# Subtraction

The concept of subtraction helps us to find the difference between two numbers or quantities. In real-life instances, we come across many situations where we find the difference between two quantities.. For example, you have 10 candies and you share 6 candies with your friends. To find the number of candies remaining with you, you will find the difference between the number of candies you had and the number of candies you gave away.

To understand subtraction better, let us use some examples.

Example 1: Sarah bought 5 donuts. She gave 2 donuts to her brother. How many donuts are left with Sarah?

Solution: Out of 5 donuts, Sarah gave 2 donuts to her brother. Strike off 2 donuts and count the remaining donuts.

Therefore, there are 3 donuts left with Sarah.

Example 2: In a juice shop, there were 8 watermelons. If 3 watermelons are used to make juice, how many watermelons are left in the shop?

Solution: Out of 8 watermelons, 3 are used to make juice. Strike off 3 watermelons and count the remaining.

Therefore, there are 5 watermelons left in the juice shop.

From the above two examples, you observe that a certain number of objects are taken away to find the difference. “The act or skill of taking one number or amount away from another is called subtraction.”

A large group of objects or a higher valued number is a minuend. The smaller group of objects or the small number which we takeaway is a subtrahend. Minuend and subtrahend are separated by a “–“ symbol or “minus” symbol. The result obtained is considered as the difference.

Similar to the addition process, we write the subtraction sentence as

Minuend – Subtrahend = Difference

For example, the difference between 9 and 4 is 5. We express this in a subtraction sentence as 9 – 4 = 5; we read it as “Nine minus four equals five”.

## Subtraction using Number Line

Understanding the concept of subtraction using the number line serves as a fundamental concept. Let us now understand the process of subtraction using a number line, as we are already familiar with the notion.

Example 3: Find the difference between 11 and 5 using the number line.

Solution: To understand the process of subtraction of positive numbers using the number line, follow the steps shown.

Step 1: Draw a number line as given below:

Step 2: Mark the minuend on the number line or starting from 0, move the number of steps equal to minuend in the positive direction. Here, it is 11.

Step 3: To find a difference between minuend and subtrahend, move steps equal to the subtrahend in the negative direction. Here, the subtrahend is 5. So, move 5 steps starting from 11 in the negative direction. After moving 5 steps in the negative direction, the number obtained is 6. Hence, the difference 11 – 5 = 6.

Example 4: Find the difference between -7 and -2 using the number line.

Solution: To understand the process of subtraction of two negative numbers using the number line, follow the steps shown.

Step1: Draw a number line as given below.

Step 2: Mark the minuend on the number line or starting from 0, move the number of steps equal to minuend in the negative direction. Here, it is -7.

Step 3: To find a difference between minuend and subtrahend, move steps equal to the subtrahend in the positive direction (when a negative number is a subtrahend, then the sign of the subtrahend changes to positive. Here, -(-2) = +2). Thus, the subtrahend is 2. So, move 2 steps starting from -7 in the positive direction. After moving 2 steps in the positive direction, the number obtained is -5. Hence, the difference -7 – (-2) = -5.

Try it yourself!

Represent the given subtraction statements on the number line.

1. 10 – 7

2. -14 – 9

## Rules for subtracting numbers

There are certain rules to be understood about the numbers used for subtraction.

When you subtract two numbers, change the – sign to + sign and the second number’s sign to the opposite. After that, add the numbers together. The signs of minuend and subtrahend play a very important role in finding the difference.

Two numbers are natural, whole numbers, or positive integers. For example, consider the numbers 5 and 2. The difference can be written as 5 – 2 = 5 + (–2) = 3.

Two numbers are integers. For example, consider the numbers 6 and –3. To find the difference, we write 6 + [–(–3)]. When two negative signs come together, we consider it as positive (– × – = +). Therefore, 6 + 3 = 9.

## Properties of Subtraction

Four properties of subtraction are

1. Closure property
2. Commutative property
3. Associative property
4. Identity property

Closure Property
For any two whole numbers, if the minuend is lesser than that of subtrahend, then the difference obtained will be an integer. Only if the minuend is greater than that of subtrahend, then the difference obtained will be a whole number.

For example, 7 – 2 = 5; 7, 2 and 5 all are whole numbers. Contrarily, 4 – 9 = – 5; 4 and 9 are whole numbers, but –5 is an integer.

