Subtraction With Regrouping

What is Subtraction?

Subtractions is the method of removing something from other. In mathematics, it is the difference between two numbers.

If we say x – y or

then x is called ‘minuend’ and y is called ‘subtrahend’. Their outcome/result is called ‘difference’.

Minuend – subtrahend = difference

Subtraction of Same Numbers

Subtraction of same numbers is always zero. In this case minuend and subtrahend are the same.

Regrouping in Subtraction

Regrouping is to borrow a one from tens place to one’s place or from hundreds place to tens place. We do regroup in mathematics only when the minuend is less than the subtrahend.

Regrouping means to split one ten so that we have ten ones. For example, in subtraction, if we have to subtract 12 ones from 21. In one’s place we must subtract 2 from 1. It does not seem correct. So, what we do is we divide one ten into ten number of ones. So now we have eleven ones, and we can subtract 2 ones from it.

Explanation

To Subtract 17 apples from 45 apples

To solve this problem, we will use two steps.

Step – ONE

In this example, we have sack which has 10 apples, and we have total 4 sacks. Further, we also have 5 apples which are not in the sack. Now, if we must give 7 apples from the apples which are not in the sack. Simply we cannot.

Step – TWO

We must open one of the sacks so that we have fifteen apples, from which we can give seven apples to someone. We will be left with eight apples. So, we will write 8 beneath the one’s column.

The condition is that we must give one sack of apple also to the person. But since we already have open one sack, so now we have total 3 sacks and if we give one sack to the person. We will be left with 2 sacks. So, we will write 2 beneath the ten’s column.

Use of Subtraction with regrouping

Subtraction with regrouping is useful in everyday problems such as.

  • To count money
  • To count fruits such as bananas, apples, guavas, etc.
  • To measure the covered distance and total distance
  • To measure the remaining time and total time
  • To measure obtained marks by the student.

Symbol

The symbol used for subtraction is – (minus). To regroup a ten to ones or a hundred to tens we use apply the slant, slash (/ ) over that number.

Consistent Unit Rule

One basic rule in subtraction is same unit. To subtract an amount/quantity from another quantity, they both must have same units.

We cannot subtract 7 pencils from 10 markers or 5 apples from 12 mangoes. The units must be consistent/homogenous i.e., we always subtract apples from apples, pencils from pencils, and markers from markers.

Properties of Subtraction

Anti-Commutative

Subtraction with regrouping (or any other method) is anti-commutative that is, a – b ≠ b – a

Minuend – subtrahend ≠ subtrahend – minuend
Swapping the minuend and subtrahend will result in the same answer but the sign will be change.

5 – 3 = 2

3 – 5 = – 2

Thus (a – b) = – (b – a)

Therefore, we must pay attention to the problem to know which number is subtracting from which one.

Non – Associative

Subtraction of numbers is not associative unlike addition. Here, a – (b – c) ≠ (a – b) – c. Unlike the above property, even the answers are different.

(4 – 2) – 1 = 2 – 1 = 1

4 – (2 – 1) = 4 – 1 = 3

So, law of association with respect to subtraction is not possible.

Techniques and Tips

Cross the line – Trick

In subtraction, if your minuend is less than subtrahend, then you must regroup. After doing regrouping, we must subtract the subtrahend from minuend. To do so, we draw straight lines as much as number of minuends. Then cross the lines equal to number of subtrahends. The remaining straight line is the answer.

Example

Suppose if we have to subtract 18 pencils from 25 pencils. In this case.

  • 25 pencils – Minuend
  • 18 pencils – Subtrahend

Count your fingers – Trick

If you are subtracting with regrouping, your minuend must be less than subtrahend. Once, you break your ten into ones then instead of drawing the lines and crossing them you can use your fingers.

Example

If we have to subtract 18 cars from 20 cars. Then in one’s column, 0 – 8 does not make any sense, so we must regroup one of the two tens such the 0 now becomes 10. Now 10 – 8 = 2.

You can count it using your hands. Count fingers of your both hands, 1,2,3,4,5,6,7,8,9, and 10 and slowly raise each finger while counting.

Next, we must subtract eight from it. So, start folding your fingers 1,2,3,4,5,6,7,8 and stop.

Now, count the number of raised fingers; 1 and 2. So, 10 – 8 is 2. We must write 2 beneath the one’s column.

In Tens column, it is now 1, so 1 – 1 is zero because minuend and subtrahend are same. We can write 0 in tens column, or we can neglect it since it is insignificant.

Other Methods of Subtraction

In this article, we are studying about subtraction with regrouping. However, there are other methods as well. These methods are used in elementary school to teach children basic subtraction. They vary from region to region.

Austrian Method

There is a method called Austrian method of subtraction. In this method subtraction is done by using addition, therefore this method is also called addition method. There is no regrouping or borrowing in this method. It is widely used in schools of Europe.

