Standard Form & Percentage Calculations: Examples & Solutions

Hey guys! Let's dive into some math problems today. We're going to tackle expressing numbers in standard form and calculating percentages. These are fundamental concepts in mathematics, super useful in everyday life, and crucial for acing your exams. So, let's get started!

Expressing Numbers in Standard Form

First off, let's break down what standard form actually means. Standard form, also known as scientific notation, is a way of writing very large or very small numbers concisely. The general form is $a \times 10^n$, where 'a' is a number between 1 and 10 (including 1 but excluding 10), and 'n' is an integer (a positive or negative whole number).

Why Use Standard Form?

You might be wondering, why bother with standard form? Well, imagine you're dealing with the distance to a galaxy or the size of a virus. These numbers are either incredibly huge or infinitesimally small. Writing them out in their full form would be cumbersome and prone to errors. Standard form gives us a neat and efficient way to represent these numbers.

Now, let's get into the examples you provided:

a) 6000 in Standard Form

Okay, so we have the number 6000. To express this in standard form, we need to rewrite it as a number between 1 and 10 multiplied by a power of 10. The number between 1 and 10 is 6.0. Now, we need to figure out what power of 10 we need to multiply 6.0 by to get 6000. Think of it this way: we moved the decimal point three places to the left, so the exponent will be 3. Therefore, 6000 in standard form is $6.0 \times 10^3$. Remember, that zero after the decimal point is important to show that we've correctly placed the decimal, but it doesn't change the value.

b) 7000 in Standard Form

This one's super similar to the last one! We've got 7000. We want to get it into the form $a \times 10^n$. The 'a' part is going to be 7.0. Again, we moved the decimal point three places to the left from the end of the number to get 7.0. So, the exponent is 3. Therefore, 7000 in standard form is $7.0 \times 10^3$. See how standard form makes these large numbers easier to handle?

c) 80,000 in Standard Form

Alright, let's step it up a notch with 80,000. Same drill here, guys! We need a number between 1 and 10, which in this case is 8.0. Now, count how many places we move the decimal point to the left to get from 80,000 to 8.0. We move it five places! That means our exponent is 5. So, 80,000 in standard form is $8.0 \times 10^5$. Notice how the exponent tells us the magnitude of the number.

d) $6 \times 10^5$ in Standard Form

Hold up! This one's already in standard form! $6 \times 10^5$ perfectly fits the $a \times 10^n$ format, where 'a' is 6 (which is between 1 and 10), and 'n' is 5 (an integer). So, we don't need to do anything here. It's already $6 \times 10^5$. This just emphasizes how standard form is a standardized way of writing numbers, and this number is already playing by the rules!

Calculating Percentages

Now, let's switch gears and talk about percentages. Percentages are a way of expressing a number as a fraction of 100. The word "percent" literally means "per hundred." Understanding percentages is super important for calculating discounts, taxes, interest rates, and all sorts of things in daily life.

The Basics of Percentage Calculations

To find a percentage of a number, we convert the percentage to a decimal or a fraction and then multiply it by the number. For example, to find 50% of 100, we can convert 50% to 0.50 (by dividing by 100) and then multiply 0.50 by 100, which gives us 50. Or, we can convert 50% to the fraction 1/2 and multiply that by 100, again giving us 50. Let's apply this to the examples you provided.

a) 5% of 2500

Okay, so we need to find 5% of 2500. First, let's convert 5% to a decimal. To do this, we divide 5 by 100, which gives us 0.05. Now, we multiply 0.05 by 2500: 0. 05 * 2500 = 125. So, 5% of 2500 is 125. Another way to think about it is that 1% of 2500 is 25 (2500 / 100), and 5% is five times that amount, so 5 * 25 = 125. Both ways work, so choose the one that clicks best for you!

b) 10% of 37000

Next up, we have 10% of 37000. This one's actually pretty easy! Ten percent is equivalent to one-tenth, so we can simply divide 37000 by 10. 37000 / 10 = 3700. Therefore, 10% of 37000 is 3700. You can also convert 10% to the decimal 0.10 and multiply: 0.10 * 37000 = 3700. Knowing that 10% is one-tenth is a handy shortcut to remember!

c) $10:100 + 15:35$

This one looks a little different, but we can totally handle it! The colon (:) here represents a ratio, which can also be expressed as a fraction. So, $10:100$ is the same as $\frac{10}{100}$, and $15:35$ is the same as $\frac{15}{35}$.

Let's simplify these fractions first. $\frac{10}{100}$ simplifies to $\frac{1}{10}$. And $\frac{15}{35}$ simplifies to $\frac{3}{7}$ (both divisible by 5). Now we have: $\frac{1}{10} + \frac{3}{7}$.

To add fractions, we need a common denominator. The least common multiple of 10 and 7 is 70. So, let's convert both fractions to have a denominator of 70:

• $\frac{1}{10} = \frac{1 \times 7}{10 \times 7} = \frac{7}{70}$
• $\frac{3}{7} = \frac{3 \times 10}{7 \times 10} = \frac{30}{70}$

Now we can add them: $\frac{7}{70} + \frac{30}{70} = \frac{37}{70}$. So, the answer to $10:100 + 15:35$ is $\frac{37}{70}$. This demonstrates how ratios and fractions are interconnected, and how simplifying fractions makes calculations much easier.

Conclusion

So, there you have it, guys! We've covered expressing numbers in standard form and calculating percentages. These are essential mathematical skills that you'll use all the time. Remember the key concepts: standard form helps us write large and small numbers efficiently, and percentages are just fractions out of 100. Practice these concepts, and you'll be math whizzes in no time! Keep up the great work, and don't hesitate to ask if you have more questions! Practice makes perfect, and the more you work with these concepts, the more comfortable you'll become. Good luck, and keep learning!