Given:$ \begin{array}{l} e F = 3 : 7 \ f : G = 2 : 3 \end{array }$Work Out { E : G $}$.Give Your Answer In Its Simplest Form.Optional WorkingAnswer: { E : G = \boxed{\phantom{0}} $}$

Introduction

In mathematics, proportional relationships are used to describe the relationship between two or more quantities. These relationships are often represented using ratios, which are a way of comparing two or more numbers. In this article, we will explore how to work out a proportional relationship given two or more ratios.

Given Ratios

We are given two ratios:

${ e : f = 3 : 7 }$ ${ f : g = 2 : 3 }$

Our goal is to work out the ratio ${ e : g }$.

Step 1: Understanding the Given Ratios

Let's start by understanding the given ratios. The first ratio, ${ e : f = 3 : 7 }$, tells us that for every 3 units of ${ e }$, there are 7 units of ${ f }$. Similarly, the second ratio, ${ f : g = 2 : 3 }$, tells us that for every 2 units of ${ f }$, there are 3 units of ${ g }$.

Step 2: Finding a Common Link

To work out the ratio ${ e : g }$, we need to find a common link between the two given ratios. In this case, the common link is the variable ${ f }$, which appears in both ratios.

Step 3: Using Cross Multiplication

We can use cross multiplication to find the relationship between ${ e }$ and ${ g }$. Cross multiplication involves multiplying the two ratios together and then dividing to find the relationship between the variables.

${ \frac{e}{f} \times \frac{f}{g} = \frac{e}{g} }$

Substituting the given ratios, we get:

${ \frac{3}{7} \times \frac{2}{3} = \frac{e}{g} }$

Step 4: Simplifying the Expression

To simplify the expression, we can cancel out any common factors. In this case, the 3 in the numerator and denominator can be cancelled out.

${ \frac{3}{7} \times \frac{2}{3} = \frac{2}{7} }$

Therefore, the ratio ${ e : g }$ is equal to ${ 2 : 7 }$.

Conclusion

In this article, we have shown how to work out a proportional relationship given two or more ratios. We used cross multiplication to find the relationship between the variables and then simplified the expression to find the final answer.

Answer

The final answer is:

${ e : g = 2 : 7 }$

Optional Working

Here is the optional working:

${ \frac{e}{f} \times \frac{f}{g} = \frac{e}{g} }$

Substituting the given ratios, we get:

${ \frac{3}{7} \times \frac{2}{3} = \frac{e}{g} }$

Simplifying the expression, we get:

${ \frac{2}{7} = \frac{e}{g} }$

Therefore, the ratio ${ e : g }$ is equal to ${ 2 : 7 }$.

Discussion

This problem is a great example of how proportional relationships can be used to solve real-world problems. In this case, we used cross multiplication to find the relationship between two variables. This technique can be used in a variety of situations, such as calculating the cost of goods or determining the amount of time it takes to complete a task.

Real-World Applications

Proportional relationships are used in a variety of real-world applications, such as:

• Calculating the cost of goods
• Determining the amount of time it takes to complete a task
• Finding the relationship between two or more variables
• Solving problems involving ratios and proportions

Conclusion

Introduction

In our previous article, we explored how to work out a proportional relationship given two or more ratios. In this article, we will answer some common questions related to proportional relationships.

Q: What is a proportional relationship?

A proportional relationship is a relationship between two or more quantities that can be represented using a ratio. Ratios are a way of comparing two or more numbers.

Q: How do I know if two ratios are proportional?

To determine if two ratios are proportional, you can use the following steps:

1. Write down the two ratios.
2. Cross multiply the two ratios.
3. Simplify the expression.
4. If the simplified expression is equal to 1, then the two ratios are proportional.

Q: What is cross multiplication?

Cross multiplication is a technique used to find the relationship between two or more variables. It involves multiplying the two ratios together and then dividing to find the relationship between the variables.

Q: How do I use cross multiplication to find a proportional relationship?

To use cross multiplication to find a proportional relationship, follow these steps:

1. Write down the two ratios.
2. Multiply the two ratios together.
3. Divide the result by the product of the two ratios.
4. Simplify the expression.
5. The result is the proportional relationship.

Q: What are some real-world applications of proportional relationships?

Proportional relationships have many real-world applications, such as:

• Calculating the cost of goods
• Determining the amount of time it takes to complete a task
• Finding the relationship between two or more variables
• Solving problems involving ratios and proportions

Q: How do I simplify a proportion?

To simplify a proportion, follow these steps:

1. Write down the proportion.
2. Look for any common factors in the numerator and denominator.
3. Cancel out any common factors.
4. Simplify the expression.

Q: What is the difference between a proportion and a ratio?

A proportion is a statement that two ratios are equal, while a ratio is a comparison of two or more numbers.

Q: How do I determine if a proportion is true?

To determine if a proportion is true, follow these steps:

1. Write down the proportion.
2. Cross multiply the two ratios.
3. Simplify the expression.
4. If the simplified expression is equal to 1, then the proportion is true.

Q: What are some common mistakes to avoid when working with proportions?

Some common mistakes to avoid when working with proportions include:

• Not simplifying the expression
• Not canceling out common factors
• Not using cross multiplication correctly

Conclusion

In conclusion, proportional relationships are an important concept in mathematics that can be used to solve a variety of problems. By understanding how to work with proportions and avoiding common mistakes, you can become proficient in using proportional relationships to solve real-world problems.

Additional Resources

For more information on proportional relationships, check out the following resources:

• Khan Academy: Proportional Relationships
• Mathway: Proportional Relationships
• Wolfram Alpha: Proportional Relationships

Practice Problems

Try the following practice problems to test your understanding of proportional relationships:

1. If ${ a : b = 2 : 3 }$ and ${ b : c = 4 : 5 }$, what is the value of ${ a : c }$?
2. If ${ x : y = 3 : 4 }$ and ${ y : z = 5 : 6 }$, what is the value of ${ x : z }$?
3. If ${ a : b = 2 : 3 }$ and ${ b : c = 4 : 5 }$, what is the value of ${ a : c }$?

Answer Key

1. ${ a : c = 6 : 10 }$
2. ${ x : z = 15 : 24 }$
3. ${ a : c = 8 : 15 }$