**What are Decimals?**

The word decimal comes from the Latin word “Decem” which means 10. In algebra, a decimal number can be defined as a number whose complete part and the fractional part are separated by a decimal point. Before we start the discussion on how to add and subtract the decimals, it is important to understand two key concepts related to decimals –

- Place Value System of Decimals
- Like and Unlike Decimals

Let us learn about these two concepts first

**Place Value System of Decimals**

We know that each place in the place value table has a value ten times the value of the next place on its right. In other words, the value of a place is one-tenth of the value of the next place on its left. We observe that if one digit moves one place left to right its value becomes one-tenth ($\frac{1}{10}$) of its previous value and when it moves two places left to right its value comes one-hundredth ($\frac{1}{100}$) of its previous term and so on. Therefore, if we wish to move beyond ones place which is the case of decimals, we will have to extend the place value table by introducing the places of tenths ($\frac{1}{10}$), hundredths ($\frac{1}{100}$), thousandths ($\frac{1}{1000}$) and so on.

Therefore, the place value table in case of a decimal number will be of the form –

Thousands( 1000 ) | Hundreds( 100 ) | Tens( 10 ) | Ones( 1 ) | Tenths($\frac{1}{10}$) | Hundredths($\frac{1}{100}$) | Thousandths($\frac{1}{1000}$) |

For example, the decimal number 257.32 in the place value system will be written as –

Hundreds | Tens | Ones | Tenths | Hundredths |

2 | 5 | 7 | 3 | 2 |

This concept is important to learn as before addition or subtraction the decimal values will be required to be placed according to their place values.

Now, let us understand what we mean by like and unlike decimals

**Like and Unlike Decimals**

Consider the following group of decimals –

- 0.3, 0.7, 2.5, 12.7, 101.5
- 1.14, 22535, 0.6, 0.98

The number of digits on the right of the decimal place in a number is known as the number of decimal places.

**The decimals having the same number of digits on the right of the decimals point are known as like decimals. For example, the first group of numbers above is a group of like decimals.**

**Similarly, the decimals having a different number of digits on the right of the decimals point are known as like decimals. . For example, the second group of numbers above is a group of unlike decimals.**

Is it possible to convert, unlike decimals into like decimals? Let us find out.

**Conversion of Unlike Decimals into Like Decimals**

Let us consider two unlike decimals 3.5 and 4.75. It is important to note here that the first decimal number 3.5 has one digit after the decimal point while the second decimal number 4.75 has two digits after the decimal point.

To convert these unlike decimals into like decimals, we should have two decimal places in 3.5. We know that by adding zero on the right of extreme right digits in the decimal part of a number does not alter its value.

Therefore, 3.5 can be written as 3.50

Now, 3.50 and 4.75 are like decimals.

Therefore, we can convert unlike decimals into like decimals by adding zeros to the right of the decimal point or by finding their equivalent decimal.

Now let us use the above concepts and learn about the addition and subtraction of decimals.

**Addition in Decimals**

Addition of decimals is as simple as the addition of whole numbers. The only difference is that we ensure the alignment of the decimal points of given numbers before their addition. The following steps are used to add numbers with decimals –

- Convert the given decimals to like decimals.
- Write the decimals in columns with the decimal points directly below each other so that tenths come under tenths, hundredths come under hundredths and so on.
- Add as we add whole numbers.
- Place the decimal point in the answer directly below the other decimal points.

Let us understand the above steps with an example.

**Example**

Suppose we want to add the decimal numbers

15.44, 7.524 and 25

**Solution**

We have been given the decimal numbers 15.44, 7.524 and 25

First of all, we will have to convert these decimals into like decimals. Therefore,

15.44 = 15.440

7.524 is already a like decimal so no conversion is required.

25 = 25.000

Now, that all the three decimals are like decimals, we will proceed ahead and write these decimals in column form, as shown below –

15.444

+ 7.524

+25.000

———-

———-

Next, we will add these fractions just as we add the whole numbers. We will get,

15.444

+ 7.524

+25.000

———-

+47.964

———-

**Hence, the addition of 15.44, 7.524 and 25 is 47.964**

**Subtraction in Decimals**

To subtract a decimal number from another decimal number, we follow the same procedure as we did in addition. The following steps for the subtraction of a decimal number from another decimal number –

- Convert the given decimals to like decimals.
- Write the decimals in columns with the decimal points directly below each other so that tenths come under tenths, hundredths come under hundredths and so on.
- Subtract as usual ignoring the decimal points.
- Place the decimal point in the answer directly below the other decimal points.

