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Chapter 1: Problem 60

Simplify each expression. $$ \left(\frac{4}{5}\right)(-0.73)\left(\frac{5}{4}\right) $$

Short Answer

Expert verified
-0.73

Step by step solution

01

- Multiply the Fractions Together

Combine \(\frac{4}{5}\) and \(\frac{5}{4}\) by multiplying the numerators together and the denominators together. This results in \(\frac{4 \times 5}{5 \times 4} = \frac{20}{20}\), which simplifies to \(1\).
02

- Identify the Remaining Factors

After multiplying and simplifying the fractions, the expression simplifies to \((1)(-0.73)\).
03

- Multiply the Result with the Remaining Number

Now, multiply \(1\) by \(-0.73\): \(1 \times -0.73 = -0.73\).

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Key Concepts

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

Multiplying Fractions
Multiplying fractions is a fundamental concept in arithmetic, crucial for many other areas of mathematics. When multiplying fractions, you combine the fractions' numerators and denominators. Let鈥檚 take a closer look at how this works using the given problem.

First, we start with the fractions: \(\frac{4}{5}\) and \(\frac{5}{4}\). To multiply these, we follow these steps:

  • Multiply the numerators: 4 \times 5 = 20
  • Multiply the denominators: 5 \times 4 = 20
The result is \(\frac{20}{20}\), which simplifies to 1, because the numerator and the denominator are the same.

Key takeaways:
  • Always multiply the numerators together.
  • Always multiply the denominators together.
  • Simplify the fraction by dividing the numerator and denominator by their greatest common divisor (GCD).

In our exercise, the fraction \(\frac{20}{20}\) simplifies perfectly to 1, making the rest of the calculations easier.
Simplification
Simplification is all about making an expression easier to work with. This can involve reducing fractions, cancelling out terms, or performing basic arithmetic. Let's break down the simplification in our exercise.

After multiplying the fractions \(\frac{4}{5}\) and \(\frac{5}{4}\), we get \(\frac{20}{20}\). Since 20 divided by 20 equals 1, the fraction simplifies to 1. This is a critical step because:
  • Simplified expressions are easier to handle.
  • They make subsequent calculations straightforward.
  • They help in understanding the problem's core structure.

In the next step, we have to multiply this simplified result (1) by \(-0.73\). Thus, \((1)(-0.73) = -0.73\). This showcases the power of simplification - turning a complex problem into a simple arithmetic operation.
Negative Numbers
Understanding negative numbers is essential for solving many problems in mathematics. Negative numbers are numbers less than zero, and they have unique rules when used in arithmetic.

In our problem, the presence of \(-0.73\) is crucial. Here鈥檚 what you need to know when working with negative numbers:
  • Adding a negative number is the same as subtraction.
  • Multiplying two negative numbers gives a positive result.
  • Multiplying a positive number by a negative number gives a negative result.

In the final step of our solution, we multiply 1 by \(-0.73\). Since anything multiplied by 1 remains unchanged except for sign considerations, the result remains \(-0.73\). This step illustrates how understanding negative numbers can help interpret and simplify results accurately.

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Most popular questions from this chapter

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