The Basics of Percentages

A percentage is a way of expressing a part of a whole as a fraction of 100. The symbol "%" means "out of one hundred." It’s important to know how to convert between percentages, fractions, and decimals fluently. Here are some tricks to know with conversions:

• Percentage to Fraction: Divide the percentage by 100 and simplify (e.g., 25% = 25/100 = 1/4).
• Percentage to Decimal: Divide the percentage by 100 (e.g., 25% = 25/100 = 0.25).
• Fraction to Percentage: Multiply the fraction by 100 (e.g., 3/4 = (3/4) * 100% = 75%)
• Decimal to Percentage: Multiply the decimal by 100 (e.g., 0.6 = 0.6 * 100% = 60%)

Let’s do some very simple practice questions:

1. Express 60% as a fraction in its simplest form.
2. Convert 45% to a decimal.
3. What percentage is equivalent to the fraction 2/5?
4. Convert the decimal 0.125 to a percentage.
5. In a class of 20 students, 8 are boys. What percentage of the class are girls?

Solutions:

1. 3/5
2. 0.45
3. 40%
4. 12.5%
5. 60% (12 girls out of 20 is 12/20 = 60/100 = 60%)

Calculating Percentages

There are two methods that you can do to find the percentage of a number.

1. Fraction method: Convert the percentage to a fraction and then multiply the fraction by the number. For example: Find 20% of 150 => 20% = 1/5; (1/5) * 150 = 30
2. Decimal Method: Convert the percentage to a decimal and then multiply the decimal by the number. For example: Find 20% of 150 => 20% = 0.20; 0.20 * 150 = 30

Let’s calculate some percentages:

1. A shirt costs $60, and there's a 25% discount. How much money will you save?
2. In a school of 500 students, 55% are girls. How many girls are there in the school?

Solution:

1. $15
2. 275

Percentage Increase and Decrease

You might want to find the percentage increase or decrease from one original value to another new value. For example, a temperature of 30 degrees Celsius today compared to a temperature of 25 degrees Celsius yesterday is a difference that is an increase. This difference can be represented by a percentage. If the temperature today is lower than the one yesterday, the difference would be a decrease.

The formulas for percentage increases and decreases are as follows:

1. Percentage Increase = ((New Value - Original Value) / Original Value) * 100%. (1) Find the difference between the new value and the original value (this is the increase). (2) Divide the increase by the original value. (3) Multiply the result by 100 to express it as a percentage.
2. Percentage Decrease = ((Original Value - New Value) / Original Value) * 100%. (1) Find the difference between the original value and the new value (this is the decrease). (2) Divide the decrease by the original value. (3) Multiply the result by 100 to express it as a percentage.

Let’s look at some examples:

• The price of a toy increased from $20 to $25. What is the percentage increase?
• A population decreased from 800 to 600. What is the percentage decrease?

Solutions:

• Percentage Increase = (($25 - $20) / $20) * 100% = (5/20) * 100% = 25%
• Percentage Decrease = ((800 - 600) / 800) * 100% = (200/800) * 100% = 25%

Percentage Error

We’ve seen that differences that increase and decrease. Another difference is in terms of errors. Percentage error measures the accuracy of an estimated or measured value compared to the actual or accepted value. For example, a student estimated a length to be 12 cm, but the actual length was 10 cm. What is the percentage error?

If there was a huge difference between what the student estimated and what the correct length was, this would result in a larger percentage error. This means that a smaller percentage error indicates greater accuracy.

The formula for this is:

• Percentage Error = (|Estimated Value - Actual Value| / Actual Value) * 100%. The vertical bars | | mean "absolute value," which is always positive.

You can see that the formula is pretty much similar to the percentage increase/decrease formula, BUT the original value is the ACTUAL value. Why? Because that’s the correct starting point where we want to calculation the difference.

Let’s do some questions:

1. A student estimated the number of marbles in a jar to be 80, but there were actually 100. What is the percentage error?
2. A carpenter measured a piece of wood to be 2.5 meters long, but the actual length was 2.4 meters. What is the percentage error?

Solutions:

1. 20%
2. 4.17% (approximately)

Percentages in Application

Let’s now do some questions to see how percentages could be tested in your exam. Remember, it’s not enough just to know the formula, it’s about knowing when to use percentages and what pathway to follow to use them to effectively answer the question.

Question 1: The Discount Dilemma

A store is having a sale on all its items. Sarah buys a dress and a pair of shoes. The dress was originally priced at $80, but it has a discount. The shoes were originally priced at $60, but they have a different discount. Sarah pays a total of $104 for both items.

If the dress was discounted by 20%, what is the percentage discount, 'x', on the shoes?

Thought Process:

• Calculate the discount on the dress: 20% of $80 = $16 discount.
• Calculate the sale price of the dress: $80 - $16 = $64.
• Calculate the sale price of the shoes: $104 (total paid) - $64 (dress sale price) = $40.
• Calculate the discount on the shoes: $60 (original price) - $40 (sale price) = $20.
• Calculate the percentage discount on the shoes: ($20 (discount) / $60 (original price)) * 100% = 33.33% (approximately).

Answer:

The percentage discount on the shoes, 'x', is approximately 33.33%.

Question 2: The Mixing Mistake

A baker is making a cake that requires a batter with 40% flour. The baker accidentally mixes 300g of flour with 400g of other ingredients.

How much more of the other ingredients, represented by 'x', in grams, does the baker need to add to the mixture to achieve the correct percentage of flour in the batter?

Thought Process:

• Calculate the total weight of the current mixture: 300g (flour) + 400g (other ingredients) = 700g. The key here is that the amount of flour (300g) remains constant. We are only adding other ingredients.
• The flour (300g) must represent 40% of the final mixture weight. Let 'T' be the final total weight of the mixture. We can set up the equation: 0.40 * T = 300g, T = 300g / 0.40, T = 750g
• Calculate Additional 'Other' Ingredients: Current total weight: 300g (flour) + 400g (other) = 700g. Additional 'other' ingredients needed (x): 750g (final total) - 700g (current total) = 50g

Practice time!

Now, it's your turn to practice.

The questions in this checkpoint are provided to give you an introduction to possible questions you may see in your exam. Don't worry too much as you'll continue to build your skills throughout the course.

Click on the button below and start your practice questions. We recommend doing untimed mode first, and then, when you're ready, do timed mode.

Every question has a suggested solutions videos after you complete the question. This video explains to you the steps to take to answer the question and provides tips and tricks.

Once you're done with the practice questions, move on to the next checkpoint.

Now, let’s get started on your practice questions.

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