Problem 70 - Fundamentals of Mathematics

Question

For the following problems, find the products. Be sure to reduce. $$\frac{3}{5} \cdot 20$$

Step-by-Step Solution

Answer

The product is 12.

Step 1: Write down the expression

We need to find the product of \(\frac{3}{5}\) and 20. The expression can be written as: \(\frac{3}{5} \cdot 20\).

Step 2: Convert whole number to fraction

To multiply a whole number by a fraction, it's helpful to write the whole number as a fraction. So, rewrite 20 as \(\frac{20}{1}\).

Step 3: Multiply the numerators

Multiply the numerators of the fractions together. The numerators are 3 and 20. So, \(3 \times 20 = 60\).

Step 4: Multiply the denominators

Multiply the denominators of the fractions together. The denominators are 5 and 1. So, \(5 \times 1 = 5\).

Step 5: Write the product as a fraction

The product of the numerators and the product of the denominators give us a new fraction: \(\frac{60}{5}\).

Step 6: Simplify the fraction

Divide the numerator and the denominator by the greatest common divisor. The greatest common divisor of 60 and 5 is 5. Divide each by 5 to simplify: \(\frac{60}{5} = 12\).

Step 7: Final Step: State the simplified product

The product of \(\frac{3}{5}\) and 20, simplified, is 12.

Key Concepts

Reduce Fractions

Reducing fractions, also known as simplifying fractions, is the process of making a fraction as simple as possible. This involves dividing the numerator and the denominator by their greatest common divisor (GCD). When you reduce a fraction, you make the numbers smaller but the value of the fraction remains the same.

For example, let's consider the fraction \(\frac{60}{5}\). The greatest common divisor of 60 and 5 is 5. By dividing both 60 and 5 by their GCD, the fraction reduces to 12, or \(\frac{12}{1}\).

• Always check both the numerator and denominator for common factors.
• Use division to simplify the fraction until no further reduction is possible.

Reducing fractions is important as it makes them easier to understand and work with in calculations.

Numerator and Denominator

In a fraction, the numerator and the denominator are two very important elements. They help define what the fraction represents.

The numerator is the top part of the fraction. It tells us how many parts of the whole we have. For example, in the fraction \(\frac{3}{5}\), the numerator is 3, which means we have 3 parts of something that is divided into 5 parts.

The denominator is the bottom part of the fraction. It shows how many equal parts the whole is divided into. Using the same example of \(\frac{3}{5}\), the 5 in the fraction tells us that there are 5 equal parts in total.

• The numerator describes the current state or quantity.
• The denominator sets the total value or quantity.

Understanding the roles of the numerator and denominator is key to performing arithmetic operations such as addition, subtraction, multiplication, and division with fractions.

Simplifying Fractions

Simplifying a fraction means making it as simple as possible while maintaining its intended value or proportion.

To simplify fractions, identify the greatest common divisor (GCD) of the numerator and the denominator and divide both by this number. This ensures you are breaking them down to their simplest form.

For example, with \(\frac{60}{5}\), finding that the GCD is 5 allows you to divide both numbers by 5 to get \(\frac{12}{1}\), which is simply 12.

• Always seek the greatest number that divides both components of the fraction without leaving a remainder.
• Simplifying fractions makes calculations easier and results clearer.

Understanding how to simplify fractions will help you in various mathematical tasks, ensuring you always have the easiest form of a number to work with.