Addition and Subtraction of Decimals: The word decimal comes from the Latin word Decem meaning \(10.\) A system used to express the whole number and fraction together is known as a decimal number system. We will separate a whole number from the fraction by inserting a dot known as a decimal point in the decimal system. Example: If we buy a sandwich, the seller tells us the price of the sandwich is \(₹45\) and \(50\) paise.

Now, if we want to express this whole amount in one figure, we will say that the price of the sandwich is \(₹45.50.\) So, we will use this decimal in many such real-life situations without even knowing it. In this article, we will learn about the decimal and the addition and subtraction of the decimals.

What is Decimal?

There are types of numbers that have a whole number part and a fractional part separated by a decimal point. Those numbers are known as decimal numbers. The dot present in between the whole number part and fractions part is called the decimal point. 

Decimals are the extension of our number system. We know that decimals can be considered fractions whose denominators are \(10, 100, 1000,\) etc. We may say that the numbers expressed in decimal forms are called decimals.
Example: \(14.265, 0.158, 130.007\) etc., are decimal numbers or decimals.

Also, \(64.6\) is a decimal number.
Here, \(64\) is a whole number part, and \(6\) is the fractional part.

Decimal Places: The number of digits in the decimal part of a decimal number is known as the number of decimal places.
For example, \(2.65\) has two decimal places, and \(93.235\) has three decimal places.

Decimal as a Fraction

Decimals are another name for fractions whose denominators are \(10, 100, 1000,\) etc. In the below section, let us define tenths, hundredths, thousandths etc.

Tenths

Observe the below figure. It is divided into ten equal parts, and one part is shaded. The shaded part represents one-tenth of the whole figure. It is written as \(\frac{1}{{10}}.\) Also \(\frac{1}{{10}}\) is written as \(.1\) or \(0.1,\) which is read as point one.

Thus, the fraction \(\frac{1}{{10}}\) is called one-tenth and written as \(0.1.\)

Similarly, \(\frac{8}{{10}}\) can be written as \(0.8,\frac{5}{{10}}\) can be written as \(0.5.\)

Hundredths

The below figure is divided into \(100\) equal parts, out of which one part is shaded. The shaded part represents one-hundredth of the whole figure and is written as \(\frac{1}{{100}}.\frac{1}{{100}}\) is also written as \(.01\) or \(0.01\) and is read as one-hundredth or point zero one.

Similarly, \(\frac{{19}}{{100}} = 0.19,\frac{{42}}{{100}} = 0.42\)

Also,
\(\frac{{145}}{{100}} = \frac{{100 + 45}}{{100}} = \frac{{100}}{{100}} + \frac{{45}}{{100}} = 1.45\)

Thousandths

If an object is divided into \(1000\) equal parts, then each part is one-thousandths of the whole. One thousandth is written as \(\frac{1}{{1000}}.\)

\(\frac{1}{{1000}}\) is also written as \(0.001\)

If we take \(7\) parts out of \(1000\) equal parts of an object, then \(7\) parts make \(\frac{7}{{1000}}\) of the whole, and it is written as \(0.007.\)

Similarly, we have \(\frac{{14}}{{1000}} = 0.014,\frac{{345}}{{1000}} = 0.345\)

Expressing Fractions as Decimals

By using the definition of tenths, hundredths, thousandths etc. We can always express fractions with denominators \(10, 100\) or \(1000\) as decimals, as shown below.

\(\frac{7}{{10}} = 0.7,\frac{4}{{10}} = 0.4,\frac{3}{{10}} = 0.3\)
\(\frac{{44}}{{10}} = \frac{{40 + 4}}{{10}} = \frac{{40}}{{10}} + \frac{4}{{10}} = 4 + \frac{4}{{10}} = 4.4\)
\(\frac{8}{{100}} = 0.08,\frac{5}{{100}} = 0.05,\frac{4}{{100}} = 0.04\)
\(\frac{{54}}{{100}} = 0.54,\frac{{65}}{{100}} = 0.65,\frac{{96}}{{100}} = 0.96\)
\(\frac{{174}}{{100}} = \frac{{100 + 74}}{{100}} = \frac{{100}}{{100}} + \frac{{74}}{{100}} = 1 + \frac{{74}}{{100}} = 1.74\)
\(\frac{5}{{1000}} = 0.005,\frac{7}{{1000}} = 0.007\)
\(\frac{{24}}{{1000}} = 0.024,\frac{{76}}{{1000}} = 0.076\)

Like and Unlike Decimals

Decimals having the same number of places are known as like decimals, i.e. decimals having the same number of digits on the right of the decimal point are known as like decimals. Otherwise, the decimals are unlike decimals.

