Problem 200 - Intermediate Algebra

Question

In the following exercises, simplify. $$ \frac{8(9-2)-4(14-9)}{7(8-3)-3(16-9)} $$

Step-by-Step Solution

Answer

The simplified fraction is \( \frac{18}{7} \).

Step 1 - Simplify the Numerator

First, simplify the terms in the numerator: \[ \frac{8(9-2)-4(14-9)}{7(8-3)-3(16-9)} \] Calculate inside the parentheses first: \[ 9-2 = 7 \] and \[ 14-9 = 5 \] So the numerator becomes: \[ 8(7) - 4(5) \]

Step 2 - Calculate the Numerator

Now perform the multiplications in the numerator: \[ 8(7) = 56 \] \[ 4(5) = 20 \] Then subtract the results: \[ 56 - 20 = 36 \] So the simplified numerator is 36.

Step 3 - Simplify the Denominator

Next, simplify the terms in the denominator: \[ \frac{8(9-2)-4(14-9)}{7(8-3)-3(16-9)} \] Calculate inside the parentheses first: \[ 8-3 = 5 \] and \[ 16-9 = 7 \] So the denominator becomes: \[ 7(5) - 3(7) \]

Step 4 - Calculate the Denominator

Now perform the multiplications in the denominator: \[ 7(5) = 35 \] \[ 3(7) = 21 \] Then subtract the results: \[ 35 - 21 = 14 \] So the simplified denominator is 14.

Step 5 - Simplify the Fraction

Finally, simplify the fraction by dividing the numerator by the denominator: \[ \frac{36}{14} \] To simplify, divide both the numerator and the denominator by their greatest common divisor, which is 2: \[ \frac{36 \div 2}{14 \div 2} = \frac{18}{7} \]

Key Concepts

Numerator

The numerator is the top part of a fraction. In the exercise, the numerator is the part above the division line. It shows how many parts of the whole we have. For example, in \(\frac{36}{14}\), the numerator is 36. To simplify, we first solved inside the parentheses and then performed the multiplications: \ 8(9-2)-4(14-9) = 8(7)-4(5) = 56-20 = 36 \.

Denominator

The denominator is the bottom part of a fraction. It shows the total number of equal parts the whole is divided into. In \(\frac{36}{14}\), the denominator is 14. Following similar steps as with the numerator, we simplified the denominator by solving inside the parentheses first and then performing the multiplications: \ 7(8-3)-3(16-9) = 7(5)-3(7) = 35-21 = 14 \.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that divides both the numerator and the denominator without leaving a remainder. For \(\frac{36}{14}\), the GCD is 2. By dividing both the numerator and the denominator by the GCD, we simplify \(\frac{36}{14}\) to \(\frac{18}{7}\). Finding the GCD helps in reducing fractions to their simplest form.

Fractions

A fraction represents parts of a whole. It consists of a numerator and a denominator. For example, \(\frac{3}{4}\) means 3 parts out of 4 total parts. Simplifying fractions involves reducing them to their simplest form, as shown in the exercise: \ \frac{36}{14} \Rightarrow \frac{18}{7} \.