Problem 50 - Prealgebra

Question

Find each product. $$ \frac{1}{5} \times 15 $$

Step-by-Step Solution

Answer

The product is 3.

Step 1: Identify the Multiplication Problem

We need to multiply \( \frac{1}{5} \) by 15. This can be written as \( \frac{1}{5} \times 15 \).

Step 2: Multiply the Fraction by the Whole Number

To multiply \( \frac{1}{5} \) by 15, think of 15 as \( \frac{15}{1} \). Thus, our problem becomes \( \frac{1}{5} \times \frac{15}{1} \).

Step 3: Multiply the Numerators

The numerator of the product is the product of the numerators: \( 1 \times 15 = 15 \).

Step 4: Multiply the Denominators

The denominator of the product is the product of the denominators: \( 5 \times 1 = 5 \).

Step 5: Write the Result as a Fraction

Putting it all together, we have \( \frac{15}{5} \).

Step 6: Simplify the Fraction

Divide the numerator and the denominator by their greatest common divisor, which is 5: \( \frac{15 \div 5}{5 \div 5} = \frac{3}{1} \), which simplifies to 3.

Key Concepts

Understanding Fractions

Fractions are essential building blocks in math that help us represent numbers that are not whole. A fraction consists of two parts:

• Numerator: The top number, which represents how many parts you have.
• Denominator: The bottom number, which shows how many parts make up a whole.

For instance, in the fraction \( \frac{1}{5} \), 1 is the numerator, and 5 is the denominator. This fraction tells you that you have 1 part out of a total of 5 equal parts. When you multiply fractions, it involves interacting these parts with other numbers, including whole numbers. Understanding how numerators and denominators work is key to grasping how fractions function in different mathematical operations.

Whole Numbers and Fractions

Whole numbers are numbers without fractions or decimals, like 1, 2, 3, etc. When we multiply a fraction by a whole number, we're essentially multiplying the whole number by a certain part or portion, as defined by the fraction. For example, multiplying \( \frac{1}{5} \) by 15 is asking you to take 15 and find out how many parts of it equal one-fifth of something. To conduct this multiplication, treat the whole number as a fraction itself, with a denominator of 1. Thus, 15 becomes \( \frac{15}{1} \). This conversion makes it easier to multiply with the original fraction.

Simplifying Fractions

Once you've multiplied fractions, the resulting number might not be as simple as it could be. Simplifying fractions makes them easier to interpret and use. To simplify a fraction, you divide both the numerator and the denominator by their greatest common divisor (GCD). The GCD is the largest number that divides both without leaving a remainder. For instance, the fraction \( \frac{15}{5} \) can be simplified by dividing both parts by 5 (the GCD). This yields \( \frac{3}{1} \), which is simply 3. By practicing simplification, you ensure the fractions are always in their most manageable form.

Working with Numerators and Denominators

Successfully multiplying fractions and whole numbers relies on your understanding of numerators and denominators. During multiplication:

• Multiply the numerators: Calculate the product of the top numbers. This gives you the new numerator.
• Multiply the denominators: Calculate the product of the bottom numbers. This creates the new denominator.

For example, to multiply \( \frac{1}{5} \) by \( \frac{15}{1} \), you multiply: - Numerators: \( 1 \times 15 = 15 \)- Denominators: \( 5 \times 1 = 5 \)Thus, the new fraction is \( \frac{15}{5} \). From here, simplifying it shows you exactly how the pieces fit together in whole numbers, leading to the answer of 3. Being proficient with numerators and denominators allows you to tackle more complex fraction problems with confidence.