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Chapter 4: Problem 238

In the following exercises, simplify. $$\frac{5 \cdot 6-3 \cdot 4}{4 \cdot 5-2 \cdot 3}$$

Short Answer

Expert verified
The simplified form is \(\frac{9}{7}\).

Step by step solution

01

Simplify the Numerator

First, calculate the expressions in the numerator: \(5 \times 6 - 3 \times 4\). First, compute the multiplication: \(5 \times 6 = 30\) and \(3 \times 4 = 12\). Next, subtract the results: \(30 - 12 = 18\). Therefore, the simplified numerator is 18.
02

Simplify the Denominator

Next, calculate the expressions in the denominator: \(4 \times 5 - 2 \times 3\). First, compute the multiplication: \(4 \times 5 = 20\) and \(2 \times 3 = 6\). Next, subtract the results: \(20 - 6 = 14\). Therefore, the simplified denominator is 14.
03

Form the Simplified Fraction

Combine the simplified numerator and denominator to form the fraction: \(\frac{18}{14}\).
04

Reduce the Fraction

To simplify \(\frac{18}{14}\), find the greatest common divisor (GCD) of 18 and 14, which is 2. Divide both the numerator and the denominator by 2: \(\frac{18 \div 2}{14 \div 2} = \frac{9}{7}\).

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Key Concepts

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

prealgebra
Prealgebra is the foundation of all advanced math. It helps students develop the basic skills they need for algebra. It covers fundamental concepts such as operations with numbers, fractions, and integers. By mastering prealgebra, students can easily tackle more complex problems. This includes simplifying fractions like in our exercise. Understanding prealgebra ensures a solid math foundation.
numerator and denominator
In any fraction \(\frac{a}{b}\), \(a\) is the numerator and \(b\) is the denominator. The numerator represents the number of parts we have. The denominator shows the total number of equal parts. Together, they form a fraction. For example, in \(\frac{18}{14}\), 18 is the numerator and 14 is the denominator. Simplifying fractions involves working on both numerator and denominator.
greatest common divisor
The greatest common divisor (GCD) is the largest number that can divide two numbers without leaving a remainder. It is crucial when simplifying fractions. To simplify \(\frac{18}{14}\), we identified the GCD of 18 and 14, which is 2. Dividing both the numerator and the denominator by their GCD reduces the fraction. Thus, \(\frac{18}{14}\) simplifies to \(\frac{9}{7}\).
step-by-step math solutions
Solving problems step-by-step helps in understanding and solving complex math problems. Here's how to simplify a fraction step-by-step:
  • Step 1: Simplify the numerator. Calculate the operations to get the result.
  • Step 2: Simplify the denominator similarly.
  • Step 3: Form the fraction with the simplified numerator and denominator.
  • Step 4: Reduce the fraction by finding and dividing by the GCD.
This clear, structured approach ensures no steps are missed and the final solution is correct.

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Most popular questions from this chapter

In the following exercises, simplify. $$ -\frac{5}{6} \div \frac{5}{12} $$

In the following exercises, solve the equation. $$ -\frac{2}{3} c=18 $$

In the following exercises, perform the indicated operation. $$ 8 \frac{6}{11}-2 \frac{9}{11} $$

In the following exercises, simplify. $$ \left(-15 \frac{5}{6}\right) \div\left(-3 \frac{1}{6}\right) $$

In the following exercises, simplify. $$ -5 \frac{7}{12} \cdot 4 \frac{4}{11} $$
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