Subtracting 491 from 753 using Austrian method is shown as.

1 + ___ = 3

9 + ___ = 5

Since, the sum 5 is small. We will add 1 in the next place subtrahend.

9 + ___ = 15

(1+4) + ___ = 7

Trade First

It is just like the borrowing or regrouping method which we are discussing in this article. However, all the borrowing or regrouping also called trade is done before subtraction.
Every time a trade occurs when the minuend is less than the subtrahend.

For example, subtracting 383 from 852 using trade first method.

In this case 852 is minuend and 383 is subtrahend. Now, in one’s place, 2-3, minuend is less than subtrahend so we will trade, and it becomes 12-3. Next, in tens place, 4-2, again minuend is less than subtrahend, so we will trade again. Thus, it become 14-8, which is feasible. In hundreds place, it becomes 7-3.

After Trading we can easily solve this as.

So, the subtraction of 852 and 383 using trade first method gives us a difference of 469.

Partial Differences

The method of partial difference is very much different to that of regrouping or borrowing method. However, it is very much popular, and we can discuss it in methods of subtraction.

In this method, if minuend is greater than subtrahend, subtraction is carried out and a plus sign is written with the difference. However, if minuend is lesser than the subtrahend then instead of subtracting subtrahend from minuend, minuend is subtracted from the subtrahend and a minus sign is place with the difference or result.

Remember, this method is always from left to right.

For example, we have to subtract 391 from 852. Now using the partial differences method of subtraction, we can solve it as,

  • 852 is minuend
  • 391 is subtrahend.

8 – 3 = +5

5 – 9 = – 4 

2 – 1 = + 1

Then just algebraically add/subtract the results to get the final difference.

+ 500 – 40 + 1 = + 461

Counting – Method

In this method, we find the difference by counting the numbers between two digits that are to be subtracted.

For example, 1561 – 263. The difference between minuend and subtrahend can be found using following method.

  • 263 + 7 = 270
  • 270 + 30 = 300
  • 300 + 700 = 1000
  • 1000 + 500 = 1500
  • 1500 + 61 = 1561

Adding the numbers 7, 30, 700, 500, and 61 will give us the difference between 1561 and 263.

7 + 30 + 700 + 500 + 61 = 1298

Breaking up – Method

It is another method by which we can find the difference between two numbers.

For example, 1561 – 963

  • 1561 – 561 = 1000
  • 1000 – 30 = 970
  • 970 – 7 = 963

The difference between minuend (1561) and subtrahend (963) can be found as,

561 + 30 + 7 = 598

Example No. 1

In a cricket test match between strikers and gladiators. Strikers batted first scoring 351 runs. In second innings, the gladiators in response managed to score only 322 runs. By how many runs, team strikers are leading from the gladiators?

Solution

Strikers Batting first – 351 – minuend Gladiators Batting second – 322 – subtrahend

So, team strikers lead by 29 runs.

Example No. 2

Plantation was carried out in a garden where 671 trees were planted. Among them 122 trees were of fig, 231 were apple trees. The remaining trees were that of peach. Calculate the number of peach trees planted in the garden.

Solution

In this example we are dealing with three digits subtraction.

  • 671 total trees – minuend
  • 122 fig trees – subtrahend
  • 231 apple trees – subtrahend

Step 1: In the one’s column, we have 1 – 2 – 1. This can also be written as 1 – (2 + 1) = 1 – 3. Since minuend is less, we must regroup and borrow 1 from tens making it eleven. Now 11 – 3 = 8.

Step 2: Since we have already regroup/ borrowed the tens place, we now have only 6 as the minuend. Then we can write the subtraction for the tens place as 6 – 2 – 3 = 6 – (2 + 3) or simply 6 – 5 which is equal to 1.

Step 3: In the hundreds place we have 6 as the minuend, 1 and 2 are subtrahend. We add the subtrahend to get one subtrahend i.e., 1 + 2 = 3. Now basic subtraction is 6 – 3 which is equal to 3. We have 3 in the hundreds place, 1 in the tens place, and 8 in the units/ones place.

The number of peach trees planted in the garden are 318.

Example No. 3

Elena customized a laptop for 3841 USD. She was charged 1121 USD for SSD modifications and another 1990 USD for graphics card and RAM modifications. What was the actual price of laptop before modifications?

Solution

We are dealing with 4 columns of subtraction. An additional column is that of thousands. We now have unit or ones, tens, hundreds, and thousands from right to left.

In our case the laptop was bought for 3841 so it is the minuend, whereas 1121 (SSD modifications) and 1990 (graphics modifications) are subtrahend.

Step 1: In the ones column we have minuend and subtrahend the same i.e., 1 – 1. So, we will write 0 beneath one’s place.

Step 2: In the ten’s column, we have 4 as minuend and 2 + 9 = 11 as the subtrahend. Hence, minuend is less than the subtrahend. So, we will regroup and borrow. Now we have fourteen as the minuend and subtraction becomes 14 – 11 and the difference is 3.

Step 3: In hundreds column, we have 7 now, after giving one to tens column. Since minuend is 7 and subtrahend is 1 + 9 = 10. Again, minuend is less than subtrahend and we will regroup. Hence, minuend becomes seventeen and subtraction can be stated as, 17 – 10 = 7. So, seven is the difference.

Step 4: In thousands place, we have minuend 2 after regrouping. Subtrahend on the other hand is 1 + 1 = 2. The subtraction is 2 – 2 = 0. Hence, we may write 0 in the thousand’s column, or we can ignore it since it is less significant.  

Example No. 4

John played for England men’s cricket team from 1995-2018. He has scored 5678 runs throughout his career playing in test cricket, One-Day Internationals, and T-20 Internationals. His score in One Day Internationals was 1521, whereas 3377 runs in test match cricket. Calculate how many runs john has scored in T-20 internationals?

Solution

In this example, we will be dealing again with 4 columns of subtraction, and we have 3 digits subtraction in which one is minuend and two are subtrahend.

  • 5678 total runs – Minuend
  • 1521 One Day International Runs – Subtrahend
  • 3377 – Test Match Runs – Subtrahend

Step 1: In the one’s column, we have 8 as minuend and 7 + 1 = 8 as the subtrahend. Since, both are same, we will get zero and write 0 in the column of ones.

Step 2: In the ten’s column, minuend is 7, subtrahend is 2 + 7 = 9 which is greater than the minuend. Hence, we must regroup and borrow one from hundreds place. Now the minuend becomes seventeen, and the subtraction is 17 – 9 = 8. The difference is eight.

Step 3: In the hundred’s column, we are now left with 5 which is the minuend. Subtrahend is 5 + 3 = 8 which is greater than the minuend. We will again borrow one from thousands place. The minuend becomes fifteen and the subtraction is like 15 – 8 = 7. Difference is seven.

Step 4: In the thousands column we are left with 4 as minuend. 1 + 3 = 4 is subtrahend. Hence, the subtraction is 4 – 4 = 0. The difference is 0 which can be neglected since it is not significant.

Example No. 5

Adam was inducted in the armed forces in the year 1975. He got retired in the year 2010. How many years has Adam served the armed forces?

Solution

It is a four-column subtraction of two digits. We are to calculate the services (in years) of Adam for the Armed forces.

  • 2010 – year of retirement – Minuend
  • 1975 – year of induction – Subtrahend

This is because, the overall minuend must be greater than subtrahend to have a positive difference.

Step 1: In the unit’s column, minuend is 0 and subtrahend is 5. Hence, minuend is less than subtrahend. We regroup and borrow 1 from tens column. The minuend becomes 10 now. Now we can have the difference of unit’s column as 10 – 5 as 5.

Step 2: In the ten’s column, now we have 0, because we regroup it in previous step. 0 is the minuend and 7 is subtrahend. Again, minuend is less than subtrahend and for positive difference we must regroup and borrow. But wait! The hundred’s place has 0. So now we will borrow 1 from thousand’s place to hundred’s place so that it becomes 10. Further, we borrow 1 from hundred’s place to make the ten’s place minuend as 10. Now 10 – 7 and we have a difference of 7.

Step 3: In the hundred’s column we have, 9 as the minuend because of regrouping and borrowing. Subtrahend is also 9. The difference is zero. We write it beneath the hundred’s column.

Step 4: In the thousand’s column, we have same minuend and subtrahend that is 1. The difference is 0. We can write it beneath the column, or we can ignore it since there is no significant digit (non-zero) to the left of it.

Example No. 6

Ali bought a phone for 700 USD. After using it for almost five years, he decided to sell it. He lost an amount of 379 USD on selling his old phone. Calculate the price on which he sold his old phone?

Solution

It is a three-column subtraction of two digits. We must calculate the amount Ali received on selling his old phone.

  • 700 USD – Original price of mobile phone – Minuend
  • 379 USD – Lost amount – Subtrahend

Step 1: In the unit’s column, we have 0 as the minuend and 9 as the subtrahend. The minuend is less than subtrahend so we must regroup and borrow. We will regroup the hundred’s place so that the ten’s place becomes 10 and again regroup the ten’s place so that the minuend of unit’s place becomes 10. Then the difference is 10 – 9 = 1. We will write 1 beneath the one’s column.

Step 2: In the ten’s column, after regrouping we have 9 which is the minuend. Subtrahend is 7 so the difference is 9 – 7 = 2. We will write 2 beneath the ten’s column.

Step 3: In the ten’s column, after regrouping, we have 6 as the minuend, whereas 3 is subtrahend. The difference is 6 – 3 = 3 so we will write 3 beneath the hundred’s column.

Worksheets

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Subtraction Games

You will find some subtraction games here alongside some games on addition as well.