Problem 64 - Prealgebra

Question

Multiply. $$\frac{1}{4} \cdot 4$$

Step-by-Step Solution

Answer

\( \frac{1}{4} \cdot 4 = 1 \)

Step 1: Identify the operation

The operation involved is multiplication between a fraction, \( \frac{1}{4} \), and a whole number, \( 4 \).

Step 2: Rewrite the whole number

Convert the whole number \( 4 \) into a fraction. Any whole number \( a \) can be written as a fraction \( \frac{a}{1} \). Thus, \( 4 = \frac{4}{1} \).

Step 3: Set up the multiplication

Write down the multiplication of the fractions: \( \frac{1}{4} \cdot \frac{4}{1} \).

Step 4: Multiply the numerators

Multiply the numerators of both fractions: \( 1 \times 4 = 4 \).

Step 5: Multiply the denominators

Multiply the denominators of both fractions: \( 4 \times 1 = 4 \).

Step 6: Simplify the fraction

Combine the results of Step 4 and Step 5 to get a new fraction: \( \frac{4}{4} \). Simplify \( \frac{4}{4} \) to get \( 1 \).

Key Concepts

Whole Numbers in Fraction Multiplication

When dealing with multiplication involving fractions and whole numbers, we need to remember that a whole number can also be written as a fraction. Understanding this concept makes operations between fractions and whole numbers seamless.

• Whole numbers like 3, 4, or 5 have an implicit denominator of 1. For example, 4 can be rewritten as \( \frac{4}{1} \).
• This transformation allows us to treat whole numbers the same way as fractions during multiplication, making it easier to perform calculations.

By converting a whole number into its fraction form, we can multiply it straightforwardly with any other fraction.

Simplifying Fractions

After performing an operation with fractions, the resulting fraction may need to be simplified. Simplifying a fraction means converting it to its simplest form, where the numerator and the denominator have no common divisors other than 1.

• The simplification process helps in making the result more readable and understandable.
• In the multiplication of \( \frac{1}{4} \) and 4 (written as \( \frac{4}{1} \)), the result is \( \frac{4}{4} \).
• Since 4 is both the numerator and denominator, they are their own greatest common divisor, simplifying to 1, the simplest whole number.

Always check if further simplification is possible by assessing if the numerator and denominator share any common factors.

Understanding Numerator and Denominator

In any fraction, the top number is called the numerator, and the bottom number is called the denominator. The numerator represents how many parts we have, while the denominator shows how many parts make up a whole.

• When multiplying fractions, the operation involves multiplying the numerators together and the denominators together.
• For the expression \( \frac{1}{4} \times \frac{4}{1} \), the numerators (1 and 4) are multiplied together to give 4, and the denominators (4 and 1) are multiplied to give 4.
• The fraction \( \frac{4}{4} \) indicates that we have 4 parts out of the total 4 parts, simplifying to 1.

This result shows how understanding the roles of numerators and denominators can help clarify and resolve fraction multiplication efficiently.