Show Your Work.Solve For $m$:$\frac{2}{5} = \frac{m}{70}$

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Introduction

When working with proportions, it's essential to show your work and follow a step-by-step approach to ensure accuracy and understanding. In this article, we'll focus on solving for $m$ in the proportion $\frac{2}{5} = \frac{m}{70}$. We'll break down the solution into manageable steps, making it easier to grasp the concept and apply it to similar problems.

Understanding the Problem

The given proportion is $\frac{2}{5} = \frac{m}{70}$. Our goal is to solve for $m$, which means we need to isolate the variable $m$ on one side of the equation. To do this, we'll use the properties of proportions and perform the necessary operations to simplify the equation.

Step 1: Cross-Multiplying

To solve for $m$, we can start by cross-multiplying the proportion. This involves multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa. In this case, we'll multiply $2$ by $70$ and $5$ by $m$.

$\frac{2}{5} = \frac{m}{70}$

Cross-multiplying:

$2 \times 70 = 5 \times m$

$140 = 5m$

Step 2: Dividing Both Sides by 5

Now that we have the equation $140 = 5m$, we can isolate $m$ by dividing both sides of the equation by $5$. This will give us the value of $m$.

$140 = 5m$

Dividing both sides by 5:

$\frac{140}{5} = \frac{5m}{5}$

$28 = m$

Conclusion

By following the steps outlined above, we've successfully solved for $m$ in the proportion $\frac{2}{5} = \frac{m}{70}$. The value of $m$ is $28$. It's essential to show your work and follow a step-by-step approach when working with proportions to ensure accuracy and understanding.

Tips and Tricks

• When working with proportions, it's essential to follow the order of operations (PEMDAS) to ensure accuracy.
• Cross-multiplying is a useful technique for solving proportions, but it's not the only method. You can also use other techniques, such as multiplying both sides of the equation by a common multiple.
• When solving for a variable, make sure to isolate the variable on one side of the equation. This will give you the correct value for the variable.

Real-World Applications

Proportions are used in various real-world applications, such as:

• Cooking: When a recipe calls for a certain ratio of ingredients, you can use proportions to scale up or down the recipe.
• Building: Architects use proportions to design buildings and ensure that the different components are in the correct ratio.
• Finance: Financial analysts use proportions to calculate interest rates and investment returns.

Common Mistakes

• Failing to show work: When working with proportions, it's essential to show your work and follow a step-by-step approach to ensure accuracy and understanding.
• Not following the order of operations: When working with proportions, it's essential to follow the order of operations (PEMDAS) to ensure accuracy.
• Not isolating the variable: When solving for a variable, make sure to isolate the variable on one side of the equation. This will give you the correct value for the variable.

Conclusion

Solving for $m$ in a proportion requires a step-by-step approach and attention to detail. By following the steps outlined above and avoiding common mistakes, you can ensure accuracy and understanding when working with proportions. Remember to show your work and follow a step-by-step approach to ensure that you're solving the problem correctly.

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Introduction

In our previous article, we discussed how to solve for $m$ in the proportion $\frac{2}{5} = \frac{m}{70}$. We broke down the solution into manageable steps and provided tips and tricks for working with proportions. In this article, we'll answer some common questions related to solving proportions and provide additional examples to help solidify your understanding.

Q&A

Q: What is the difference between a proportion and an equation?

A: A proportion is a statement that two ratios are equal, while an equation is a statement that two expressions are equal. In the case of the proportion $\frac{2}{5} = \frac{m}{70}$, we're stating that the ratio of 2 to 5 is equal to the ratio of $m$ to 70.

Q: How do I know which method to use when solving a proportion?

A: When solving a proportion, you can use either the cross-multiplication method or the multiplication method. The cross-multiplication method involves multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa. The multiplication method involves multiplying both sides of the equation by a common multiple. Choose the method that works best for you and the problem you're working on.

Q: What if I have a proportion with a variable in the denominator?

A: If you have a proportion with a variable in the denominator, you can use the same methods we discussed earlier. However, you may need to use algebraic manipulation to isolate the variable. For example, if you have the proportion $\frac{2}{x} = \frac{m}{70}$, you can multiply both sides of the equation by $x$ to get rid of the variable in the denominator.

Q: Can I use proportions to solve problems with decimals?

A: Yes, you can use proportions to solve problems with decimals. When working with decimals, make sure to follow the order of operations (PEMDAS) and use the correct notation for decimals. For example, if you have the proportion $\frac{2.5}{5} = \frac{m}{70}$, you can solve for $m$ using the same methods we discussed earlier.

Q: How do I know if a proportion is true or false?

A: To determine if a proportion is true or false, you can use the following steps:

1. Write the proportion as an equation.
2. Cross-multiply the equation.
3. Simplify the equation.
4. Check if the equation is true or false.

If the equation is true, then the proportion is true. If the equation is false, then the proportion is false.

Q: Can I use proportions to solve problems with fractions?

A: Yes, you can use proportions to solve problems with fractions. When working with fractions, make sure to follow the rules of fraction arithmetic and use the correct notation for fractions. For example, if you have the proportion $\frac{2}{5} = \frac{m}{70}$, you can solve for $m$ using the same methods we discussed earlier.

Additional Examples

Example 1: Solving a Proportion with a Variable in the Denominator

Solve the proportion $\frac{2}{x} = \frac{m}{70}$ for $m$.

Solution:

$\frac{2}{x} = \frac{m}{70}$

Cross-multiplying:

$2 \times 70 = x \times m$

$140 = xm$

Dividing both sides by $x$:

$\frac{140}{x} = m$

Example 2: Solving a Proportion with Decimals

Solve the proportion $\frac{2.5}{5} = \frac{m}{70}$ for $m$.

Solution:

$\frac{2.5}{5} = \frac{m}{70}$

Cross-multiplying:

$2.5 \times 70 = 5 \times m$

$175 = 5m$

Dividing both sides by 5:

$\frac{175}{5} = m$

$35 = m$

Conclusion

Solving proportions is an essential skill in mathematics, and it has many real-world applications. By following the steps outlined in this article and practicing with additional examples, you can become proficient in solving proportions and apply this skill to a variety of problems. Remember to show your work and follow a step-by-step approach to ensure accuracy and understanding.