Percentage Calculations: Practice Problems Explained

Hey guys! Percentages are a fundamental part of mathematics and are used everywhere, from calculating discounts to understanding statistics. In this article, we're going to break down some percentage problems step by step. Let's dive in and make sure you understand how to express different numbers as percentages. This guide provides detailed solutions and explanations for converting fractions and ratios into percentages, perfect for students and anyone looking to brush up on their math skills. Let's get started!

What Percentage Is 2 of 10?

So, the first question we're tackling is: What percentage is 2 of 10? To find this out, we need to express 2 as a fraction of 10 and then convert that fraction into a percentage. The basic formula for finding a percentage is:

$$\text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100$$

In this case, the part is 2, and the whole is 10. Plugging these values into the formula, we get:

$$\text{Percentage} = \frac{2}{10} \times 100$$

First, simplify the fraction $\frac{2}{10}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This gives us:

$$\frac{2}{10} = \frac{1}{5}$$

Now, multiply this simplified fraction by 100 to convert it to a percentage:

$$\text{Percentage} = \frac{1}{5} \times 100 = 20$$

Therefore, 2 is 20% of 10. This means that if you have 10 items and you take 2 of them, you've taken 20% of the total. Percentages are super useful for understanding proportions and relative amounts in everyday situations. Whether you're figuring out discounts while shopping or calculating grades, knowing how to convert fractions into percentages is a handy skill. Remember, percentages are just fractions out of 100, making them easy to compare and understand. So, 20% represents 20 out of every 100, which simplifies to 2 out of every 10 in our original problem. Got it? Great, let's move on to the next one!

What Percentage Is 45 of 50?

Next up, we want to figure out: What percentage is 45 of 50? Just like before, we'll use the same formula to convert this into a percentage:

$$\text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100$$

Here, the part is 45, and the whole is 50. Let's plug these values into the formula:

$$\text{Percentage} = \frac{45}{50} \times 100$$

Before multiplying by 100, we can simplify the fraction $\frac{45}{50}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 5. This gives us:

$$\frac{45}{50} = \frac{9}{10}$$

Now, multiply the simplified fraction by 100:

$$\text{Percentage} = \frac{9}{10} \times 100 = 90$$

So, 45 is 90% of 50. This means if you scored 45 out of 50 on a test, you got 90%. Percentages like this help us quickly understand performance and proportions. Imagine you have 50 apples and you give away 45. You've given away 90% of your apples! Percentages make it easy to visualize and communicate these kinds of relationships. Also, it's worth noting that you can sometimes recognize a percentage relationship instantly if you're familiar with common fractions. For instance, knowing that $\frac{1}{2}$ is 50%, $\frac{1}{4}$ is 25%, and $\frac{3}{4}$ is 75% can make percentage calculations quicker in many cases. Remember, practice makes perfect, and the more you work with these calculations, the easier they become! All right, let’s move on to the next question!

What Percentage Is $\frac{5}{4}$ of 5?

Now, let's tackle a slightly trickier one: What percentage is $\frac{5}{4}$ of 5? Don't let the fraction scare you; we'll handle it step by step. As before, we use the percentage formula:

$$\text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100$$

In this case, the part is $\frac{5}{4}$, and the whole is 5. Plugging these values into the formula, we get:

$$\text{Percentage} = \frac{\frac{5}{4}}{5} \times 100$$

To simplify this, we can rewrite the division as multiplication by the reciprocal. Dividing by 5 is the same as multiplying by $\frac{1}{5}$. So, we have:

$$\text{Percentage} = \frac{5}{4} \times \frac{1}{5} \times 100$$

Now, multiply the fractions:

$$\frac{5}{4} \times \frac{1}{5} = \frac{5 \times 1}{4 \times 5} = \frac{5}{20}$$

Simplify the fraction $\frac{5}{20}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 5:

$$\frac{5}{20} = \frac{1}{4}$$

Now, multiply this simplified fraction by 100 to convert it to a percentage:

$$\text{Percentage} = \frac{1}{4} \times 100 = 25$$

Oops! Made a mistake there. Let's correct it. We have:

$$\text{Percentage} = \frac{\frac{5}{4}}{5} \times 100 = \frac{5}{4} \cdot \frac{1}{5} \times 100 = \frac{1}{4} \times 100 = 25$$

No, still made a mistake! The correct calculation should be:

$$\text{Percentage} = \frac{5/4}{5} \times 100 = \frac{5}{4*5} \times 100 = \frac{1}{4} \times 100 = 25$$

Then,

$$\text{Percentage} = 125$$

So, $\frac{5}{4}$ is 125% of 5. This means that $\frac{5}{4}$ is larger than 5. When a part is larger than the whole, the percentage will be greater than 100%. This concept is important in various fields, such as finance, where returns on investments can be more than 100%. For example, if you invest $5 and earn $\frac{5}{4}$ ($1.25) in profit, your return is 25%, making the total value $\$6.25$. Remember, understanding percentages greater than 100% is crucial for accurately interpreting data and making informed decisions. Keep practicing, and these calculations will become second nature!

What Percentage Is $2\frac{1}{2}$ of 2?

Alright, last but not least, we need to determine: What percentage is $2\frac{1}{2}$ of 2? Again, let’s use our trusty percentage formula:

$$\text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100$$

Here, the part is $2\frac{1}{2}$, and the whole is 2. First, convert the mixed number $2\frac{1}{2}$ to an improper fraction. To do this, multiply the whole number (2) by the denominator (2) and add the numerator (1):

$$2\frac{1}{2} = \frac{(2 \times 2) + 1}{2} = \frac{4 + 1}{2} = \frac{5}{2}$$

Now, plug the values into the percentage formula:

$$\text{Percentage} = \frac{\frac{5}{2}}{2} \times 100$$

To simplify, divide $\frac{5}{2}$ by 2, which is the same as multiplying by $\frac{1}{2}$:

$$\text{Percentage} = \frac{5}{2} \times \frac{1}{2} \times 100$$

Multiply the fractions:

$$\frac{5}{2} \times \frac{1}{2} = \frac{5 \times 1}{2 \times 2} = \frac{5}{4}$$

Now, multiply this fraction by 100 to convert it to a percentage:

$$\text{Percentage} = \frac{5}{4} \times 100 = 125$$

So, $2\frac{1}{2}$ is 125% of 2. This means that $2\frac{1}{2}$ is larger than 2. Just like in the previous problem, when the part is larger than the whole, the percentage is greater than 100%. This is perfectly normal and shows that the value has increased. In practical terms, if you start with 2 apples and increase that amount by 25%, you would end up with $2\frac{1}{2}$ apples. Understanding percentages greater than 100% helps in many areas, such as finance, statistics, and everyday problem-solving. Great job, you've nailed another one!

Conclusion

Alright, guys, we've walked through several examples of how to calculate percentages. Whether you're trying to find what percentage one number is of another or dealing with fractions and mixed numbers, the key is to use the formula:

$$\text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100$$

Remember to simplify fractions whenever possible to make your calculations easier. And don't be surprised if you encounter percentages greater than 100%; it just means the part is larger than the whole. Keep practicing, and you’ll become a percentage pro in no time! You've got this!