Introduction to Subtraction

Subtraction is one of the four basic operations in arithmetic, helping us find out how much one quantity is less than another. Simply put, it is the process of taking away or finding the difference between two numbers.

Understanding subtraction is crucial in daily life and in competitive exams where swift and accurate calculations are needed. Whether it's calculating how much money remains after spending or determining the distance left to travel, subtraction is everywhere.

Let's begin by learning some important terms used in subtraction:

• Minuend: The number from which another number is to be subtracted.
• Subtrahend: The number that is to be subtracted from the minuend.
• Difference: The result obtained after subtraction.

For example, in the subtraction \( 95 - 37 \), 95 is the minuend, 37 is the subtrahend, and the difference is what we find by subtracting 37 from 95.

In mathematical form, subtraction is expressed as:

where \( D \) is the difference, \( M \) is the minuend, and \( S \) is the subtrahend.

Subtraction plays an important role especially in contexts involving metric units like meters and kilograms, and in financial calculations involving Indian Rupees (INR) and paise. We will explore these applications in this section.

Basic Subtraction Without Borrowing

When subtracting numbers digit by digit, if every digit of the minuend is greater than or equal to the corresponding digit of the subtrahend, subtraction becomes straightforward. This means no digit borrowing (or regrouping) is needed.

Remember, subtraction needs to happen place by place starting from the units (rightmost digit), then tens, hundreds, and so on. Proper place value alignment is crucial.

Consider this subtraction:

Each digit in 742 is greater than or equal to the corresponding digit in 536:

• Units: 2 - 6 (Here 2 is less than 6, so normally we borrow, but for this example, choose numbers where borrowing isn't needed as we proceed.)

To demonstrate subtraction without borrowing properly, let's pick a different example where all top digits are greater or equal:

Subtraction here proceeds digit by digit:

• Units: 5 - 2 = 3
• Tens: 6 - 3 = 3
• Hundreds: 8 - 4 = 4

The final difference is 433.

Subtraction with Borrowing (Regrouping)

Often, in subtraction, the digit in the minuend at a certain place value is smaller than the digit in the same place of the subtrahend. In such cases, direct subtraction isn't possible, because we cannot subtract a larger digit from a smaller one without going into negative numbers.

This is where the borrowing or regrouping method comes into play. Borrowing means taking 1 from the next higher place value to make the current digit larger by 10 and then performing subtraction.

Let's understand this with an example:

We start from units place:

• Units: 2 (minuend) is less than 7 (subtrahend), so we need to borrow 1 from tens place.
• Tens place has 5, after borrowing 1 (which equals 10 units), tens place becomes 4, and units place becomes \(2 + 10 = 12\).
• Now, \(12 - 7 = 5\).
• Moving to tens place: 4 (after borrowing) - 8. Since 4 is less than 8, borrow 1 from hundreds place.
• Hundreds place 6 becomes 5, tens place becomes \(4 + 10 = 14\).
• Then \(14 - 8 = 6\).
• Finally, hundreds place: 5 - 3 = 2.

So, difference = \(265\).

graph TD    Start[Start subtraction from units place]    CheckUnits{Is digit in minuend ≥ subtrahend?}    BorrowUnits[Borrow 1 from next higher place; add 10 to current digit]    SubtractUnits[Subtract digits at units place]    MoveTens[Move to tens place]    CheckTens{Is digit in minuend (after borrowing) ≥ subtrahend?}    BorrowTens[Borrow 1 from hundreds; add 10 to tens digit]    SubtractTens[Subtract digits at tens place]    MoveHundreds[Move to hundreds place]    SubtractHundreds[Subtract digits at hundreds place]    End[Write down the difference]    Start --> CheckUnits    CheckUnits -- No --> BorrowUnits --> SubtractUnits    CheckUnits -- Yes --> SubtractUnits    SubtractUnits --> MoveTens    MoveTens --> CheckTens    CheckTens -- No --> BorrowTens --> SubtractTens    CheckTens -- Yes --> SubtractTens    SubtractTens --> MoveHundreds    MoveHundreds --> SubtractHundreds    SubtractHundreds --> End

Subtraction Involving Decimals

Subtraction works the same way with decimal numbers, but there is a vital rule: Align the decimal points vertically before subtracting to ensure digits in the same place value (tenths, hundredths, etc.) are subtracted correctly.

If necessary, add zeros to make the decimal parts of both numbers have the same length.

For example, subtract:

Write as:

  45.80- 23.75

Now subtract as whole numbers digit by digit, borrowing where needed, then place the decimal point in the difference in the same vertical line.

Worked Examples