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Chapter 1: Problem 66

Simplify. \(\frac{5(12-7)-4}{5^{2}-18}\)

Short Answer

Expert verified
The simplified value of the expression is 3.

Step by step solution

01

Simplify the Numerator

The numerator of the given fraction is \(5(12-7) - 4\). First, solve the expression inside the parenthesis: \(12 - 7 = 5\). Then, multiply the result by 5: \(5 imes 5 = 25\). Lastly, subtract 4: \(25 - 4 = 21\). This simplifies the numerator to 21.
02

Simplify the Denominator

The denominator is \(5^2 - 18\). First, calculate \(5^2 = 25\). Then subtract 18 from 25: \(25 - 18 = 7\). This simplifies the denominator to 7.
03

Simplify the Entire Fraction

Now that both the numerator and denominator are simplified, the fraction becomes \(\frac{21}{7}\). Divide 21 by 7 to simplify the fraction: \(21 \div 7 = 3\). The simplified form of the expression is 3.

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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, understanding the roles of the numerator and denominator is essential. The **numerator** is the top part of a fraction, representing how many parts we have. Meanwhile, the **denominator** is the bottom part, indicating the total number of equal parts the whole is divided into.
For example, in the fraction \(\frac{a}{b}\), \(a\) is the numerator, and \(b\) is the denominator.
In our original exercise, we started with a complex fraction \(\frac{5(12-7)-4}{5^{2}-18}\).
  • The expression \(5(12-7)-4\) was the numerator. We simplified it step-by-step to end up with just 21.
  • The denominator \(5^2 - 18\) was also simplified to just 7.
Understanding how each part functions separately helps in systematically simplifying fractions and solving more complex problems.
Order of Operations
The **order of operations** is crucial in solving mathematical expressions accurately. It follows a specific sequence often remembered by the acronym PEMDAS:
  • **P**arentheses
  • **E**xponents
  • **M**ultiplication and **D**ivision
  • **A**ddition and **S**ubtraction
When we applied it to our exercise, we began by resolving operations inside the parentheses.
This meant calculating \(12 - 7\) first.
Next, we tackled any potential exponents鈥攆or the denominator鈥攃alculated \(5^2\).
Finally, we performed multiplication, division, addition, or subtraction, as needed.
By strictly adhering to this sequence, we ensure every calculation leads to the correct simplification of the expression.
Division in Fractions
Once both the numerator and denominator are simplified, the next step often involves **division**. This is how we truly "simplify" a fraction. When the numerator and denominator share a common factor, division can shrink the fraction to its simplest form.
In our example, the fraction \(\frac{21}{7}\) simplifies because 21 can be divided evenly by 7.
The calculation is straightforward:
  • Divide 21 by 7 to get 3.
  • The fraction \(\frac{21}{7}\) simplifies directly to 3.
This process reduces the fraction to its smallest size and often leads to an integer.
Understanding division in fractions helps make problems less daunting and allows solutions to be more manageable.

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Most popular questions from this chapter

Identify which number is the minuend and which number is the subtrahend. $$ \text { Subtract } 7 \text { from } 70 $$

The smallest commercially available jigsaw puzzle is a 1000 -piece puzzle manufactured in Spain. It is in the shape of a rectangle with length of 18 inches and width of 12 inches. Find the perimeter of this rectangular-shaped puzzle.

Without computing it, tell whether the average of \(126,135,198,\) and 113 is \(86 .\) Explain why it is possible or why it is not.

Write each whole number in expanded form. $$209$$

In Example 12 in this section, we found that the average of \(75,96,81,\) and 88 is \(85 .\) If the number 96 is removed from the list of numbers, does the average increase or decrease? Explain why.
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