Fractions

Fractions

In the real world, there are not only whole numbers. E.g. bread can be cut into two halves and pizza into eight slices. To describe them we use fractions. They simply tell us how many parts of a whole we have.

Fractions show parts of whole numbers. Each fraction consists of a numerator and a denominator. The numerator is the top digit that represents how many equal parts the fraction contains. The denominator is the bottom digit that represents how many equal parts the whole is broken into:

There are many ways to write the same fraction. In the picture, you can see that $1/2$, $2/4$, $3/6$ and $4/8$ all represent the same amount. Even though the numerators and denominators of fractions are different, the colored parts of the boxes always show the same half of the whole.

We can also think of fractions as numbers on a number line. To represent $1/8$ on the number line, first draw a line from 0 to 1. Then divide the interval between 0 and 1 into eight equal parts (denominator=8). This gives fractions from $1/8$ to $8/8$ which is actually equal to 1 (and can also be written as $4/4$ or $2/2$).

Fractions

In the real world, there are not only whole numbers. E.g. bread can be cut into two halves and pizza into eight slices. To describe them we use fractions. They simply tell us how many parts of a whole we have.

Fractions show parts of whole numbers. Each fraction consists of a numerator and a denominator. The numerator is the top digit that represents how many equal parts the fraction contains. The denominator is the bottom digit that represents how many equal parts the whole is broken into:

There are many ways to write the same fraction. In the picture, you can see that $1/2$, $2/4$, $3/6$ and $4/8$ all represent the same amount. Even though the numerators and denominators of fractions are different, the colored parts of the boxes always show the same half of the whole.

We can also think of fractions as numbers on a number line. To represent $1/8$ on the number line, first draw a line from 0 to 1. Then divide the interval between 0 and 1 into eight equal parts (denominator=8). This gives fractions from $1/8$ to $8/8$ which is actually equal to 1 (and can also be written as $4/4$ or $2/2$).

Fractions

In the real world, there are not only whole numbers. E.g. bread can be cut into two halves and pizza into eight slices. To describe them we use fractions. They simply tell us how many parts of a whole we have.

Fractions show parts of whole numbers. Each fraction consists of a numerator and a denominator. The numerator is the top digit that represents how many equal parts the fraction contains. The denominator is the bottom digit that represents how many equal parts the whole is broken into:

There are many ways to write the same fraction. In the picture, you can see that $1/2$, $2/4$, $3/6$ and $4/8$ all represent the same amount. Even though the numerators and denominators of fractions are different, the colored parts of the boxes always show the same half of the whole.

We can also think of fractions as numbers on a number line. To represent $1/8$ on the number line, first draw a line from 0 to 1. Then divide the interval between 0 and 1 into eight equal parts (denominator=8). This gives fractions from $1/8$ to $8/8$ which is actually equal to 1 (and can also be written as $4/4$ or $2/2$).

Fractions

In the real world, there are not only whole numbers. E.g. bread can be cut into two halves and pizza into eight slices. To describe them we use fractions. They simply tell us how many parts of a whole we have.

Fractions show parts of whole numbers. Each fraction consists of a numerator and a denominator. The numerator is the top digit that represents how many equal parts the fraction contains. The denominator is the bottom digit that represents how many equal parts the whole is broken into:

There are many ways to write the same fraction. In the picture, you can see that $1/2$, $2/4$, $3/6$ and $4/8$ all represent the same amount. Even though the numerators and denominators of fractions are different, the colored parts of the boxes always show the same half of the whole.

We can also think of fractions as numbers on a number line. To represent $1/8$ on the number line, first draw a line from 0 to 1. Then divide the interval between 0 and 1 into eight equal parts (denominator=8). This gives fractions from $1/8$ to $8/8$ which is actually equal to 1 (and can also be written as $4/4$ or $2/2$).

Fractions

In the real world, there are not only whole numbers. E.g. bread can be cut into two halves and pizza into eight slices. To describe them we use fractions. They simply tell us how many parts of a whole we have.

