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Chapter 4: Problem 236

In the following exercises, simplify. $$\frac{15 \cdot 5-5^{2}}{2 \cdot 10}$$

Short Answer

Expert verified
2.5

Step by step solution

01

Simplify the Numerator

Evaluate the expression in the numerator. Compute the product and the exponentiation separately, then subtract.\[15 \times 5 - 5^2 = 75 - 25 = 50\]
02

Simplify the Denominator

Evaluate the expression in the denominator. Compute the product.\[2 \times 10 = 20\]
03

Divide the Simplified Numerator by the Simplified Denominator

Divide the result from the numerator by the result from the denominator.\[\frac{50}{20} = 2.5\]

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Key Concepts

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

Numerator and Denominator
In any fraction, you will find two main parts: the numerator and the denominator. The numerator is the number above the fraction line, and it represents the part of the whole you have.
On the other hand, the denominator is the number below the fraction line, and it represents the total number of equal parts the whole is divided into. Here is a simple example:
  • In the fraction \(\frac{3}{4}\), 3 is the numerator, and 4 is the denominator.
  • The numerator tells us that we are considering 3 parts, while the denominator tells us that the whole is divided into 4 parts.
Having a clear understanding of these two components is crucial for working with and simplifying fractions.
In the given exercise, we start by simplifying the numerator and the denominator separately for easier calculation.
Exponentiation
Exponentiation is a mathematical operation involving numbers called bases and exponents. When a number is raised to an exponent, it means that the base is multiplied by itself as many times as the exponent indicates. Let's break it down:
  • The expression \(5^2\) means that 5 is multiplied by itself 2 times: \(5 \times 5 = 25\).
  • Similarly, \(3^3\) means that 3 is multiplied by itself 3 times: \(3 \times 3 \times 3 = 27\).
In our exercise, we have the expression \(5^2\), which equals 25. This was subtracted from the product of 15 and 5. By handling the exponentiation first, we simplify the process of solving the numerator.
Fraction Simplification
Simplifying a fraction means reducing it to its simplest form. This involves dividing both the numerator and the denominator by their greatest common divisor (GCD). Let's look at a few steps for simplifying a fraction:
  • Step 1: Find the GCD of the numerator and the denominator.
  • Step 2: Divide both the numerator and the denominator by the GCD.
  • Step 3: Write the simplified fraction.
In our exercise, after simplifying the numerator and denominator separately, we had the fraction \(\frac{50}{20}\). Since 10 is the GCD of both 50 and 20, dividing both by 10 reduces the fraction to \(\frac{5}{2}\), which is equivalent to 2.5.
This final value, 2.5, is the result of simplifying the original fraction. Understanding these steps allows you to manipulate and simplify fractions confidently.

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Most popular questions from this chapter

In the following exercises, evaluate. 6\(m n^{2}\) when $$ m=\frac{3}{4} \text { and } n=-\frac{1}{3} $$

In the following exercises, find the least common denominator. $$\frac{1}{3} \text { and } \frac{1}{12}$$

In the following exercises, find the least common denominator. $$\frac{3}{4}, \frac{1}{6}, \text { and } \frac{5}{10}$$

In the following exercises, add. $$\frac{15}{32}+\frac{9}{32}$$

In the following exercises, simplify. $$12 \left(\frac{9}{20}-\frac{4}{15}\right)$$
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