Solve For $b$ In The Proportion.$\frac{42}{36} = \frac{7}{b}$b =$ $\square$
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Introduction
In mathematics, proportions are used to compare the relationships between different quantities. A proportion is a statement that two ratios are equal. In this article, we will focus on solving for b in a proportion, specifically the equation . We will use algebraic techniques to isolate the variable b and find its value.
Understanding the Equation
The given equation is a proportion, where the ratio of 42 to 36 is equal to the ratio of 7 to b. To solve for b, we need to isolate the variable b on one side of the equation. We can start by cross-multiplying the two ratios, which means multiplying the numerator of the first ratio by the denominator of the second ratio, and vice versa.
Cross-Multiplying
Cross-multiplying the two ratios gives us the equation:
42b = 7(36)
Expanding the Equation
Next, we can expand the right-hand side of the equation by multiplying 7 by 36.
42b = 252
Isolating b
Now, we can isolate the variable b by dividing both sides of the equation by 42.
b = 252/42
Simplifying the Fraction
To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 6.
b = 42/7
Evaluating the Expression
Finally, we can evaluate the expression by dividing 42 by 7.
b = 6
Conclusion
In this article, we have solved for b in the proportion . By cross-multiplying, expanding, and isolating the variable b, we have found that b = 6. This is the value of b that satisfies the given proportion.
Applications of Proportions
Proportions have many real-world applications, including finance, science, and engineering. For example, in finance, proportions are used to calculate interest rates and investment returns. In science, proportions are used to describe the relationships between different physical quantities, such as the ratio of a substance's mass to its volume. In engineering, proportions are used to design and optimize systems, such as bridges and buildings.
Tips for Solving Proportions
When solving proportions, it's essential to follow these tips:
- Read the problem carefully: Make sure you understand what the problem is asking for.
- Identify the variables: Determine which variables are given and which one needs to be solved for.
- Use algebraic techniques: Use techniques such as cross-multiplication, expansion, and isolation to solve for the variable.
- Simplify the expression: Simplify the expression by dividing both the numerator and the denominator by their greatest common divisor.
- Evaluate the expression: Evaluate the expression by performing the necessary calculations.
Common Mistakes to Avoid
When solving proportions, it's essential to avoid these common mistakes:
- Not reading the problem carefully: Failing to read the problem carefully can lead to misunderstandings and incorrect solutions.
- Not identifying the variables: Failing to identify the variables can lead to incorrect solutions.
- Not using algebraic techniques: Failing to use algebraic techniques can lead to incorrect solutions.
- Not simplifying the expression: Failing to simplify the expression can lead to incorrect solutions.
- Not evaluating the expression: Failing to evaluate the expression can lead to incorrect solutions.
Conclusion
In conclusion, solving proportions is an essential skill in mathematics that has many real-world applications. By following the tips and avoiding common mistakes, you can solve proportions with confidence. Remember to read the problem carefully, identify the variables, use algebraic techniques, simplify the expression, and evaluate the expression to find the correct solution.
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Introduction
In our previous article, we solved for b in the proportion . We used algebraic techniques to isolate the variable b and find its value. In this article, we will answer some frequently asked questions about solving proportions.
Q&A
Q: What is a proportion?
A: A proportion is a statement that two ratios are equal. It is a way of comparing the relationships between different quantities.
Q: How do I solve a proportion?
A: To solve a proportion, you need to isolate the variable on one side of the equation. You can use algebraic techniques such as cross-multiplication, expansion, and isolation to solve for the variable.
Q: What is cross-multiplication?
A: Cross-multiplication is a technique used to solve proportions. It involves multiplying the numerator of the first ratio by the denominator of the second ratio, and vice versa.
Q: How do I simplify a fraction?
A: To simplify a fraction, you need to divide both the numerator and the denominator by their greatest common divisor.
Q: What is the greatest common divisor (GCD)?
A: The greatest common divisor (GCD) is the largest number that divides both the numerator and the denominator of a fraction.
Q: How do I evaluate an expression?
A: To evaluate an expression, you need to perform the necessary calculations to simplify the expression.
Q: What are some common mistakes to avoid when solving proportions?
A: Some common mistakes to avoid when solving proportions include not reading the problem carefully, not identifying the variables, not using algebraic techniques, not simplifying the expression, and not evaluating the expression.
Q: How do I apply proportions in real-world situations?
A: Proportions have many real-world applications, including finance, science, and engineering. For example, in finance, proportions are used to calculate interest rates and investment returns. In science, proportions are used to describe the relationships between different physical quantities, such as the ratio of a substance's mass to its volume. In engineering, proportions are used to design and optimize systems, such as bridges and buildings.
Q: What are some tips for solving proportions?
A: Some tips for solving proportions include reading the problem carefully, identifying the variables, using algebraic techniques, simplifying the expression, and evaluating the expression.
Conclusion
In conclusion, solving proportions is an essential skill in mathematics that has many real-world applications. By following the tips and avoiding common mistakes, you can solve proportions with confidence. Remember to read the problem carefully, identify the variables, use algebraic techniques, simplify the expression, and evaluate the expression to find the correct solution.
Additional Resources
For more information on solving proportions, you can refer to the following resources:
- Mathway: A online math problem solver that can help you solve proportions and other math problems.
- Khan Academy: A free online learning platform that offers video lessons and practice exercises on math and other subjects.
- Wolfram Alpha: A online calculator that can help you solve proportions and other math problems.
Final Thoughts
Solving proportions is an essential skill in mathematics that has many real-world applications. By following the tips and avoiding common mistakes, you can solve proportions with confidence. Remember to read the problem carefully, identify the variables, use algebraic techniques, simplify the expression, and evaluate the expression to find the correct solution.