For example, \(6.35, 17.08, 423.98\) are like decimals and \(8.5, 9.32, 23.546\) are unlike decimals.

It is to note that annexing zeros on the right side of the extreme right digit of the decimal part of a number does not alter the value of the number. So, the, unlike decimals, can always be converted into like decimals by annexing the required number of zeros on the right side of the extreme right digit in the decimal part.

For example, \(9.4, 15.95, 48.106\) are unlike decimals. These decimals can be re-written as \(9.400, 15.950, 48.106.\) Now, these are like decimals.

Comparing Decimals

To compare decimal numbers, we will follow the below steps:

1. Obtain the decimal numbers.
2. Compare the whole number parts. The number with the greater whole part will be more significant. If the whole parts are equal, go to the next step.
3. Compare the extreme left digits of the decimal parts of two numbers. The number with a greater extreme left digit will be greater. If the extreme digits of decimal parts are equal, then compare the following digits and so on.

Example: Which is greater, \(48.23\) and \(39.35?\)

Solution: The given decimals have distinct whole number parts, so we compare whole number parts only.
In \(48.23,\) the whole number part is \(48.\)
In \(39.35,\) the whole number part is \(39.\)
Since \(48>39\)
Therefore, \(48.23>39.35\)

Addition and Subtraction of Decimals

To add or subtract decimals, we will use the following steps:

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

Example: Add \(16.43\) and \(8.454\)

Solution: \(16.43+8.454\)

Therefore, \(16.43+8.454=24.884\)

Addition and Subtraction of Decimals and Whole Numbers

To add or subtract a decimal and a whole number, the whole number is changed into a decimal number. It is done by placing a decimal after the whole number and then writing the required number of zeroes after the decimal point. In other words, we need to convert the whole number as a like decimal by adding zeroes.

For example, the whole number \(6\) is written in decimal form as \(6.0.\) Below steps are the methods to add and subtract decimals and whole Numbers: 

1. Write the numbers such that the decimals are lined perfectly.
2. Put zeroes equal to the other decimal number, making it a decimal number.
3. Now add the decimal numbers and find the solution to addition.

Example: Subtract \(15\) from \(27.56\)

Solution: Here, \(15\) is a whole number. To subtract \(15\) from \(27.56,\) we will convert \(15\) as a like decimal by adding two zeroes after a decimal point.
\(⇒15=15.00\)
So,

Therefore, \(27.56-15.00=12.56\)

Solved Examples – Addition and Subtraction of Decimals

Q.1. Aakash bought vegetables weighing \({\rm{10}}\,{\rm{kg}}{\rm{.}}\) Out of this \({\rm{3}}\,{\rm{kg}}\,{\rm{500}}\,{\rm{g}}\) is onion, \({\rm{2kg}}\,{\rm{75g}}\) is tomato, and the rest is potato. What is the weight of the potato?
Ans: We have,
Weight of onion \( = 3\;{\rm{kg}}\,500\;{\rm{g}} = 3.500\;{\rm{kg}}\)
Weight of tomato \( = 2\;{\rm{kg}}\,75\;{\rm{g}} = 2.075\;{\rm{kg}}\)
Therefore, the total weight of onion and tomato is

Therefore, the total weight of vegetables\( = 10\;{\rm{kg}}\)
So, the weight of the potato is \(10\;{\rm{kg}} – 5.575\;{\rm{kg}}\)

Therefore, the weight of the potato is \(4.425\;{\rm{kg}}.\)

Q.2. Rashid spent \(₹35.75\) on maths books and \(₹32.60\) on science books. Find the total amount spent by Rashid.
Ans: We have,
Money spent on maths books\(=₹35.75\)
Money spent on science books\(=₹32.60\)
Therefore, total money spent\(=₹35.75+₹32.60\)

So, total amount spent by Rashid\(=₹68.35\)

