Decimals

Fractions and decimals

Decimal numbers are another way to represent numbers that are smaller than the unit 1. Decimals are written to the right of the units place separated by a period called decimal point. The first digit to the right of the decimal point represents tenths, the second represents hundredths, the third represents thousandths, etc.

If you want to compare or do calculations with fractions and decimals, you need to convert all of them to one type (either decimals or fractions).

Convert decimals to fractions:

1. Convert the decimal to fraction using tenths, hundredths, thousandths, etc. depending on the number of decimal places:

• one place=tenth, e.g. $0.5=5/10; ; 3.3=33/10$
• two places=hundredths, e.g. $0.02=2/100 ;; 0.12=12/100 ;; 1.38=138/100$
• three places=thousandth, e.g. $0.002=2/1000; ; 0.304=304/1000$

2. Simplify the fraction to the lowest common term:

Convert fractions to decimals

1. Find a number you can multiply by the denominator (bottom of the fraction) to make it 10, or 100, or 1000, etc.

2. A fraction in this form can be converted to a decimal - just put the decimal point in the correct position (one space from the right hand side for every zero in the bottom number):
E.g. $5/10=0.5;$ ; $25/100=0.25$
For some numbers there is no way to multiply them to become tens, hundreds or thousands. For these numbers you can calculate an approximate decimal (finding a denominator close to 10, 100 or 1000):

Convert whole numbers to fractions

Just put 1 below the whole number:

Dividing by whole numbers

In a division such as $9 ÷ 2$, the number being divided (9) is called the dividend, the number we divide by (2) is the divisor, and the result is the quotient.

When the divisor is a whole number, we can divide in the usual way. The quotient does not have to be a whole number – it can be a decimal number.

The number 2 goes into 9 four whole times ($4 × 2 = 8$), and 1 is left over. We do not stop at the remainder – we keep dividing. The remaining 1 is half of 2, and half is $0.5$, so the result is $4 + 0.5 = 4.5$.

The same rule works when the dividend is already a decimal number. We divide as usual and keep the decimal point in the quotient directly above the decimal point in the dividend:

Here $4 ÷ 4 = 1$ and $0.8 ÷ 4 = 0.2$, so together we get $1.2$.

Dividing by decimal numbers

When the divisor is a decimal number, dividing directly is awkward. It is much easier to first change the divisor into a whole number. To do this, multiply both the dividend and the divisor by the same number: 10, 100, 1000 and so on – whichever is needed to remove the decimal point from the divisor.

Why does this not change the result? A division is like a fraction: $7 ÷ 3.5$ is the same as $7/3.5$. If we multiply the numerator and the denominator by the same number, the fraction keeps the same value – exactly as $1/2$ and $10/20$ are equal. So the quotient stays the same, only the numbers become easier to work with.

For example:

We multiplied both numbers by 10. The divisor 3.5 became the whole number 35, and $70 ÷ 35$ is easy: the answer is 2. Check: $2 × 3.5 = 7$

Moving the decimal point in both numbers

Instead of thinking about multiplication by 10, 100 or 1000, we can imagine moving the decimal point in both numbers by the same number of places to the right. Multiplying by 10 moves the point one place, by 100 two places, by 1000 three places.

A simple rule: count the decimal places in the divisor. That is how many places you must move the decimal point in both numbers.

For example:

The divisor 0.35 has two decimal places, so we move the decimal point two places to the right in both numbers. $0.35$ becomes $35$, and $0.7$ becomes $70$ (we add a zero, because $0.7 = 0.70$):

It is important to move the decimal point by the same number of places in both the dividend and the divisor. If you move it by different amounts, the result will be wrong.

Dividing as fractions

There is another way to see the same idea. Every decimal number can be written as a fraction with 10, 100 or 1000 in the denominator:

To divide by a fraction, we multiply by its reciprocal – we turn the second fraction upside down and multiply:

We get the same result, $2$, as before. Look at the middle step: the 100 from the denominator of $0.35$ moves up and cancels the 10 from the denominator of $0.7$. That is exactly why "moving the decimal point in both numbers" works – the powers of ten cancel each other.

One more example:

Both denominators are 10, so they cancel completely and only $48 ÷ 12 = 4$ remains.

Finding the easy division first

Sometimes the divisor is a whole number, but the dividend is a small decimal, for example $0.07 ÷ 35$. Nobody can do this directly in their head. Or the numbers become large after we move the decimal point, for example $7 ÷ 0.035 = 7000 ÷ 35$.

In such cases we use a two-step method:

Step 1. Ignore the decimal point and the zeros for a moment and find the easy division hidden inside. In all the examples below it is $70 ÷ 35 = 2$.