Let us understand the above steps with an example –

**Example**

Suppose we want to find the value of 11.6 – 9.847

**Solution**

We have been given the decimals, 11.6 and 9.847 and we are required to find their difference.

First of all, we will have to convert these decimals into like decimals. Therefore,

11.6 = 11.600

9.847 is already a like decimal so no conversion is required.

Next, we will subtract these fractions just as we add the whole numbers. We will get,

11.600

-9.847

———-

1.753

———-

**Therefore, 11.6 – 9.847 = 1.753**

**Addition and Subtraction of Decimals in real Life Situations**

We perform addition and subtraction of decimals in many situations in our day-to-day lives. For example, you may need to add up your grocery bill before paying at the counter. Have you ever notice that most of the prices are in decimals? Let us discuss through an example where we use addition or subtraction of decimals in our daily life.

**Example**

Peter had £7.45 from his pocket money. He used it to buy candies for £5.30. How much pocket money was he left with?

**Solution**

We have been given that, Peter had £7.45 from his pocket money. He used it to buy candies for £5.30.

In order to find out the pocket money he was left with, we will need to find the difference in the given values. Therefore, we have

Total pocket money with Peter = £7.45

Money Peter spent on buying candies = £5.30

Pocket money left with Peter = £7.45- £5.30

7.45

-5.30

———-

2.15

———-

**Hence, Peter is left with £2.15 pocket money.**

**Solved Examples**

**Example 1** Sam earned £ 320.50. He purchased a bag for £ 98.25. How much money does he have now?

**Solution** Let us first define what is given and what needs to be calculated. We have been given that Sam earned £ 320.50 and he purchased a bag for £ 98.25. We are required to find how much money Sam has now. Therefore,

Earnings of Sam = £ 320.50

Price of the bag purchased by Sam = £ 98.25

Both the decimal numbers are like terms; hence we can go ahead with the calculations.

Money left with Sam = £ 320.50 – £ 98.25 = £ 222.25

**Hence, money left with Sam after purchasing a bag of £ 98.25 = £ 222.25**

**Example 2** Edward spent £ 35.75 for Maths book and £ 32.60 for Science book. Find the total amount spent by Edward.

**Solution** Let us first define what is given and what needs to be calculated. We have been given that Edward spent £ 35.75 for Maths book and £ 32.60 for Science book. We are required to find the total amount spent by Edward. Therefore,

Money spent by Edward on Maths book = £ 35.75

Money spent by Edward on Science book = £ 32.60

Both the decimal numbers are like terms; hence we can go ahead with the calculations.

Total money spent by Edward = £ 35.75 + £ 32.60 = £ 68.35

**Therefore, total money spent of Edward on maths and science books = £ 68.35.**

**Example 3**** **Sophie had £ 305.80 in her bank account. She deposited £ 250.25 more and then withdrew £ 317.50 from her account. What is the balance now in her account?

**Solution**** **Let us first define what is given and what needs to be calculated. We have been given that Sophie had £ 305.80 in her bank account. She deposited £ 250.25 more and then withdrew £ 317.50 from her account. We are required to find the balance in her account now. Therefore,

Initial amount in Sophie’s account = £ 305.80 …………………. ( 1 )

Amount deposited by Sophie = £ 250.25 …………………. ( 2 )

Both the decimal numbers are like terms; hence we can go ahead with the calculations.

Total amount in Sophie’s account = ( 1 ) + ( 2 )

= £ 305.80 + £ 250.25

= £ 556.05 …………………………… ( 3 )

Now, from this total amount, Sophie had withdrawn £ 317.50

Therefore, money left in her account = ( 3 ) – £ 317.50

Again, both the decimal numbers are like terms; hence we can go ahead with the calculations.