Fractions show parts of whole numbers. Each fraction consists of a numerator and a denominator. The numerator is the top digit that represents how many equal parts the fraction contains. The denominator is the bottom digit that represents how many equal parts the whole is broken into:

To find the fraction that is missing to make 1 whole, look at how many equal parts the whole is divided into. A whole means that all parts are filled. So if the whole is divided into 4 parts, then the whole is 4/4. If 3/4 is shaded, then 1/4 is missing. The denominator stays the same, and we only find how many parts are still needed.

There are many ways to write the same fraction. In the picture, you can see that $1/2$, $2/4$, $3/6$, and $4/8$ all represent the same amount. Even though the numerators and denominators are different, the shaded parts of the boxes always show the same half of the whole. To make one whole, we need the other half, which is also $1/2$.

Comparing fractions

A fraction is a part of a whole. To find the greater fraction you need to find the fraction that contains a greater part of the whole.

If fractions have the same denominator, you can compare numerators:

It is possible to draw it to see the result:

If there is a common numerator, the fraction with the greater denominator is actually smaller:

You can draw it as well:

If two fractions do not have common numerator or denominator, you can find equivalent fractions with the same denominator.
E.g. compare $3/8$ and $1/3$:

If you compare a fraction with a whole number, you can turn the whole number into a fraction with the same denominator:

Compare 3 and $9/4$:

Comparing fractions

A fraction is a part of a whole. To find the greater fraction you need to find the fraction that contains a greater part of the whole.

If fractions have the same denominator, you can compare numerators:

It is possible to draw it to see the result:

If there is a common numerator, the fraction with the greater denominator is actually smaller:

You can draw it as well:

If two fractions do not have common numerator or denominator, you can find equivalent fractions with the same denominator.
E.g. compare $3/8$ and $1/3$:

If you compare a fraction with a whole number, you can turn the whole number into a fraction with the same denominator:

Compare 3 and $9/4$:

Fractions and whole numbers

If you compare a fraction with a whole number, you can turn the whole number into a fraction with the same denominator:

Compare 3 and $9/4$:

Equivalent fractions

Equivalent fractions are equal in value, even though they look different.

E.g.: $;2/4=1/2$
We can draw it:

When we multiply or divide the numerator (the top number) and the denominator (the bottom number) of a fraction by another fraction with the same numerator or denominator, it keeps the same value. A fraction with the same numerator and denominator is actually equal to 1 and by multiplying or dividing by 1 we won't change the value of original fraction.
E.g.:

If there is a missing part of a fraction in an equation, we need to find the equivalent fraction:

We need to multiply 1 by 2 to get 2. We need to do the same with the denominator (3). And 3×2=6

If there is a missing part of a fraction in an inequality, we need to find the equivalent fraction at first and then decrease/increase it to have it greater or smaller. E.g.:

Since $3/8$ are equivalent to $6/16$, we need to find a fraction that has 6 as a numerator (top number) and at the same time it is smaller than $6/16$. This is any fraction with a denominator greater than 16, e.g. $6/17 ; 6/18 ; 6/19$

Fractions of whole numbers

To find a fraction of a whole number (for example $3/4$ of 12), we can use a simple idea: a fraction of a number means that we divide the number into equal parts and then take some of those parts.

For example, finding half of a number means dividing it by 2. So half of 12 is:

This also helps us understand phrases like increase by a half and decrease by a half. If we increase 12 by a half, we first find half of 12, which is 6, and then add it:

If we decrease 12 by a half, we again find half of 12, but this time we subtract it:

The same idea works for other fractions too. For example, if we want to find one quarter of 12, we divide 12 into 4 equal parts:

So $1/4$ of 12 is 3. If we want $3/4$ of 12, we take three of these quarters:

Mathematically, finding a fraction of a whole number is really the same as multiplying by the fraction. So:

This works because multiplying by a fraction means: divide by the denominator and take the number of parts shown by the numerator. So a fraction of a whole number is just a special kind of multiplication.