Commutative Property
If the order of minuend and subtrahend are interchanged, the result obtained will be different. Subtraction of numbers does not satisfy the commutative property.

For example, 6 – 2 = 4, but 2 – 6 = –4. Therefore, 6 – 2 ≠ 2 – 6.

Associative Property
If the order of operating the numbers is changed, then the result obtained will be different. Subtraction of numbers does not satisfy the associative property.

For example, 15 – (7 – 3) = 11, but (15 – 7) – 3 = 6. Therefore, 15 – (7 – 3) ≠ (15 – 7) – 3.

Identity Property
The difference between any number and zero is always the number itself.

For example, 6 – 0 = 6

## Subtraction by Regrouping

Subtraction by regrouping is a process of finding the difference of numbers that have more than one digit by arranging them in their respective place values. As we are familiar with addition, subtraction also follows the same procedure of vertically subtracting digits in their place values. First, we subtract the digits at ones place, followed by the digits in tens place and so on.

Suppose the digit in the minuend is smaller than the digit in the subtrahend, borrow a number from the digit in the neighboring place value and increase that value of smaller digit by 10. Let us see examples to understand the process of finding the difference.

Example 5: Find the difference between 73 and 41 by regrouping.

Solution: To find the difference, first we write the given numbers using place values as follows:

73 = 7 tens + 3 ones
41 = 4 tens + 1 ones

The numbers can also be arranged in a column as shown below.

Subtract the numbers in the place values and write the difference below them. The difference of digits in ones place is 3 – 1 = 2; the difference of digits in tens place is 7 – 4 = 3.

Example 6: Find the difference between 492 and 86 by regrouping.

Solution: The numbers can be written in place values as shown:
492 = 4 hundreds + 9 tens + 2 ones
86 = 8 tens + 6 ones

Subtract the numbers in the place values and write the difference below them. When the digit in the minuend is lesser than the digit in the subtrahend, we borrow 1 from the next highest place value and increase the value of the smaller digit by 10. Thus, 9 tens is reduced by 1 tens becomes 8 tens and 2 is increased by 10 becomes 12 and 12 – 6 = 6. The process continues in the same order, we get

To subtract higher value numbers the same procedure can be followed. Let us solve an example of higher value numbers.

Example 7: Write the numbers in their place values and subtract 51,490 from 65,812.

Solution: In the given numbers, 65812 is a minuend and 51490 is a subtrahend. The numbers can be written in their place value as shown.

Subtract the numbers in the place values and write the difference below them.

Therefore, 65812 – 51490 = 14322.

Subtraction with Stories
In a real-life, subtraction plays a tremendous role in money transactions – buying and selling goods, debit and credit card balance, and many more.

Follow the steps to solve word problems.

Step 1: Read the problem carefully.
Step 2: Think about what is happening.
Step 3: Decide how to solve the problem and solve it.

Let us solve some of our real-life applications of subtraction.

Example 8: There were 585 students in class VI of a school. Out of them 498 passed and were promoted to the next class. How many students did not get promoted?

Solution: Total number of students in a class VI = 585
Number of students passed and were promoted to the next class = 498

To get the number of students who did not get promoted, subtract 498 from 585.

585 – 498 = 87

Therefore, 87 students did not get promoted to the next class.

Example 9: A cement merchant had 9876 bags of cement in his store. He sold 3200 bags of cement on Monday and 2179 bags of cement on Tuesday. How many bags are left unsold in the store?

Solution: Total number of cement bags in a store = 9876
Number of cement bags sold on Monday = 3200
Number of cement bags sold on Tuesday = 2179

To find the number of cement bags unsold, subtract the total number of bags sold from the total number of cement bags there were in the store.

Therefore, there are 4497 bags of cement unsold in the store.

## Math Facts

• In vedic sutras, the numbers from 1 to 10 are paired to form 10 and called them as pooraks. For example, 1 and 9 add up to 10; 2 and 8 add up to 10, and so on. This can be observed in the image shown below.

The same ten point circle can be used to show the subtraction pairs of numbers. Observe the image below.

## Mathemagician

Srinivasa Ramanujan, the mathematical genius, is recognised for his extraordinary contributions to mathematics. Srinivasa Ramanujan (1887-1920), who died at the young age of 32, made significant contributions to mathematics that only a few people could match in their lifetime.