The one thing to know:
Fractions help us understand and work with parts of a whole, showing how many pieces we have and how many pieces make up the entire thing.
- 1A fraction shows a part of a whole, with a top number (numerator) telling you how many parts you have, and a bottom number (denominator) telling you how many parts make up the whole.
- 2You can add, subtract, multiply, and divide fractions, but often need to make sure they are talking about the same 'size' of parts first.
- 3Fractions are used everywhere, from recipes to finances, and can be written in different ways like decimals or percentages.
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Part 1 of 7Think of it like:
Think of a pizza cut into equal slices. A fraction tells you how many slices you have (the top number) and how many slices were in the whole pizza (the bottom number). If you have 3 slices from a pizza cut into 8, that's 3/8.
Key idea: Fractions are a way to represent parts of a whole or any number of equal pieces.
Imagine you have a delicious cake, and you want to share it. If you cut the cake into equal pieces, a helps you describe how much of that cake you or someone else gets. It is a way to talk about parts of a whole thing.
When you say "one half" or "three quarters," you are using fractions. They tell us how many pieces of a certain size we have. For example, if you eat "one half" of a cake, you are eating one piece out of two equal pieces that made up the whole cake.
Key idea: The numerator tells you how many parts you have, and the denominator tells you how many parts make up the whole.
Every simple fraction has two main parts: the top number and the bottom number. The top number is called the , and it tells you how many parts you are counting. The bottom number is called the , and it tells you how many equal parts make up the entire whole.
For example, in the fraction 3/4, the '3' (numerator) means you have 3 parts. The '4' (denominator) means that 4 of these equal parts make up the whole thing. So, 3/4 of a cake means you have 3 slices from a cake that was cut into 4 equal slices.
Fractions can also show (like comparing 3 apples to 4 oranges) or (like 3 divided by 4).
Sometimes, you might see negative fractions, like -1/2. This just means the opposite of a positive fraction. If 1/2 means a half dollar profit, then -1/2 means a half dollar loss.
“The numerator tells you how many parts you are counting, and the denominator tells you how many equal parts make up the entire whole.”
Quick check
What do the numerator and denominator tell you in a fraction like 2/5?
Key idea: Fractions can be simple, unit, proper (less than a whole), or improper (a whole or more).
Fractions come in different forms. A (also called a common fraction) is one where both the numerator and denominator are whole numbers, and the denominator is not zero. Examples are 1/2 or 8/5.
A is a simple fraction where the numerator is 1, like 1/7. This means you have one single piece of something that was divided into 7 equal parts.
Fractions can also be 'proper' or 'improper'. A is when the top number (numerator) is smaller than the bottom number (denominator), like 2/3. This means you have less than a whole.
An is when the top number is equal to or larger than the bottom number, like 9/4 or 3/3. This means you have one whole or more than one whole.
A common misconception is that improper fractions are 'wrong' or 'bad'. They are not! They are just another way to express a quantity that is equal to or greater than one whole. For example, 9/4 means you have nine quarter pieces, which is more than two whole things.
Any whole number can also be written as a fraction by putting '1' as its denominator. For example, the number 5 can be written as 5/1. The '1' is sometimes called the .
“A proper fraction is when the numerator is smaller than the denominator, meaning you have less than a whole.”
Quick check
What is the difference between a proper and an improper fraction?
Key idea: Mixed numbers combine a whole number and a fraction, and you can convert them to and from improper fractions.
A combines a whole number and a proper fraction. For example, if you have 2 whole cakes and 3/4 of another cake, you would write it as 2 3/4.
You can easily switch between mixed numbers and improper fractions. To change a mixed number like 2 3/4 into an improper fraction, you multiply the whole number (2) by the denominator (4), then add the numerator (3). Keep the same denominator. So, 2 × 4 + 3 = 11, and the improper fraction is 11/4.
To change an improper fraction like 11/4 back to a mixed number, you divide the numerator (11) by the denominator (4). The whole number part of the answer is your whole number, and any leftover (remainder) becomes the new numerator over the original denominator. 11 divided by 4 is 2 with 3 left over, so it is 2 3/4.
Key idea: To add or subtract fractions, they must have a common denominator; then you add or subtract the numerators.
When you want to add or subtract fractions, a key rule is that you can only combine 'like' quantities. Imagine trying to add quarters and dimes directly; it is easier if you convert them all to cents first. For fractions, this means they need to have the same denominator.
If fractions already have the same denominator, you just add or subtract the numerators and keep the denominator the same. For example, 2/4 + 3/4 = 5/4.
If fractions have different denominators, you need to find a . This is a number that both original denominators can divide into evenly. A simple way to find one is to multiply the two denominators together. Then, you change each fraction to an equivalent fraction with this new common denominator. For example, to add 1/4 and 1/3, you can use 12 as the common denominator (4 × 3 = 12).
You change 1/4 to 3/12 (by multiplying top and bottom by 3) and 1/3 to 4/12 (by multiplying top and bottom by 4). Now you can add: 3/12 + 4/12 = 7/12.
Quick check
How do you add or subtract fractions with different denominators?
Key idea: To multiply fractions, multiply the numerators and denominators; to divide, flip the second fraction and then multiply.
Multiplying fractions is often simpler than adding or subtracting them! You just multiply the numerators together (the top numbers) and multiply the denominators together (the bottom numbers).
For example, to multiply 2/3 by 3/4: multiply 2 × 3 to get 6 (new numerator), and multiply 3 × 4 to get 12 (new denominator). So, 2/3 × 3/4 = 6/12. This fraction can then be simplified to 1/2.
A neat trick called can make multiplication easier. Before you multiply, if a numerator and a denominator share a common factor, you can divide both by that factor. In 2/3 × 3/4, the '3' in the first denominator and the '3' in the second numerator can cancel out. Also, the '2' in the first numerator and the '4' in the second denominator can be divided by 2, becoming '1' and '2'. This leaves you with 1/1 × 1/2 = 1/2.
To divide fractions, you "flip" the second fraction (find its ) and then multiply. For example, to divide 1/2 by 3/4, you flip 3/4 to become 4/3, and then multiply 1/2 × 4/3 = 4/6, which simplifies to 2/3.
Key idea: Fractions can be expressed as decimals (parts of ten) or percentages (parts of one hundred).
Fractions can also be written as or . These are just different ways to show parts of a whole.
A decimal fraction is a fraction where the denominator is a power of ten (like 10, 100, 1000). We usually write these using a decimal point. For instance, 3/4 is the same as 0.75. The '75' means 75 hundredths, so the implied denominator is 100.
To change a fraction to a decimal, you divide the numerator by the denominator. For example, 1 divided by 4 equals 0.25. Some fractions, like 1/3, create repeating decimals (0.333...), which means the decimal goes on forever.
A is a special type of fraction where the denominator is always 100. The word "percent" means "per hundred." So, 51% means 51/100. Percentages are very common for showing things like discounts or statistics.
Why does this matter?
- Fractions are essential for everyday tasks, like following recipes (1/2 cup of flour) or understanding measurements (a board that is 3/4 inch thick).
- They help you understand financial concepts, such as discounts (25% off) or interest rates (a fraction of your savings).
- Fractions are the building blocks for more advanced math, like algebra and calculus, and are used in science and engineering to describe relationships and proportions.
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1 / 6What is the name for the bottom number in a fraction?
Can you explain these?
Try to explain each in your own words, without looking. The ones you stumble on are exactly where to re-read.
- 1Parts of a whole
- 2Numerator and denominator
- 3Types of fractions
- 4Arithmetic operations
- 5Alternative representations
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