(ii) If $a : B = 9 : 5$, Using Properties Of Proportion Only, Find $\frac{10a + 9b}{10a - 9b}$.

Introduction

Proportions are a fundamental concept in mathematics, used to describe the relationship between two or more quantities. In this article, we will explore how to use properties of proportion to solve a specific problem involving ratios. We will use the given ratio $a : b = 9 : 5$ to find the value of $\frac{10a + 9b}{10a - 9b}$.

Understanding Proportions

A proportion is a statement that two ratios are equal. It can be written in the form $\frac{a}{b} = \frac{c}{d}$, where $a$, $b$, $c$, and $d$ are numbers. Proportions can be used to solve a wide range of problems, from simple ratios to more complex equations.

Given Ratio

The given ratio is $a : b = 9 : 5$. This means that the ratio of $a$ to $b$ is equal to the ratio of 9 to 5. We can write this as an equation:

$$\frac{a}{b} = \frac{9}{5}$$

Using Properties of Proportion

To solve the problem, we will use the properties of proportion. One of the key properties of proportion is that if two ratios are equal, then the corresponding parts of the ratios are in proportion. This means that if $\frac{a}{b} = \frac{c}{d}$, then $a : b = c : d$.

**Step 1: Find the Value of $\frac{a}{b}$

We are given the ratio $a : b = 9 : 5$. We can write this as an equation:

$$\frac{a}{b} = \frac{9}{5}$$

**Step 2: Find the Value of $\frac{10a + 9b}{10a - 9b}$

We are asked to find the value of $\frac{10a + 9b}{10a - 9b}$. To do this, we can use the given ratio $a : b = 9 : 5$.

We can start by multiplying both sides of the equation by 10:

$$10 \cdot \frac{a}{b} = 10 \cdot \frac{9}{5}$$

This gives us:

$$\frac{10a}{b} = \frac{90}{5}$$

Simplifying, we get:

$$\frac{10a}{b} = 18$$

**Step 3: Find the Value of $\frac{10a + 9b}{10a - 9b}$

Now, we can use the value of $\frac{10a}{b}$ to find the value of $\frac{10a + 9b}{10a - 9b}$.

We can start by multiplying both sides of the equation by $b$:

$$\frac{10a}{b} \cdot b = 18 \cdot b$$

This gives us:

$$10a = 18b$$

**Step 4: Find the Value of $\frac{10a + 9b}{10a - 9b}$

Now, we can use the value of $10a = 18b$ to find the value of $\frac{10a + 9b}{10a - 9b}$.

We can start by adding $9b$ to both sides of the equation:

$$10a + 9b = 18b + 9b$$

This gives us:

$$10a + 9b = 27b$$

**Step 5: Find the Value of $\frac{10a + 9b}{10a - 9b}$

Now, we can use the value of $10a + 9b = 27b$ to find the value of $\frac{10a + 9b}{10a - 9b}$.

We can start by subtracting $9b$ from both sides of the equation:

$$10a = 27b - 9b$$

This gives us:

$$10a = 18b$$

**Step 6: Find the Value of $\frac{10a + 9b}{10a - 9b}$

Now, we can use the value of $10a = 18b$ to find the value of $\frac{10a + 9b}{10a - 9b}$.

We can start by dividing both sides of the equation by $10a$:

$$\frac{10a}{10a} = \frac{18b}{10a}$$

This gives us:

$$1 = \frac{18b}{10a}$$

**Step 7: Find the Value of $\frac{10a + 9b}{10a - 9b}$

Now, we can use the value of $1 = \frac{18b}{10a}$ to find the value of $\frac{10a + 9b}{10a - 9b}$.

We can start by multiplying both sides of the equation by $10a - 9b$:

$$1 \cdot (10a - 9b) = \frac{18b}{10a} \cdot (10a - 9b)$$

This gives us:

$$10a - 9b = \frac{18b}{10a} \cdot (10a - 9b)$$

**Step 8: Find the Value of $\frac{10a + 9b}{10a - 9b}$

Now, we can use the value of $10a - 9b = \frac{18b}{10a} \cdot (10a - 9b)$ to find the value of $\frac{10a + 9b}{10a - 9b}$.

We can start by dividing both sides of the equation by $10a - 9b$:

$$\frac{10a + 9b}{10a - 9b} = \frac{18b}{10a}$$

Conclusion

In this article, we used the properties of proportion to solve a specific problem involving ratios. We started with the given ratio $a : b = 9 : 5$ and used it to find the value of $\frac{10a + 9b}{10a - 9b}$. We broke down the problem into smaller steps and used the properties of proportion to simplify the equation. Finally, we arrived at the solution $\frac{10a + 9b}{10a - 9b} = \frac{18b}{10a}$.

Final Answer

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Introduction

In our previous article, we explored how to use properties of proportion to solve a specific problem involving ratios. We used the given ratio $a : b = 9 : 5$ to find the value of $\frac{10a + 9b}{10a - 9b}$. In this article, we will answer some common questions related to solving proportions.

Q: What is a proportion?

A proportion is a statement that two ratios are equal. It can be written in the form $\frac{a}{b} = \frac{c}{d}$, where $a$, $b$, $c$, and $d$ are numbers.

Q: How do I use properties of proportion to solve a problem?

To use properties of proportion to solve a problem, you need to follow these steps:

1. Write the given ratio as an equation.
2. Use the properties of proportion to simplify the equation.
3. Solve for the unknown variable.

Q: What are some common properties of proportion?

Some common properties of proportion include:

• If $\frac{a}{b} = \frac{c}{d}$, then $a : b = c : d$.
• If $\frac{a}{b} = \frac{c}{d}$, then $a \cdot d = b \cdot c$.
• If $\frac{a}{b} = \frac{c}{d}$, then $\frac{a + b}{a - b} = \frac{c + d}{c - d}$.

Q: How do I use the properties of proportion to solve a problem involving ratios?

To use the properties of proportion to solve a problem involving ratios, you need to follow these steps:

1. Write the given ratio as an equation.
2. Use the properties of proportion to simplify the equation.
3. Solve for the unknown variable.

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

Some common mistakes to avoid when solving proportions include:

• Not writing the given ratio as an equation.
• Not using the properties of proportion to simplify the equation.
• Not solving for the unknown variable.

Q: How do I check my answer when solving a proportion?

To check your answer when solving a proportion, you need to follow these steps:

1. Write the given ratio as an equation.
2. Use the properties of proportion to simplify the equation.
3. Solve for the unknown variable.
4. Check that the solution satisfies the original equation.

Q: What are some real-world applications of proportions?

Some real-world applications of proportions include:

• Finance: Proportions are used to calculate interest rates and investment returns.
• Science: Proportions are used to calculate the concentration of a solution.
• Engineering: Proportions are used to calculate the size and shape of a structure.

Conclusion

In this article, we answered some common questions related to solving proportions. We covered the definition of a proportion, how to use properties of proportion to solve a problem, and some common mistakes to avoid. We also discussed some real-world applications of proportions. By following the steps outlined in this article, you can become proficient in solving proportions and apply them to a wide range of problems.

Final Answer

The final answer is $\boxed{\frac{18}{1}}$.