Write A Proportion For The Statement.\$7.25 Per Hour Is Proportional To \$72.50 Per 10 Hours.The Proportion Is: $\[ \frac{\$7.25}{1 \text{ Hour}} = \frac{\$72.50}{10 \text{ Hours}} \\]

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Introduction

In mathematics, a proportion is a statement that two ratios are equal. It is a way to express a relationship between two quantities in the form of a fraction. Proportions are used to solve problems involving ratios, rates, and percentages. In this article, we will explore how to write a proportion for a given statement and discuss its significance in mathematics.

What is a Proportion?

A proportion is a mathematical statement that two ratios are equal. It is written in the form:

ab=cd\frac{a}{b} = \frac{c}{d}

where aa and bb are the antecedent and consequent of the first ratio, and cc and dd are the antecedent and consequent of the second ratio.

Writing a Proportion

To write a proportion, we need to identify the ratios involved in the statement. Let's consider the statement: "$7.25 per hour is proportional to $72.50 per 10 hours."

The first step is to identify the ratios involved. In this case, the ratios are:

  • $7.25 per hour
  • $72.50 per 10 hours

We can write these ratios as fractions:

$7.251 hour=$72.5010 hours\frac{\$7.25}{1 \text{ hour}} = \frac{\$72.50}{10 \text{ hours}}

This is the proportion we need to write.

Simplifying the Proportion

To simplify the proportion, we can divide both sides by the greatest common divisor (GCD) of the two fractions. In this case, the GCD is $0.25.

$7.251 hour=$72.5010 hours\frac{\$7.25}{1 \text{ hour}} = \frac{\$72.50}{10 \text{ hours}}

$7.251 hour=$72.50÷0.2510 hours÷0.25\frac{\$7.25}{1 \text{ hour}} = \frac{\$72.50 \div 0.25}{10 \text{ hours} \div 0.25}

$7.251 hour=$2904 hours\frac{\$7.25}{1 \text{ hour}} = \frac{\$290}{4 \text{ hours}}

Interpreting the Proportion

The proportion $7.251 hour=$2904 hours\frac{\$7.25}{1 \text{ hour}} = \frac{\$290}{4 \text{ hours}} tells us that if we work for 1 hour at a rate of $7.25 per hour, we will earn $7.25. Similarly, if we work for 4 hours at a rate of $290 per 4 hours, we will also earn $290.

Real-World Applications

Proportions have many real-world applications. For example, in finance, proportions are used to calculate interest rates, investment returns, and loan payments. In science, proportions are used to calculate rates of change, such as the rate of decay of a radioactive substance. In engineering, proportions are used to design and optimize systems, such as bridges, buildings, and machines.

Conclusion

In conclusion, proportions are an essential concept in mathematics that helps us understand and express relationships between two quantities. By writing a proportion, we can identify the ratios involved and simplify the proportion to make it easier to understand and work with. Proportions have many real-world applications and are used in various fields, including finance, science, and engineering.

Additional Examples

Example 1

A car travels 250 miles in 5 hours. If the car travels at a constant rate, how many miles will it travel in 10 hours?

To solve this problem, we can write a proportion:

250 miles5 hours=x miles10 hours\frac{250 \text{ miles}}{5 \text{ hours}} = \frac{x \text{ miles}}{10 \text{ hours}}

Simplifying the proportion, we get:

250 miles5 hours=x miles10 hours\frac{250 \text{ miles}}{5 \text{ hours}} = \frac{x \text{ miles}}{10 \text{ hours}}

250 miles5 hours=250 miles×210 hours\frac{250 \text{ miles}}{5 \text{ hours}} = \frac{250 \text{ miles} \times 2}{10 \text{ hours}}

250 miles5 hours=500 miles10 hours\frac{250 \text{ miles}}{5 \text{ hours}} = \frac{500 \text{ miles}}{10 \text{ hours}}

Therefore, the car will travel 500 miles in 10 hours.

Example 2

A bakery sells 500 loaves of bread per day at a price of $2 per loaf. If the bakery wants to sell 1000 loaves of bread per day, how much will it need to charge per loaf?

To solve this problem, we can write a proportion:

500 loaves$2 per loaf=1000 loavesx per loaf\frac{500 \text{ loaves}}{\$2 \text{ per loaf}} = \frac{1000 \text{ loaves}}{x \text{ per loaf}}

Simplifying the proportion, we get:

500 loaves$2 per loaf=1000 loavesx per loaf\frac{500 \text{ loaves}}{\$2 \text{ per loaf}} = \frac{1000 \text{ loaves}}{x \text{ per loaf}}

500 loaves$2 per loaf=500 loaves×2x per loaf\frac{500 \text{ loaves}}{\$2 \text{ per loaf}} = \frac{500 \text{ loaves} \times 2}{x \text{ per loaf}}

500 loaves$2 per loaf=1000 loavesx per loaf\frac{500 \text{ loaves}}{\$2 \text{ per loaf}} = \frac{1000 \text{ loaves}}{x \text{ per loaf}}

Therefore, the bakery will need to charge $4 per loaf to sell 1000 loaves of bread per day.

Final Thoughts

Frequently Asked Questions

Q: What is a proportion?

