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Chapter 1: Problem 409

Simplify. $$ \frac{7 \cdot 4-2(8-5)}{9 \cdot 3-3 \cdot 5} $$

Short Answer

Expert verified
\( \frac{11}{6} \)

Step by step solution

01

Simplify the Numerator

First, simplify the numerator: \( 7 \times 4 - 2(8 - 5) \). Calculate inside the parentheses: \( 8 - 5 = 3 \). Then, substitute back: \( 7 \times 4 - 2 \times 3 \). Multiply to get: \( 28 - 6 \). Finally, subtract: \( 28 - 6 = 22 \). So, the simplified numerator is 22.
02

Simplify the Denominator

Next, simplify the denominator: \( 9 \times 3 - 3 \times 5 \). Multiply to get: \( 27 - 15 \). Finally, subtract: \( 27 - 15 = 12 \). So, the simplified denominator is 12.
03

Divide the Simplified Numerator by the Simplified Denominator

Now, divide the simplified numerator by the simplified denominator: \( \frac{22}{12} \). Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2: \( \frac{22 \div 2}{12 \div 2} = \frac{11}{6} \). Therefore, the simplified form of the given expression is \( \frac{11}{6} \).

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Key Concepts

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

Fraction Simplification
Simplifying fractions involves making them as simple as possible. This means breaking down the fraction into its smallest components. It's important because it makes the fraction easier to understand and use.

To simplify a fraction:
  • Find the greatest common divisor (GCD) of the numerator and the denominator.
  • Divide both the numerator and the denominator by their GCD.
In our case, we simplified \(\frac{22}{12}\) to \(\frac{11}{6}\) by dividing both 22 and 12 by their GCD, which is 2. This process makes fractions manageable and easy to work with in further calculations.
Order of Operations
The order of operations is a set of rules used to clarify which procedures should be performed first in a given mathematical expression. It's symbolized as PEMDAS (Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)).

In the exercise:
  • First, we calculated inside the parentheses: \(8 - 5 = 3\).
  • Then, we performed the multiplication and subtraction in the numerator to get 22.
  • Similarly, we handled the multiplication and subtraction in the denominator to get 12.
  • Lastly, we simplified the division of the resulting numerator by the denominator.
This method ensures that calculations are done in a consistent and correct manner, avoiding ambiguity.
Numerator and Denominator
When working with fractions, it is essential to understand the terms numerator and denominator.

The numerator is the top number of a fraction. It represents how many parts of the whole you have.

The denominator is the bottom number. It represents the number of equal parts the whole is divided into.

In the given problem:
  • The original numerator is \(7 \times 4 - 2(8 - 5) = 22\).
  • The original denominator is \(9 \times 3 - 3 \times 5 = 12\).
Once we simplified both, we divided the numerator by the denominator to arrive at the simplified fraction \(\frac{11}{6}\). Knowing these terms helps in understanding the fraction's value and making further calculations more intuitive.

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