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Chapter 2: Problem 80

Simplify each expression. $$ \frac{5}{6}\left(\frac{6}{5}\right) $$

Short Answer

Expert verified
1

Step by step solution

01

Review the expression

Consider the given expression: \(\frac{5}{6}\frac{6}{5}\)
02

Multiply the fractions

Multiply the numerators together and the denominators together. This gives: \(\frac{5 \times 6}{6 \times 5}\)
03

Simplify the fraction

Since both the numerator and the denominator are the same (\(30\)), the fraction simplifies to \(1\). So, \(\frac{5 \times 6}{6 \times 5} = 1\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fraction Multiplication
When multiplying fractions, it's important to understand the basic principle: you multiply the numerators together and the denominators together. Here's the basic formula: \(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\).
In the given problem, \(\frac{5}{6} \times \frac{6}{5}\), we follow these steps:
  • Multiply the numerators: \(5 \times 6\)
  • Multiply the denominators: \(6 \times 5\)
Applying these steps, we get: \(\frac{5 \times 6}{6 \times 5}\).
Fraction multiplication is straightforward if these rules are followed, making it easier to manage even more complex problems.
Numerator
The numerator is the top part of a fraction. It represents how many parts of the whole you have. In fraction multiplication, the numerators of the two fractions are multiplied first.
In the example \(\frac{5}{6} \times \frac{6}{5}\), both 5 and 6 are numerators. Multiply them together: \5 \times 6 = 30\.
  • The top part of the resulting fraction is 30.
Understanding the role of the numerator helps to see that you're combining parts of different wholes.
Denominator
The denominator is the bottom part of a fraction. It tells you how many parts the whole is divided into. In fraction multiplication, the denominators of the two fractions are multiplied together.
For \(\frac{5}{6} \times \frac{6}{5}\), both 6 and 5 are denominators. Multiply them: \6 \times 5 = 30\.
  • The bottom part of the resulting fraction is 30.
Knowing the role of the denominator helps to see how the parts relate to the whole.
Simplification
Simplification is the process of reducing a fraction to its simplest form. After multiplying numerators and denominators, check if the fraction can be reduced.
In \(\frac{5 \times 6}{6 \times 5}\), the fraction becomes \(\frac{30}{30}\). Since both the numerator and the denominator are the same, the fraction simplifies to 1.
Therefore, \(\frac{5 \times 6}{6 \times 5} = 1\). Simplifying fractions makes it easier to understand the true value of the fraction. Always look for opportunities to simplify!

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