Step 2. Fix the decimal point in the result. Two rules tell us how:

Rule	Example
If we made the dividend 10 times smaller, the quotient is also 10 times smaller – move the decimal point one place to the left. Dividing a smaller number gives a smaller result.	$70 ÷ 35 = 2$ $7 ÷ 35 = 0.2$ $0.7 ÷ 35 = 0.02$
If we made the divisor 10 times smaller, the quotient is 10 times larger – move the decimal point one place to the right. Dividing by a smaller number gives a larger result.	$70 ÷ 35 = 2$ $70 ÷ 3.5 = 20$ $70 ÷ 0.35 = 200$

Now let us use both steps on the two difficult examples.

Example 1: $0.07 ÷ 35$. The easy division is $70 ÷ 35 = 2$. But our dividend is not 70 – it is $0.07$, which is 1000 times smaller ($70 → 7 → 0.7 → 0.07$, three places). So the quotient must also be 1000 times smaller than 2. We move the decimal point three places to the left:

Check: $0.002 × 35 = 0.07$

Example 2: $7 ÷ 0.035$. First we make the divisor a whole number by moving the decimal point three places in both numbers: $7000 ÷ 35$. The easy division is again $70 ÷ 35 = 2$, but our dividend 7000 is 100 times larger than 70 ($70 → 700 → 7000$, two places). So the quotient is 100 times larger than 2 – we move the decimal point two places to the right:

Check: $200 × 0.035 = 7$

Compare this with $7 ÷ 0.35 = 700 ÷ 35 = 20$. Notice that a smaller divisor gives a larger quotient: dividing 7 by 0.035 gives a much larger result than dividing 7 by 0.35. This makes sense: the smaller the pieces we cut 7 into, the more pieces we get.

Long division

When the quotient is not obvious, we use long division. The method is the same as for whole numbers – we only need to take care of the decimal point.

First, we make the divisor a whole number. The divisor 3.2 has one decimal place, so we multiply both numbers by 10:

Now we can use long division. Since 32 is larger than 6, the number 32 goes into 6 zero times, so the quotient starts with 0 followed by a decimal point. To continue, we write 6 as 6.0000 – adding zeros after the decimal point does not change its value – and bring down one zero at a time. The decimal point in the quotient is placed directly above the decimal point in the dividend.

Therefore:

Check: $0.1875 × 3.2 = 0.6$

Dividing by whole numbers

In a division such as $9 ÷ 2$, the number being divided (9) is called the dividend, the number we divide by (2) is the divisor, and the result is the quotient.

When the divisor is a whole number, we can divide in the usual way. The quotient does not have to be a whole number – it can be a decimal number.

The number 2 goes into 9 four whole times ($4 × 2 = 8$), and 1 is left over. We do not stop at the remainder – we keep dividing. The remaining 1 is half of 2, and half is $0.5$, so the result is $4 + 0.5 = 4.5$.

The same rule works when the dividend is already a decimal number. We divide as usual and keep the decimal point in the quotient directly above the decimal point in the dividend:

Here $4 ÷ 4 = 1$ and $0.8 ÷ 4 = 0.2$, so together we get $1.2$.

Dividing by decimal numbers

When the divisor is a decimal number, dividing directly is awkward. It is much easier to first change the divisor into a whole number. To do this, multiply both the dividend and the divisor by the same number: 10, 100, 1000 and so on – whichever is needed to remove the decimal point from the divisor.

Why does this not change the result? A division is like a fraction: $7 ÷ 3.5$ is the same as $7/3.5$. If we multiply the numerator and the denominator by the same number, the fraction keeps the same value – exactly as $1/2$ and $10/20$ are equal. So the quotient stays the same, only the numbers become easier to work with.

For example:

We multiplied both numbers by 10. The divisor 3.5 became the whole number 35, and $70 ÷ 35$ is easy: the answer is 2. Check: $2 × 3.5 = 7$

Moving the decimal point in both numbers

Instead of thinking about multiplication by 10, 100 or 1000, we can imagine moving the decimal point in both numbers by the same number of places to the right. Multiplying by 10 moves the point one place, by 100 two places, by 1000 three places.

A simple rule: count the decimal places in the divisor. That is how many places you must move the decimal point in both numbers.

For example:

The divisor 0.35 has two decimal places, so we move the decimal point two places to the right in both numbers. $0.35$ becomes $35$, and $0.7$ becomes $70$ (we add a zero, because $0.7 = 0.70$):

It is important to move the decimal point by the same number of places in both the dividend and the divisor. If you move it by different amounts, the result will be wrong.

