QUESTION 5: Solving Problems5.1 Write $ $24 18$ $ In Simplest Form.5.2 Complete The Table And Determine If The Proportion Is Direct Or Indirect. Give A Reason For Your Answer.$[ \begin{tabular}{|l|c|c|c|c|c|} \hline Number Of Pens & 2 &
5.1 Simplifying Fractions
When dealing with fractions, it's essential to simplify them to their lowest terms. This involves finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by the GCD. In this case, we need to simplify the fraction $$24 $.
To simplify the fraction, we first need to find the GCD of 24 and 18. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24, while the factors of 18 are 1, 2, 3, 6, 9, and 18. The greatest common factor of 24 and 18 is 6.
Now that we have the GCD, we can simplify the fraction by dividing both the numerator and the denominator by 6.
{\frac{24}{18} = \frac{24 \div 6}{18 \div 6} = \frac{4}{3}$}$
Therefore, the simplified form of the fraction $$24 $ is {\frac{4}{3}$}$.
5.2 Direct and Indirect Proportions
A proportion is a statement that two ratios are equal. It can be written in the form {a : b = c : d$}$, where {a$}$ and {b$}$ are the antecedent and consequent of the first ratio, and {c$}$ and {d$}$ are the antecedent and consequent of the second ratio.
In this problem, we are given a table with different numbers of pens and their corresponding prices. We need to complete the table and determine if the proportion is direct or indirect.
| Number of Pens | Price (in dollars) |
|---|---|
| 2 | 6 |
| 4 | 12 |
| 6 | 18 |
| 8 | ? |
To complete the table, we need to find the price of 8 pens. Since the proportion is direct, we can set up a proportion using the first two rows of the table.
{\frac{2}{6} = \frac{4}{12}$}$
We can simplify this proportion by dividing both sides by 2.
{\frac{1}{3} = \frac{2}{6}$}$
Now, we can use this proportion to find the price of 8 pens. We can set up a proportion using the first row of the table and the unknown price of 8 pens.
{\frac{2}{6} = \frac{8}{x}$}$
We can simplify this proportion by cross-multiplying.
${2x = 48\$}
Now, we can solve for {x$}$.
{x = \frac{48}{2} = 24$}$
Therefore, the price of 8 pens is ${24\$}.
Is the Proportion Direct or Indirect?
To determine if the proportion is direct or indirect, we need to examine the relationship between the antecedent and the consequent. In a direct proportion, the antecedent and the consequent are directly proportional, meaning that as one increases, the other also increases. In an indirect proportion, the antecedent and the consequent are inversely proportional, meaning that as one increases, the other decreases.
In this case, the price of pens increases as the number of pens increases. Therefore, the proportion is direct.
Conclusion
In this problem, we simplified a fraction and determined if a proportion is direct or indirect. We used the concept of greatest common divisor to simplify the fraction and the concept of direct proportion to complete the table and determine the price of 8 pens. We also examined the relationship between the antecedent and the consequent to determine if the proportion is direct or indirect.
Key Takeaways
- To simplify a fraction, find the greatest common divisor of the numerator and the denominator and divide both by the GCD.
- A proportion is a statement that two ratios are equal.
- In a direct proportion, the antecedent and the consequent are directly proportional.
- In an indirect proportion, the antecedent and the consequent are inversely proportional.
Practice Problems
- Simplify the fraction $$36 $.
- Complete the table and determine if the proportion is direct or indirect.
| Number of Pens | Price (in dollars) |
|---|---|
| 2 | 8 |
| 4 | 16 |
| 6 | 24 |
| 8 | ? |
- Determine if the proportion {\frac{3}{4} = \frac{9}{x}$}$ is direct or indirect.
Answer Key
- {\frac{3}{2}$}$
- The price of 8 pens is ${32\$}. The proportion is direct.
- The proportion is direct.
Frequently Asked Questions: Simplifying Fractions and Proportions ====================================================================
Q: What is a fraction?
A: A fraction is a way of expressing a part of a whole. It consists of a numerator (the top number) and a denominator (the bottom number). For example, {\frac{1}{2}$}$ is a fraction where 1 is the numerator and 2 is the denominator.
Q: How do I simplify a fraction?
A: To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and the denominator and divide both by the GCD. For example, to simplify the fraction $$24 $, you would find the GCD of 24 and 18, which is 6, and then divide both the numerator and the denominator by 6 to get {\frac{4}{3}$}$.
Q: What is a proportion?
A: A proportion is a statement that two ratios are equal. It can be written in the form {a : b = c : d$}$, where {a$}$ and {b$}$ are the antecedent and consequent of the first ratio, and {c$}$ and {d$}$ are the antecedent and consequent of the second ratio.
Q: How do I determine if a proportion is direct or indirect?
A: To determine if a proportion is direct or indirect, you need to examine the relationship between the antecedent and the consequent. In a direct proportion, the antecedent and the consequent are directly proportional, meaning that as one increases, the other also increases. In an indirect proportion, the antecedent and the consequent are inversely proportional, meaning that as one increases, the other decreases.
Q: What is the difference between a direct and an indirect proportion?
A: The main difference between a direct and an indirect proportion is the relationship between the antecedent and the consequent. In a direct proportion, the antecedent and the consequent are directly proportional, while in an indirect proportion, the antecedent and the consequent are inversely proportional.
Q: How do I use proportions to solve problems?
A: To use proportions to solve problems, you need to set up a proportion using the given information and then solve for the unknown variable. For example, if you are given the proportion {\frac{2}{6} = \frac{4}{x}$}$ and you need to find the value of {x$}$, you would set up the proportion and then solve for {x$}$.
Q: What are some common mistakes to avoid when working with fractions and proportions?
A: Some common mistakes to avoid when working with fractions and proportions include:
- Not simplifying fractions to their lowest terms
- Not checking for direct or indirect proportions
- Not setting up proportions correctly
- Not solving for the unknown variable correctly
Q: How can I practice simplifying fractions and working with proportions?
A: You can practice simplifying fractions and working with proportions by:
- Simplifying fractions and proportions in your head or on paper
- Using online resources or worksheets to practice
- Working with real-world examples, such as measuring ingredients for a recipe or calculating the cost of items
- Creating your own problems and solutions to practice and reinforce your understanding
Conclusion
In this article, we have covered frequently asked questions about simplifying fractions and working with proportions. We have discussed the basics of fractions and proportions, how to simplify fractions, and how to determine if a proportion is direct or indirect. We have also provided tips and resources for practicing and reinforcing your understanding of fractions and proportions.