Problem 199
Question
In the following exercises, simplify. $$ \frac{9(8-2)-3(15-7)}{6(7-1)-3(17-9)} $$
Step-by-Step Solution
Verified Answer
\( \frac{5}{2} \)
1Step 1 - Simplify Inside Parentheses
Simplify the expressions inside each parentheses: \( 8-2 = 6 \) and \( 15-7 = 8 \), \( 7-1 = 6 \) and \( 17-9 = 8 \)
2Step 2 - Substitute Simplified Values
Place the simplified values back into the fractions: \( \frac{9(6)-3(8)}{6(6)-3(8)} \)
3Step 3 - Perform Multiplications
Multiply the coefficients with the simplified values: \( 9 \times 6 = 54 \) and \( 3 \times 8 = 24 \), \( 6 \times 6 = 36 \) and \( 3 \times 8 = 24 \)
4Step 4 - Substitute the Multiplications
Replace the results of the multiplications into the fraction: \( \frac{54-24}{36-24} \)
5Step 5 - Perform Subtractions
Subtract the numbers in the numerator and the denominator: \( 54-24 = 30 \) and \( 36-24 = 12 \)
6Step 6 - Simplify the Fraction
Divide the simplified numerator and denominator: \( \frac{30}{12} = \frac{5}{2} \)
Key Concepts
FractionsOrder of OperationsAlgebraic SimplificationNumerator and Denominator
Fractions
When dealing with algebraic fractions, it's important to understand what a fraction is. A fraction consists of a numerator, the top number, and a denominator, the bottom number.
In algebra, fractions work the same way as in arithmetic, but they can include variables and other expressions.
To simplify, you often need to perform operations like addition, subtraction, multiplication, and division on both the numerator and the denominator.
Understanding how fractions interact with algebraic expressions is key to simplifying complex problems.
In algebra, fractions work the same way as in arithmetic, but they can include variables and other expressions.
To simplify, you often need to perform operations like addition, subtraction, multiplication, and division on both the numerator and the denominator.
Understanding how fractions interact with algebraic expressions is key to simplifying complex problems.
Order of Operations
The order of operations is crucial in mathematics to get the correct answer.
It follows the PEMDAS/BODMAS rule:
In the given exercise, you see the parentheses handled first.
Next, multiplication and division are done before finally performing addition and subtraction.
Following this order ensures that each part of the equation is simplified correctly.
It follows the PEMDAS/BODMAS rule:
- Parentheses/Brackets
- Exponents/Orders
- Multiplication and Division (from left to right)
- Addition and Subtraction (from left to right)
In the given exercise, you see the parentheses handled first.
Next, multiplication and division are done before finally performing addition and subtraction.
Following this order ensures that each part of the equation is simplified correctly.
Algebraic Simplification
Algebraic simplification involves reducing equations and expressions to their simplest form.
To achieve this, follow these steps:
In our exercise, simplifying inside the parentheses came first.
Then we moved on to multiplication and finally simplification of the fraction as a whole.
To achieve this, follow these steps:
- Simplify inside all parentheses.
- Carry out any multiplication or division within the fractions.
- Perform addition and subtraction as needed.
In our exercise, simplifying inside the parentheses came first.
Then we moved on to multiplication and finally simplification of the fraction as a whole.
Numerator and Denominator
The numerator and denominator are the components of a fraction.
The numerator is the top part, and the denominator is the bottom part.
In algebra, both can be expressions that need simplification.
In our problem, we simplified the expressions within the numerator and the denominator separately before dividing.
Simplifying these parts correctly ensures that the overall fraction is simplified correctly.
Breaking down the numerator and denominator into simpler expressions often makes the problem more manageable.
The numerator is the top part, and the denominator is the bottom part.
In algebra, both can be expressions that need simplification.
In our problem, we simplified the expressions within the numerator and the denominator separately before dividing.
Simplifying these parts correctly ensures that the overall fraction is simplified correctly.
Breaking down the numerator and denominator into simpler expressions often makes the problem more manageable.
Other exercises in this chapter
Problem 197
In the following exercises, simplify. $$ \frac{7 \cdot 4-2(8-5)}{9 \cdot 3-3 \cdot 5} $$
View solution Problem 198
In the following exercises, simplify. $$ \frac{9 \cdot 7-3(12-8)}{8 \cdot 7-6 \cdot 6} $$
View solution Problem 200
In the following exercises, simplify. $$ \frac{8(9-2)-4(14-9)}{7(8-3)-3(16-9)} $$
View solution Problem 201
In the following exercises, simplify. $$ \frac{2^{3}+4^{2}}{\left(\frac{2}{3}\right)^{2}} $$
View solution