Fractions

In the real world, there are not only whole numbers. E.g. bread can be cut into two halves and pizza into eight slices. To describe them we use fractions. They simply tell us how many parts of a whole we have.

Fractions show parts of whole numbers. Each fraction consists of a numerator and a denominator. The numerator is the top digit that represents how many equal parts the fraction contains. The denominator is the bottom digit that represents how many equal parts the whole is broken into:

How to find a fraction of a whole number

Sometimes we need to find a part of a whole, even when the whole is a number greater than 1. For example:

What is 2/3 of 6?

What does it mean?

It means that the whole number 6 is divided into 3 equal parts, and then we take 2 of those parts.

We can imagine the whole number 6 as 6 equal parts. Then we want to find 2/3 of this whole.

If we divide 6 parts into 3 equal groups, each group has:

$$6 ÷ 3 = 2$$

So 1/3 of 6 is 2 parts. That means 2/3 of 6 is 2 groups, so:

$$2 × 2 = 4$$

If we drew 6 small boxes, we would shade 4 of them. So:

$$2/3 × 6 = 4$$

Rule

To find a fraction of a whole number:

Divide by the denominator, then multiply by the numerator.

For example:

$$6 ÷ 3 = 2$$

$$2 × 2 = 4$$

Fractions of whole numbers

To find a fraction of a whole number (for example $3/4$ of 12), we can use a simple idea: a fraction of a number means that we divide the number into equal parts and then take some of those parts.

For example, finding half of a number means dividing it by 2. So half of 12 is:

This also helps us understand phrases like increase by a half and decrease by a half. If we increase 12 by a half, we first find half of 12, which is 6, and then add it:

If we decrease 12 by a half, we again find half of 12, but this time we subtract it:

The same idea works for other fractions too. For example, if we want to find one quarter of 12, we divide 12 into 4 equal parts:

So $1/4$ of 12 is 3. If we want $3/4$ of 12, we take three of these quarters:

Mathematically, finding a fraction of a whole number is really the same as multiplying by the fraction. So:

This works because multiplying by a fraction means: divide by the denominator and take the number of parts shown by the numerator. So a fraction of a whole number is just a special kind of multiplication.

Adding and subtracting fractions using a picture

When we add fractions, a picture helps us see how many equal parts make up the whole. The denominator tells us into how many equal parts the whole is divided. The numerator tells us how many of these parts we have.

When fractions have the same denominator, we only add the numerators. When fractions have different denominators, we first need to rewrite them so that they use equal-sized parts.

Fractions with the same denominator

Let us look at the example:

$1/10 + 3/10 = 4/10$

The whole is divided into 10 equal parts. First, we have 1/10, which means one square. Then we add 3/10, which means three more squares. Altogether, we have 4/10.

Orange shows the first fraction, and green shows the second fraction.

Because both fractions are divided into tenths, we can add them directly:

$1 + 3 = 4$, so $1/10 + 3/10 = 4/10$

Fractions with different denominators

Now let us try this example:

$1/3 + 2/9$

Here we cannot add the fractions right away, because they do not have the same denominators. The first fraction is written in thirds, and the second one in ninths.

But we can draw it if we use 9 squares. Then one third is equal to 3 squares (the first orange row). We add two ninths in green. The whole square is divided into 9 equal parts. Altogether, we get 5/9.

With pictures like these, always imagine that you are adding the shaded squares together. The picture shows you how many equal parts you already have and how many you will have altogether after adding more.

Subtracting fractions

We can also use a picture when we subtract fractions. For example:

$7/9 - 2/9 = 5/9$

The whole is divided into 9 equal parts. First, we shade 7 parts. Then we take away 2 parts. There are 5 parts left, so the answer is 5/9.

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When fractions have the same denominator, we subtract the numerators:

$7 - 2 = 5$, so $7/9 - 2/9 = 5/9$

Adding and subtracting fractions

If the denominators (the bottom numbers) are the same, we just need to add or subtract the numerators (the top numbers).