Dividing as fractions

There is another way to see the same idea. Every decimal number can be written as a fraction with 10, 100 or 1000 in the denominator:

To divide by a fraction, we multiply by its reciprocal – we turn the second fraction upside down and multiply:

We get the same result, $2$, as before. Look at the middle step: the 100 from the denominator of $0.35$ moves up and cancels the 10 from the denominator of $0.7$. That is exactly why "moving the decimal point in both numbers" works – the powers of ten cancel each other.

One more example:

Both denominators are 10, so they cancel completely and only $48 ÷ 12 = 4$ remains.

Finding the easy division first

Sometimes the divisor is a whole number, but the dividend is a small decimal, for example $0.07 ÷ 35$. Nobody can do this directly in their head. Or the numbers become large after we move the decimal point, for example $7 ÷ 0.035 = 7000 ÷ 35$.

In such cases we use a two-step method:

Step 1. Ignore the decimal point and the zeros for a moment and find the easy division hidden inside. In all the examples below it is $70 ÷ 35 = 2$.

Step 2. Fix the decimal point in the result. Two rules tell us how:

Rule	Example
If we made the dividend 10 times smaller, the quotient is also 10 times smaller – move the decimal point one place to the left. Dividing a smaller number gives a smaller result.	$70 ÷ 35 = 2$ $7 ÷ 35 = 0.2$ $0.7 ÷ 35 = 0.02$
If we made the divisor 10 times smaller, the quotient is 10 times larger – move the decimal point one place to the right. Dividing by a smaller number gives a larger result.	$70 ÷ 35 = 2$ $70 ÷ 3.5 = 20$ $70 ÷ 0.35 = 200$

Now let us use both steps on the two difficult examples.

Example 1: $0.07 ÷ 35$. The easy division is $70 ÷ 35 = 2$. But our dividend is not 70 – it is $0.07$, which is 1000 times smaller ($70 → 7 → 0.7 → 0.07$, three places). So the quotient must also be 1000 times smaller than 2. We move the decimal point three places to the left:

Check: $0.002 × 35 = 0.07$

Example 2: $7 ÷ 0.035$. First we make the divisor a whole number by moving the decimal point three places in both numbers: $7000 ÷ 35$. The easy division is again $70 ÷ 35 = 2$, but our dividend 7000 is 100 times larger than 70 ($70 → 700 → 7000$, two places). So the quotient is 100 times larger than 2 – we move the decimal point two places to the right:

Check: $200 × 0.035 = 7$

Compare this with $7 ÷ 0.35 = 700 ÷ 35 = 20$. Notice that a smaller divisor gives a larger quotient: dividing 7 by 0.035 gives a much larger result than dividing 7 by 0.35. This makes sense: the smaller the pieces we cut 7 into, the more pieces we get.

Long division

When the quotient is not obvious, we use long division. The method is the same as for whole numbers – we only need to take care of the decimal point.

First, we make the divisor a whole number. The divisor 3.2 has one decimal place, so we multiply both numbers by 10:

Now we can use long division. Since 32 is larger than 6, the number 32 goes into 6 zero times, so the quotient starts with 0 followed by a decimal point. To continue, we write 6 as 6.0000 – adding zeros after the decimal point does not change its value – and bring down one zero at a time. The decimal point in the quotient is placed directly above the decimal point in the dividend.

Therefore:

Check: $0.1875 × 3.2 = 0.6$

Order of operations

When calculations have more than one operation, we have to follow rules for the order of operations:

Step 1:
From left to right, do all multiplication and division.

Step 2:
From left to right, do all addition and subtraction.

However, if there are parentheses or other types of brackets, do all operations that lie inside them first:
The order of operations also applies while working inside parentheses:
When you have parentheses inside brackets and braces (sometimes referred as nested parentheses), always work from the inside out. There are multiple types of brackets. The innermost parentheses/round brackets () are calculated first, followed by the brackets/box brackets [] that normally form the next layer, followed by braces/curly brackets {} that form a third layer outwards:

Repeating decimals

A recurring/repeating decimal is a number whose digits are periodic (repeating its values at regular intervals) and keep repeating forever, e.g. $1/3=0.333333 ...$ The infinitely repeated digit sequence is called the repetend. It can be denoted by a horizontal line (a vinculum) or dots above it, e.g. $0.2ov57=0.257575757 ...$

Every repeating or terminating decimal is a rational number since it can be converted to a fraction.
To convert repeating decimal to fraction follow these steps:

Step 1:
Set the repeating decimal equal to fraction x:

Step 2:
Move the repeating digit(s)/repetend to the left of the decimal point by multiplying the decimal by 10, 100, 1000 etc.
Step 3:
Subtract the number from both sides of the equation. This will help you to get rid off the decimal part:
Step 4:
Solve the equation for x: