Problem 31
Question
For exercises \(25-68\), evaluate or simplify. $$ \frac{\frac{4}{9}+\frac{3}{2}}{\frac{5}{9}-\frac{1}{6}} $$
Step-by-Step Solution
Verified Answer
5
1Step 1: Find a common denominator for addition (Numerator)
Identify the least common denominator (LCD) for the fractions in the numerator. The fractions are \( \frac{4}{9} \) and \( \frac{3}{2} \). The LCD for 9 and 2 is 18. Rewrite each fraction with this common denominator: \( \frac{4}{9} = \frac{8}{18} \) and \( \frac{3}{2} = \frac{27}{18} \).
2Step 2: Add the fractions in the numerator
Now add the fractions: \( \frac{8}{18} + \frac{27}{18} = \frac{35}{18} \). Thus, the numerator is \( \frac{35}{18} \).
3Step 3: Find a common denominator for subtraction (Denominator)
Identify the least common denominator (LCD) for the fractions in the denominator. The fractions are \( \frac{5}{9} \) and \( \frac{1}{6} \). The LCD for 9 and 6 is 18. Rewrite each fraction with this common denominator: \( \frac{5}{9} = \frac{10}{18} \) and \( \frac{1}{6} = \frac{3}{18} \).
4Step 4: Subtract the fractions in the denominator
Now subtract the fractions: \( \frac{10}{18} - \frac{3}{18} = \frac{7}{18} \). Thus, the denominator is \( \frac{7}{18} \).
5Step 5: Divide the fractions
To divide by a fraction, multiply by its reciprocal: \( \frac{\frac{35}{18}}{\frac{7}{18}} = \frac{35}{18} \times \frac{18}{7} \).
6Step 6: Simplify the expression
Simplify the resulting expression: \( \frac{35}{18} \times \frac{18}{7} = \frac{35 \times 18}{18 \times 7} = \frac{35}{7} = 5 \).
Key Concepts
least common denominatorfraction additionfraction subtractionfraction divisionsimplifying fractions
least common denominator
When adding or subtracting fractions, the first step is finding a common platform so you can operate on them directly. This is where the Least Common Denominator (LCD) comes in. The LCD is the smallest number that both denominators can divide into evenly.
For instance, in the problem given, we had the fractions \(\frac{4}{9}\) and \(\frac{3}{2}\) in the numerator. The denominators are 9 and 2. Since 18 is the smallest number that both 9 and 2 can evenly divide into, it becomes our LCD here.
Always change the fractions so that they have this common denominator. The same applies in the denominator with fractions \(\frac{5}{9}\) and \(\frac{1}{6}\); their LCD is also 18.
With this step, we ensure that all our fractions have the same denominator, simplifying our calculations.
For instance, in the problem given, we had the fractions \(\frac{4}{9}\) and \(\frac{3}{2}\) in the numerator. The denominators are 9 and 2. Since 18 is the smallest number that both 9 and 2 can evenly divide into, it becomes our LCD here.
Always change the fractions so that they have this common denominator. The same applies in the denominator with fractions \(\frac{5}{9}\) and \(\frac{1}{6}\); their LCD is also 18.
With this step, we ensure that all our fractions have the same denominator, simplifying our calculations.
fraction addition
Adding fractions with different denominators can seem tricky, but it's easy once they share a common one.
From the exercise, after converting \(\frac{4}{9}\) and \(\frac{3}{2}\) to \(\frac{8}{18}\) and \(\frac{27}{18}\) respectively, you can easily add them by adding their numerators.
This gives you \(\frac{8+27}{18} = \frac{35}{18}\).
The process is simple:
From the exercise, after converting \(\frac{4}{9}\) and \(\frac{3}{2}\) to \(\frac{8}{18}\) and \(\frac{27}{18}\) respectively, you can easily add them by adding their numerators.
This gives you \(\frac{8+27}{18} = \frac{35}{18}\).
The process is simple:
- Convert to a common denominator.
- Add the numerators.
- Keep the common denominator unchanged.
fraction subtraction
Subtracting fractions also requires a common denominator. Similar to addition, once the fractions have the same denominator, you subtract their numerators.
For the given example, we converted \(\frac{5}{9}\) and \(\frac{1}{6}\) to \(\frac{10}{18}\) and \(\frac{3}{18}\) respectively.
Then, you simply subtract the numerators: \(\frac{10-3}{18} = \frac{7}{18}\).
Just follow these steps:
For the given example, we converted \(\frac{5}{9}\) and \(\frac{1}{6}\) to \(\frac{10}{18}\) and \(\frac{3}{18}\) respectively.
Then, you simply subtract the numerators: \(\frac{10-3}{18} = \frac{7}{18}\).
Just follow these steps:
- Make denominators the same.
- Subtract the numerators.
- Keep the common denominator.
fraction division
To divide fractions, you multiply by the reciprocal of the divisor fraction. The reciprocal is simply flipping the numerator and the denominator of the fraction.
You start by taking \(\frac{35}{18}\) and dividing it by \(\frac{7}{18}\), which is equivalent to multiplying by the reciprocal of \(\frac{7}{18}\), i.e., \(\frac{18}{7}\).
So, \(\frac{35}{18} \times \frac{18}{7}\) results in \(\frac{35 \times 18}{18 \times 7}\).
Notice how the 18s cancel out, simplifying to \(\frac{35}{7} = 5\).
You start by taking \(\frac{35}{18}\) and dividing it by \(\frac{7}{18}\), which is equivalent to multiplying by the reciprocal of \(\frac{7}{18}\), i.e., \(\frac{18}{7}\).
So, \(\frac{35}{18} \times \frac{18}{7}\) results in \(\frac{35 \times 18}{18 \times 7}\).
Notice how the 18s cancel out, simplifying to \(\frac{35}{7} = 5\).
simplifying fractions
In mathematics, simplifying a fraction means reducing it to its simplest form.
Once you've performed all operations, simplify the resulting fraction if possible. In our example, after performing the division, we got \(\frac{35}{7} = 5\).
Here, we further reduce it to an integer, as there are no more common factors between the numerator and denominator other than 1.
Simply put:
Once you've performed all operations, simplify the resulting fraction if possible. In our example, after performing the division, we got \(\frac{35}{7} = 5\).
Here, we further reduce it to an integer, as there are no more common factors between the numerator and denominator other than 1.
Simply put:
- Identify common factors between the numerator and denominator.
- Divide both by their Greatest Common Divisor (GCD).
Other exercises in this chapter
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For exercises 7-32, simplify. $$ \frac{28 x^{6}+42 x^{5}}{x^{3}-x^{2}} \cdot \frac{x^{2}-1}{42 x+63} $$
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