If the denominators (the bottom numbers) are different, we need to find the least common denominator. We can do this by following these steps:

Step 1: Convert fractions to have the same denominator (by multiplying):

Step 2: Add or subtract the numerators (the top numbers), and write the result over the same denominator (the bottom number).

Sometimes is better to divide one of the fractions:

And in some cases you have to cross-multiply (multiply the numerator of each fraction by the denominator of the other):

If you are adding/subtracting fractions and whole numbers, you can turn the whole number into a fraction with the same denominator.

Adding and subtracting fractions

If the denominators (the bottom numbers) are the same, we just need to add or subtract the numerators (the top numbers).

If the denominators (the bottom numbers) are different, we need to find the least common denominator. We can do this by following these steps:

Step 1: Convert fractions to have the same denominator (by multiplying):

Step 2: Add or subtract the numerators (the top numbers), and write the result over the same denominator (the bottom number).

Sometimes is better to divide one of the fractions:

And in some cases you have to cross-multiply (multiply the numerator of each fraction by the denominator of the other):

If you are adding/subtracting fractions and whole numbers, you can turn the whole number into a fraction with the same denominator.

Mixed numbers

A proper fraction is a fraction with the numerator (top number) less than the denominator (bottom number), e.g. $1/3 ;;; 3/4 ;;; 5/7 ;;; 6/7$. It is always less than 1.

An improper fraction is a fraction with the numerator larger than (or equal to) the denominator, e.g. $7/6 ;;; 8/4 ;;; 3/3 ;;; 8/7 ;;; 34/9$. It is always greater than or equal to 1.

A mixed number is a whole number and a proper fraction combined. Thus, it is always greater than 1.

Each improper fraction can also be expressed as a mixed number.

How to convert an improper fraction to a mixed number.

To convert $15/7$ into a mixed number, we need to find out how many times 7 can fit into 15. For this divide the numerator by the denominator:
$15÷7=2$ with remainder 1 ($15=2×7+1$).
2 will be used as the whole number of mixed number and 1 will be the numerator above the denominator.

How to convert a mixed number into a fraction:

Multiply the whole number by the denominator:

Add 20 to the numerator: Write 23 over the denominator

A mixed number can be also written as: 1⁵⁄₈ ; 3³⁄₄.

Dividing fractions

To divide two fractions, find the reciprocal of the second fraction (invert it) and multiply.

You can simplify/reduce the resulting fraction (or convert to a mixed number):

Dividing fractions by a whole number

Rewrite the whole number as a fraction (just put the whole number over 1 as denominator):

Complex fractions

In general, a complex fraction is normally formed of two fractions, one on top of the other (the numerator, denominator, or both contain a fraction). It is actually a ratio of two fractions.

To convert this to a simple fraction, rewrite as division:

You can also just multiply the top numerator by the bottom denominator and the top denominator by the bottom numerator:

If only the numerator or the denominator contains a fraction, rewrite the other one as a fraction, and then divide or multiply:

How to solve a problem with a whole and one of its parts

In this problem, we work with a whole and one part of it.

We have the equation:

$1/6 Y$ means one sixth of Y. We can also write it as $1/6 × Y$. Y is some number, the whole that we are trying to find. We can draw it as one whole circle and divide it into 6 equal parts, so one part represents $1/6$ of the whole.

1. What we can already see in the equation

In the equation, we have:

On the left side, there is one sixth of the whole and also 10. Together, they make the whole Y.

That means the number 10 must represent the remaining part of the whole.

2. How large the missing part is

If we already have 1/6 Y of the whole Y, then $5/6$ of the whole Y is missing:

And according to the equation, this missing part is equal to 10.

3. How much one sixth is

If it is true that:

then we first find out how much 1/6 Y is. Five sixths has the value 10, so we find one sixth by dividing by five:

So $1/6$ of the whole Y is equal to 2.

4. How much the whole Y is

The whole has 6 sixths. If one sixth is 2, then the whole is:

Quick check

Substitute the value back into the equation:

That is correct, so the result is right.