fraction Operations

Handling fractions in mathematical operations such as multiplication, division, addition, and subtraction requires special attention.
Especially when fractions are involved with whole numbers, understanding their interaction becomes crucial.

With multiplication and division:

• Change division into multiplication using reciprocals.
• Multiply numerators and denominators separately when needed.
• Simplify fractions for clearer results.

In our specific example of \( 13 \div \frac{1}{3} \): Transform the division into \( 13 \times 3 \) to simplify the operation, yielding a more straightforward arithmetic problem.
Thus, fraction operations, when understood thoroughly, open doors to easier calculations and a deeper grasp of mathematical principles.