Q.3. Ravi purchased \(5\;{\rm{kg}}\,400\;{\rm{g}}\) rice, \(2\;{\rm{kg}}\,20\;{\rm{g}}\) sugar and \(10\;{\rm{kg}}\,850\;{\rm{g}}\) atta. Could you find the total weight of his purchases?
Ans: We have,
Quantity of rice purchased\( = 5\;{\rm{kg}}\,400\;{\rm{g}} = 5.400\;{\rm{kg}}\)
Quantity of sugar purchased\( = 2\;{\rm{kg}}\,20\;{\rm{g}} = 2.020\;{\rm{kg}}\)
Quantity of atta purchased\( = 10\;{\rm{kg}}\,850\;{\rm{g}} = 10.850\;{\rm{g}}\)
Therefore, the total weight of his purchases is:

So, the total weight of his purchases\( = 18.270\;{\rm{kg}}\)

Q.4. Abhishek had \(₹7.45.\) He bought chocolates for \(₹5.30.\) Find the balance amount left with Abhishek.
Ans: We have,
The total amount of money\(=₹7.45\)
Amount spent on chocolates\(=₹5.30\)
So, the amount left with Abhishek is:

Therefore, the amount left with Abhishek\(=₹2.15\)

Q.5. Namita travels \(20\;{\rm{km}}\,50\;{\rm{m}}\) every day. Out of this, she travels \(10\;{\rm{km}}\,200\;{\rm{m}}\) by bus and the rest by auto. How much distance does she travel by auto?
Ans: We have,
Total distance travelled in a day\( = 20\;{\rm{km}}\,50\;{\rm{m}} = 20.050\;{\rm{km}}\)
Distance travelled by bus\( = 10\;{\rm{km}}\,200\;{\rm{m}} = 10.200\;{\rm{km}}\)
Therefore, distance travelled by auto is:

Hence, Namita travelled \(9.850\;{\rm{km}}\) or \(9\;{\rm{km}}\,850\;{\rm{m}}\) by auto.

Summary

In the above article, we have learned decimals’ definition, decimal as a fraction, expressing fraction as a decimal, comparing decimals, like and unlike decimals, addition and subtraction of decimals, and addition and addition subtraction of decimals with whole number and solved some example problems on the same.

Frequently Asked Questions (FAQs) – Addition and Subtraction of Decimals

Q.1. How do you do addition and subtraction of decimals?
Ans: To add or subtract decimals, we will use the following steps:
1. Convert the given decimals to like decimals.
2. Write the decimals in columns with their decimal points directly below each other so that tenths come under tenths, hundredths come under hundredths and so on.
3. Add or subtract as we add or subtract the whole numbers.
4. Place the decimal point, in the answer, directly below the other decimal points.

Q.2. How do you subtract decimals step by step?
Ans: To subtract a decimal number from another decimal number, we follow the same procedure as an addition. We will follow the following steps to subtract a decimal number from another decimal number.
1. Convert the given decimals to like decimals.
2. Write the decimals in columns with decimal points directly below each other.
3. Subtract, as usual, ignoring the decimal points.
4. Place the decimal point in the difference directly below the other decimal points.

Q.3. How do you add and subtract decimals and mixed decimals?
Ans: We follow the below steps to add and subtract decimals and mixed decimals.
A decimal number with a whole number part is a mixed decimal.
1. Place the numbers vertically so that the decimal points all lie on a vertical line.
2. Add extra zeros to the right of the number so that each number has the same number of digits to the right of the decimal place.
3. Add or subtract the numbers as you would whole numbers.

Q.4. Do you line up the decimal points when adding and subtracting?
Ans: When we add or subtract decimals, the essential step is to line up the decimal points. If the numbers do not have the same number of digits after the decimal point, we can fill zeroes to the right of the decimal point to help us line up the numbers.

Q.5. How do you add decimals step by step?
Ans: The addition of decimals is as simple as the addition of whole numbers. The only difference is that we ensure aligning the decimal points of given numbers before their addition. We may use the following steps to add numbers with decimals.
1. Convert the given decimals to like decimals.
2. Write the decimals in columns with the decimal points directly below each other so
3. Add as we add whole numbers.
4. Place the decimal point in the answer directly below the other decimal points.