Three Numbers \[$ N, Y, Z \$\] Are In The Ratio Of \[$ 2:7:11 \$\].Calculate The Value Of \[$\frac{9n - 2y}{5y - 2z}\$\].
Introduction
In mathematics, ratios are used to compare the size of two or more quantities. Given three numbers { n, y, z $}$ in the ratio of { 2:7:11 $}$, we can use this information to calculate the value of {\frac{9n - 2y}{5y - 2z}$}$. In this article, we will explore the concept of ratios, how to express them mathematically, and how to use them to solve problems.
Understanding Ratios
A ratio is a comparison of two or more quantities. It is usually expressed as a fraction, with the first quantity as the numerator and the second quantity as the denominator. For example, if we have two quantities, { a $}$ and { b $}$, the ratio of { a $}$ to { b $}$ is written as { \frac{a}{b} $}$.
In the given problem, we have three numbers { n, y, z $}$ in the ratio of { 2:7:11 $}$. This means that the ratio of { n $}$ to { y $}$ is { \frac{2}{7} $}$, the ratio of { y $}$ to { z $}$ is { \frac{7}{11} $}$, and the ratio of { n $}$ to { z $}$ is { \frac{2}{11} $}$.
Expressing Ratios Mathematically
We can express the given ratio mathematically as:
{ \frac{n}{y} = \frac{2}{7} $}{$ \frac{y}{z} = \frac{7}{11} $}{$ \frac{n}{z} = \frac{2}{11} $}$
Solving the Problem
To solve the problem, we need to find the value of {\frac{9n - 2y}{5y - 2z}$}$. We can start by expressing { n $}$ and { z $}$ in terms of { y $}$.
From the first equation, we can express { n $}$ as:
{ n = \frac{2}{7}y $}$
From the second equation, we can express { z $}$ as:
{ z = \frac{11}{7}y $}$
Now, we can substitute these expressions into the given expression:
{\frac{9n - 2y}{5y - 2z}$ = \frac{9(\frac{2}{7}y) - 2y}{5y - 2(\frac{11}{7}y)}$
Simplifying the expression, we get:
[$\frac{9n - 2y}{5y - 2z}$ = \frac{\frac{18}{7}y - 2y}{5y - \frac{22}{7}y}$
Combining like terms, we get:
[$\frac{9n - 2y}{5y - 2z}$ = \frac{\frac{18}{7}y - \frac{14}{7}y}{\frac{35}{7}y - \frac{22}{7}y}$
Simplifying further, we get:
[$\frac{9n - 2y}{5y - 2z}$ = \frac{\frac{4}{7}y}{\frac{13}{7}y}$
Canceling out the common factor of [$ \frac{1}{7} $}$, we get:
{\frac{9n - 2y}{5y - 2z}$ = \frac{4}{13}$
Therefore, the value of [\frac{9n - 2y}{5y - 2z}\$} is { \frac{4}{13} $}$.
Conclusion
In this article, we have explored the concept of ratios and how to use them to solve problems. We have used the given ratio of { 2:7:11 $}$ to express the numbers { n, y, z $}$ mathematically and then used these expressions to solve the problem. The value of {\frac{9n - 2y}{5y - 2z}$}$ is { \frac{4}{13} $}$.
Final Answer
The final answer is { \frac{4}{13} $}$.
Additional Resources
For more information on ratios and how to use them to solve problems, please refer to the following resources:
- Khan Academy: Ratios and Proportions
- Math Is Fun: Ratios
- Wikipedia: Ratio (mathematics)
References
- "Ratios and Proportions" by Khan Academy
- "Ratios" by Math Is Fun
- "Ratio (mathematics)" by Wikipedia
Frequently Asked Questions: Ratios and Proportions =====================================================
Q: What is a ratio?
A: A ratio is a comparison of two or more quantities. It is usually expressed as a fraction, with the first quantity as the numerator and the second quantity as the denominator.
Q: How do I express a ratio mathematically?
A: To express a ratio mathematically, you can use the following format:
{ \frac{a}{b} $}$
Where { a $}$ is the first quantity and { b $}$ is the second quantity.
Q: What is the difference between a ratio and a proportion?
A: A ratio is a comparison of two or more quantities, while a proportion is a statement that two ratios are equal. For example:
{ \frac{a}{b} = \frac{c}{d} $}$
This is a proportion, where the ratio of { a $}$ to { b $}$ is equal to the ratio of { c $}$ to { d $}$.
Q: How do I solve a proportion?
A: To solve a proportion, you can use the following steps:
- Write the proportion as an equation.
- Cross-multiply the equation.
- Solve for the unknown quantity.
For example:
{ \frac{a}{b} = \frac{c}{d} $}$
Cross-multiplying, we get:
{ ad = bc $}$
Solving for { a $}$, we get:
{ a = \frac{bc}{d} $}$
Q: What is the concept of equivalent ratios?
A: Equivalent ratios are ratios that have the same value, but are expressed in different ways. For example:
{ \frac{a}{b} = \frac{c}{d} $}$
This is an equivalent ratio, where the ratio of { a $}$ to { b $}$ is equal to the ratio of { c $}$ to { d $}$.
Q: How do I simplify a ratio?
A: To simplify a ratio, you can divide both the numerator and the denominator by their greatest common divisor (GCD). For example:
{ \frac{12}{18} $}$
The GCD of 12 and 18 is 6. Dividing both the numerator and the denominator by 6, we get:
{ \frac{2}{3} $}$
Q: What is the concept of ratios in real-life situations?
A: Ratios are used in many real-life situations, such as:
- Cooking: Recipes often use ratios of ingredients to produce a desired outcome.
- Building: Architects use ratios to design buildings and ensure that they are proportional.
- Finance: Investors use ratios to evaluate the performance of stocks and bonds.
Q: How do I use ratios to solve problems?
A: To use ratios to solve problems, you can follow these steps:
- Identify the ratio that is relevant to the problem.
- Express the ratio mathematically.
- Use the ratio to solve the problem.
For example:
- If a recipe calls for a ratio of 2:3 of flour to sugar, and you want to make a batch that serves 12 people, how much flour and sugar will you need?
- If a building has a ratio of 3:4 of windows to doors, and you want to add 20 more windows, how many doors will you need to add?
Conclusion
Ratios and proportions are fundamental concepts in mathematics that have many real-life applications. By understanding how to express ratios mathematically and how to use them to solve problems, you can become proficient in using ratios to solve a wide range of problems.
Additional Resources
For more information on ratios and proportions, please refer to the following resources:
- Khan Academy: Ratios and Proportions
- Math Is Fun: Ratios
- Wikipedia: Ratio (mathematics)
References
- "Ratios and Proportions" by Khan Academy
- "Ratios" by Math Is Fun
- "Ratio (mathematics)" by Wikipedia