= £ 556.05 – £ 317.50

= £ 238.55

**Hence, money left in Sophie’s account = £ 238.55**

**Example 4** Sam, Peter and Henry bought 8.5 litres, 7.25 litres and 9.4 litres milk respectively from a milk booth. How much milk did they buy in all? If there were 30 litres of milk in both, find the quantity of milk left.

**Solution**** **Let us first define what is given and what needs to be calculated. We have been given that Sam, Peter and Henry bought 8.5 litres, 7.25 litres and 9.4 litres milk respectively from a milk booth. We need to find out –

a) How much milk did they buy in all?

b) If there were 30 litres of milk in both, find the quantity of milk left.

Let us find the answers to the above problems one by one.

Amount of milk bought by Sam = 8.5 litres

Amount of milk bought by Peter = 7.25 litres

Amount of milk bought by Henry = 9.4 litres

Total amount of milk bought by them = 8.5 litres + 7.25 litres + 9.4 litres

Note here that the given decimals are not like terms. Hence, we will first need them to be converted into like terms. Therefore, we now have,

Amount of milk bought by Sam = 8.5 litres = 8.50 litres

Amount of milk bought by Peter = 7.25 litres = 7.25 litres

Amount of milk bought by Henry = 9.4 litres = 9.40 litres

Now that the terms are like terms, we can go ahead with the calculations.

Total milk bought by Sam, Peter and Henry = 8.50 litres + 7.25 litres + 9.40 litres = 25.15 litres

**Hence, the total milk bought by Sam, Peter and Henry =25.15 litres**

Now, let us solve the second part of the question. We have been given that there were 30 litres of milk in the booth. How much milk is left after Sam, Peter and Henry bought the milk?

Therefore,

Total milk in booth = 30 litres

Milk bought by Sam, Peter and Henry = 25.15 litres

Again, we can see here that the given decimals are not like terms. Hence, we will first need them to be converted into like terms. Therefore, we now have,

Total milk in booth = 30 litres = 30.00 litres

Milk bought by Sam, Peter and Henry = 25.15 litres

Now that the terms are like terms, we can go ahead with the calculations.

Milk left in the booth = 30.00 litres – 25.15 litres

= 4.85 litres

**Hence, after Sam, Peter and Henry, 4.85 litres of milk was left in the booth.**

**Example 5**** **Find the value of 0.07 + 2.8 + 0.5

**Solution** We have been given the decimal numbers 0.07, 2.8 and 0.5 we are required to find the sum of these decimals.

First of all, we will have to convert these decimals into like decimals. We can see that there is a maximum of 2 digits after the decimal point in the three numbers. Therefore, in order to make them like terms, there must be exactly two digits after the decimal point in each of the three given numbers. Therefore,

0.07 is already a like decimal so no conversion is required.

2.8 = 2.80

0.5 = 0.50

Now that the three decimal numbers are like terms we can go ahead and add these decimals. We will get,

0.07

+2.80

+0.50

———-

3.37

———-

**Hence, 0.07 + 2.80 + 0.50 = 3.37**

**Key Facts and Summary**

- A decimal number can be defined as a number whose complete part and the fractional part are separated by a decimal point.
- Each place in the place value table has a value ten times the value of the next place on its right.
- The number of digits on the right of the decimal place in a number is known as the number of decimal places.
- The decimals having the same number of digits on the right of the decimals point are known as like decimals.
- The decimals having a different number of digits on the right of the decimals point are known as like decimals.
- By adding zero on the right of extreme right digits in the decimal part of a number does not alter its value.
- We can convert unlike decimals into like decimals by adding zeros to the right of the decimal point or by finding their equivalent decimal.
- For adding two or more decimal numbers, write down the decimal numbers, one number under the other number and line up the decimal points. Convert the given decimals to like decimals. Arrange the addends in such a way that the digits of the same place are in the same column. Add the numbers from the right as we carry addition usually.
- For finding the difference between two or more decimal numbers, write down the decimal numbers, one number under the other number and line up the decimal points. Convert the given decimals to like decimals. Arrange the addends in such a way that the digits of the same place are in the same column. Subtract the numbers from the right as we carry addition usually.

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