A: A proportion is a mathematical statement that two ratios are equal. It is written in the form:

ab=cd\frac{a}{b} = \frac{c}{d}

where aa and bb are the antecedent and consequent of the first ratio, and cc and dd are the antecedent and consequent of the second ratio.

Q: How do I write a proportion?

A: To write a proportion, you need to identify the ratios involved in the statement. Let's consider the statement: "$7.25 per hour is proportional to $72.50 per 10 hours."

The first step is to identify the ratios involved. In this case, the ratios are:

  • $7.25 per hour
  • $72.50 per 10 hours

We can write these ratios as fractions:

$7.251 hour=$72.5010 hours\frac{\$7.25}{1 \text{ hour}} = \frac{\$72.50}{10 \text{ hours}}

This is the proportion we need to write.

Q: How do I simplify a proportion?

A: To simplify a proportion, you can divide both sides by the greatest common divisor (GCD) of the two fractions. In this case, the GCD is $0.25.

$7.251 hour=$72.5010 hours\frac{\$7.25}{1 \text{ hour}} = \frac{\$72.50}{10 \text{ hours}}

$7.251 hour=$72.50÷0.2510 hours÷0.25\frac{\$7.25}{1 \text{ hour}} = \frac{\$72.50 \div 0.25}{10 \text{ hours} \div 0.25}

$7.251 hour=$2904 hours\frac{\$7.25}{1 \text{ hour}} = \frac{\$290}{4 \text{ hours}}

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

A: Proportions have many real-world applications. For example, in finance, proportions are used to calculate interest rates, investment returns, and loan payments. In science, proportions are used to calculate rates of change, such as the rate of decay of a radioactive substance. In engineering, proportions are used to design and optimize systems, such as bridges, buildings, and machines.

Q: How do I use proportions to solve problems?

A: To use proportions to solve problems, you need to identify the ratios involved and write a proportion. Then, you can simplify the proportion and use it to solve the problem.

For example, let's say you want to know how many miles a car will travel in 10 hours if it travels 250 miles in 5 hours. You can write a proportion:

250 miles5 hours=x miles10 hours\frac{250 \text{ miles}}{5 \text{ hours}} = \frac{x \text{ miles}}{10 \text{ hours}}

Simplifying the proportion, you get:

250 miles5 hours=250 miles×210 hours\frac{250 \text{ miles}}{5 \text{ hours}} = \frac{250 \text{ miles} \times 2}{10 \text{ hours}}

250 miles5 hours=500 miles10 hours\frac{250 \text{ miles}}{5 \text{ hours}} = \frac{500 \text{ miles}}{10 \text{ hours}}

Therefore, the car will travel 500 miles in 10 hours.

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

A: Some common mistakes to avoid when working with proportions include:

  • Not identifying the ratios involved
  • Not writing the proportion correctly
  • Not simplifying the proportion
  • Not using the proportion to solve the problem

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
  2. Simplify the proportion
  3. Check if the two ratios are equal

If the two ratios are equal, the proportion is true. If the two ratios are not equal, the proportion is false.

Q: Can proportions be used to solve problems involving fractions?

A: Yes, proportions can be used to solve problems involving fractions. For example, let's say you want to know how many fractions of a pizza you can eat if you eat 1/4 of a pizza in 2 hours. You can write a proportion:

1/4 pizza2 hours=x pizza10 hours\frac{1/4 \text{ pizza}}{2 \text{ hours}} = \frac{x \text{ pizza}}{10 \text{ hours}}

Simplifying the proportion, you get:

1/4 pizza2 hours=1/4 pizza×510 hours\frac{1/4 \text{ pizza}}{2 \text{ hours}} = \frac{1/4 \text{ pizza} \times 5}{10 \text{ hours}}

1/4 pizza2 hours=5/4 pizza10 hours\frac{1/4 \text{ pizza}}{2 \text{ hours}} = \frac{5/4 \text{ pizza}}{10 \text{ hours}}

Therefore, you can eat 5/4 of a pizza in 10 hours.

Q: Can proportions be used to solve problems involving decimals?

A: Yes, proportions can be used to solve problems involving decimals. For example, let's say you want to know how many dollars you will earn if you work 10 hours at a rate of $7.25 per hour. You can write a proportion:

$7.251 hour=x dollars10 hours\frac{\$7.25}{1 \text{ hour}} = \frac{x \text{ dollars}}{10 \text{ hours}}

Simplifying the proportion, you get:

$7.251 hour=$7.25×1010 hours\frac{\$7.25}{1 \text{ hour}} = \frac{\$7.25 \times 10}{10 \text{ hours}}

$7.251 hour=$72.5010 hours\frac{\$7.25}{1 \text{ hour}} = \frac{\$72.50}{10 \text{ hours}}

Therefore, you will earn $72.50 if you work 10 hours at a rate of $7.25 per hour.

Final Thoughts

Proportions are a fundamental concept in mathematics that helps us understand and express relationships between two quantities. By writing a proportion, we can identify the ratios involved and simplify the proportion to make it easier to understand and work with. Proportions have many real-world applications and are used in various fields, including finance, science, and engineering.