Problem 68 - Prealgebra

Question

Multiply. $$\frac{5}{3}\left(\frac{3}{5}\right)$$

Step-by-Step Solution

Answer

The product of \( \frac{5}{3} \times \frac{3}{5} \) is 1.

Step 1: Set up the multiplication

Write the expression as a multiplication of two fractions: \( \frac{5}{3} \times \frac{3}{5} \).

Step 2: Multiply the numerators

Multiply the numerators of the fractions: \( 5 \times 3 = 15 \).

Step 3: Multiply the denominators

Multiply the denominators of the fractions: \( 3 \times 5 = 15 \).

Step 4: Simplify the fraction

The result of the multiplication is \( \frac{15}{15} \). Since the numerator and the denominator are the same, the fraction simplifies to 1.

Key Concepts

Understanding the Numerator

When working with fractions, the numerator plays a crucial role. It's the top number in a fraction and it tells you how many parts of a whole you are focusing on. For instance, in the fraction \( \frac{5}{3} \), 5 is the numerator. This means you have 5 parts out of 3 equal divisions.

When multiplying fractions, you multiply the numerators together to get the product's numerator. This is a straightforward process. For example, in the multiplication \( \frac{5}{3} \times \frac{3}{5} \), simply multiply the two numerators: 5 and 3, which results in 15. It's like stacking up the parts from both fractions one after another.

• Numerator is the number above the fraction line.
• It represents the number of counted parts.
• In multiplication, multiply all numerators together.

Understanding the Denominator

The denominator is equally important in the fraction. It’s the number at the bottom and indicates the total number of parts that make up the whole. For example, in \( \frac{5}{3} \), the denominator is 3, meaning each whole is divided into 3 parts.

When multiplying fractions, we also multiply the denominators to get the product's denominator. In the exercise \( \frac{5}{3} \times \frac{3}{5} \), the denominators are 3 and 5. Multiplying these gives us the total size of the parts, which is 15.

• Denominator is the number below the fraction line.
• Indicates how many parts in a whole.
• In multiplication, multiply all denominators together.

Simplifying Fractions

After multiplying the numerators and denominators, the last step is to simplify the fraction. Simplifying means reducing the fraction to its simplest form, where no common factors except 1 are left between the numerator and the denominator.

In our example, the result of multiplying \( \frac{5}{3} \) and \( \frac{3}{5} \) is \( \frac{15}{15} \). Since the numerator and the denominator are equal, they can be divided by 15, simplifying the fraction to 1. Simplification makes fractions easier to understand and compare.

• Simplifying involves reducing a fraction to its simplest form.
• Look for the greatest common divisor (GCD).
• In \( \frac{15}{15} \), division by 15 simplifies to 1.