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Magazine Chapter 8: Problem 44
Divide. $$\frac{\frac{8 m-6}{m}}{\frac{40 m-30}{3 m^{2}}}$$
Short Answer
Expert verified
The short answer is \(\frac{9m^2}{10m}\).
Step by step solution
01
Write the given expression
We start by writing down the given expression:
\[\frac{\frac{8m-6}{m}}{\frac{40m-30}{3m^2}}\]
02
Reciprocal of the second fraction
To divide both fractions, we need to multiply the first fraction by the reciprocal of the second fraction:
\[\frac{8m-6}{m} \times \frac{3m^2}{40m-30}\]
03
Simplify the fractions
Now we need to simplify both fractions. We can factor out common terms in both the numerator and the denominator of the second fraction:
\[\frac{8m-6}{m} \times \frac{3m^2}{10(4m-3)}\]
04
Multiply the fractions
Now we can multiply the two simplified fractions:
\[\frac{(8m-6)(3m^2)}{m(10)(4m-3)}\]
05
Simplify further
Let's see if there are any common terms in the numerator and the denominator that we can cancel out. In the numerator, we can factor out a 3 from the term (8m-6) which results in 3(4m-3), so our expression becomes:
\[\frac{(3)(4m-3)(3m^2)}{m(10)(4m-3)}\]
Now, cancel out the common term (4m-3) from both the numerator and the denominator:
\[\frac{(3)(3m^2)}{m(10)}\]
06
Write the final answer
After simplifying the given expression, we get:
\[\frac{9m^2}{10m}\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Simplifying Fractions
When you deal with fractions, simplifying them is a way to make them easier to understand and solve. Simplifying a fraction means reducing it to its simplest form. This is done by removing any common factors between the numerator (top number) and the denominator (bottom number).
Here's how you simplify a fraction:
Often, simplifying fractions involves factorizing first to see these common terms, which we'll talk about next!
Here's how you simplify a fraction:
- First, find the greatest common factor (GCF) of the numerator and the denominator.
- Divide both the numerator and the denominator by this GCF.
Often, simplifying fractions involves factorizing first to see these common terms, which we'll talk about next!
Factorization
Factorization is a key concept in algebra and dealing with fractions, making a math problem easier by breaking down numbers or expressions into their "factors" or pieces. It's like finding the building blocks of a number!
For example, when factorizing \( 8m-6 \), notice both terms share a factor of 2. Thus, \( 8m-6 \) can be rewritten as \( 2(4m-3) \).
With polynomials or larger algebraic expressions, always look for the greatest common factor (GCF) you can "factor out," just like how 20 and 15 both "factor out" a 5.
Seeing these common factors helps simplify complex fractions by turning them into smaller, easier parts where divisors cancel out easily.
For example, when factorizing \( 8m-6 \), notice both terms share a factor of 2. Thus, \( 8m-6 \) can be rewritten as \( 2(4m-3) \).
With polynomials or larger algebraic expressions, always look for the greatest common factor (GCF) you can "factor out," just like how 20 and 15 both "factor out" a 5.
Seeing these common factors helps simplify complex fractions by turning them into smaller, easier parts where divisors cancel out easily.
Reciprocal of a Fraction
The reciprocal of a fraction simply means flipping it upside down. The numerator becomes the denominator and vice versa.
For example, if we have \( \frac{3}{4} \), its reciprocal would be \( \frac{4}{3} \).
Taking the reciprocal is especially useful in division. When you divide by a fraction, you actually multiply by its reciprocal instead. So, dividing one fraction by another can quickly turn into a multiplication problem.
In our exercise, we took the reciprocal of \( \frac{40m-30}{3m^2} \) to change the division into a multiplication problem, making it easier to handle.
For example, if we have \( \frac{3}{4} \), its reciprocal would be \( \frac{4}{3} \).
Taking the reciprocal is especially useful in division. When you divide by a fraction, you actually multiply by its reciprocal instead. So, dividing one fraction by another can quickly turn into a multiplication problem.
In our exercise, we took the reciprocal of \( \frac{40m-30}{3m^2} \) to change the division into a multiplication problem, making it easier to handle.
Multiplication of Fractions
Multiplying fractions is straightforward but requires careful handling of numerators and denominators.
To multiply two fractions:
In the previous exercise, after switching to multiplication by using reciprocals, we multiplied and then simplified the product. Multiplication plays a big role in simplifying complex algebraic expressions.
To multiply two fractions:
- Multiply the numerators (top numbers) together.
- Multiply the denominators (bottom numbers) together.
In the previous exercise, after switching to multiplication by using reciprocals, we multiplied and then simplified the product. Multiplication plays a big role in simplifying complex algebraic expressions.
