Problem 7 - Prealgebra

Question

Find each of the following products. (Multiply.) $$\frac{3}{4} \cdot 9$$

Step-by-Step Solution

Answer

The product is \( \frac{27}{4} \) or \( 6\frac{3}{4} \).

Step 1: Understand the problem

We need to multiply a fraction, \( \frac{3}{4} \), by a whole number, 9.

Step 2: Convert the whole number to a fraction

Express the whole number 9 as a fraction by placing it over 1, i.e., \( \frac{9}{1} \). This makes multiplication easier.

Step 3: Multiply the numerators

Multiply the numerators of the fractions: 3 (the numerator of \( \frac{3}{4} \)) and 9 (the numerator of \( \frac{9}{1} \)). This gives \( 3 \times 9 = 27 \).

Step 4: Multiply the denominators

Multiply the denominators of the fractions: 4 (the denominator of \( \frac{3}{4} \)) and 1 (the denominator of \( \frac{9}{1} \)). This gives \( 4 \times 1 = 4 \).

Step 5: Form the new fraction

Combine the results from Steps 3 and 4 to form the new fraction: \( \frac{27}{4} \).

Step 6: Simplify or convert to mixed number if needed

If the fraction can be simplified or converted to a mixed number, perform that step. Here, \( \frac{27}{4} \) can be written as a mixed number: 6 remainder 3, so it becomes \( 6\frac{3}{4} \).

Key Concepts

Fractions

Fractions are a way to represent parts of a whole. They are expressed as two numbers separated by a horizontal bar. The top number is the numerator, which indicates how many parts we have.
The bottom number is the denominator, which shows the total number of equal parts the whole is divided into. For example, in the fraction \( \frac{3}{4} \), 3 is the numerator, and 4 is the denominator.
Fractions allow us to work with numbers that are not whole and are especially useful in situations where exact measurements and portions are needed.

• Numerator: The number above the fraction line indicating how many parts are considered.
• Denominator: The number below the fraction line describing how many total parts make up a whole.

Fractions can be added, subtracted, multiplied, or divided, allowing us to perform a wide variety of calculations.

Whole Numbers

Whole numbers are the set of numbers without fractions or decimals. They begin from zero and go on infinitely.

• Examples: 0, 1, 2, 3, ...
• They are always non-negative.

Whole numbers are used for counting and ordering objects. When multiplying a fraction by a whole number, it is often helpful to convert the whole number into a fraction first.
This is done by putting the whole number over 1, turning it into a fraction like \( \frac{9}{1} \). This conversion simplifies the multiplication process by keeping the operation consistent.

Mixed Numbers

A mixed number is a combination of a whole number and a fraction. It is used to express fractions that are greater than 1 and offer an easy way to understand the size of the fraction.
For example, the fraction \( \frac{27}{4} \) can be converted to the mixed number \( 6\frac{3}{4} \), which indicates 6 whole parts and a fraction part \( \frac{3}{4} \).
Converting an improper fraction (where the numerator is larger than the denominator) into a mixed number tells us exactly how many times the denominator fits entirely into the numerator and what is left over.

• Improper Fraction: \( \frac{27}{4} \) tells us 27 divided by 4.
• Mixed Number: Shows the result as 6 full times and a remainder, i.e., \( 6\frac{3}{4} \).

Mixed numbers make it easier to visualize the size of a portion when dealing with improper fractions.

Numerator and Denominator

In fractions, understanding the roles of the numerator and denominator is crucial. The numerator, located on top, shows how many parts of the whole are being considered. It counts the specific pieces or shares we have in the context of division by the denominator.
The denominator, placed below, indicates the total number of equal parts the whole is divided into. It helps form the "size" of each part.

• The numerator tells "how many" parts are at play.
• The denominator sets the "size" or "total number" of parts.

When multiplying fractions, we simply multiply the numerators together and the denominators together, simplifying where possible.

• For instance, in \( \frac{3}{4} \times \frac{9}{1} \), we have \( 3 \times 9 = 27 \) (numerators) and \( 4 \times 1 = 4 \) (denominators), producing a new fraction \( \frac{27}{4} \).

Understanding this structure is essential for effectively manipulating and multiplying fractions.