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Thought-Process to Discover Knowledge

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mathsWhole NumbersWhole numbers addition and subtraction

### Subtraction: Simplified Procedure for Large Numbers

This page extends the subtraction in first principle into a simplified procedure for subtraction of large numbers, which is called as subtraction by place value with de-grouping.

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Consider the subtraction 15-12. From 15, 12 is taken away. The remaining quantity is counted.

Considering the subtraction 15 - 12. Taking away 12 from 15 is shown in the figure. The dotted lines cross the equal amount from the numbers 12 and 15. The remaining quantity is shown in the dotted ellipse. The count of remaining amount is 3.

Consider the subtraction 23 - 8. The quantities are shown in the figure. Taking away 8, from 23 is to be done. The 3 units in 23 is smaller than the 8 units. It is noted that, the tens bar is made of 10 units, so that can be split into units.

Considering the subtraction 23-8. The subtraction is illustrated in the figure. Note that 1 ten is broken-down into 10 units.

The subtrahend 8 is taken away from the 23 and the remaining is 1 ten and 5 units. The difference is 15.

A tens block is split into 10 units blocks. This is referred as de-grouping. The grouped place-value is degrouped into lower place-value.

De-grouping: When handling grouped numbers, at a place-value, 1 of a higher place-value can be de-grouped into 10 of the next lower place-value.

Considering the subtraction 23-8. A simplified procedure, Subtraction by Place-value with de-grouping is shown in the figure.

23-8 is considered.

In the units place minuend 3 is smaller than subtrahend 8. So, one ten is broken into 10 units. The units place subtraction is 13-8 = 5.

In the tens place, the subtraction is with remaining 1 ten in minuend as 1-0 =0.

The difference is calculated as 15.

Consider the subtraction of 274-132. We can quickly carryout the subtraction by the quantities in the figure. The difference is 142.

Consider the subtraction 274-132. The numbers are given in place value in the figure. The difference is to be calculated.

Considering the subtraction 274-132. A simplified procedure, Subtraction by Place-value with de-grouping is given in the figure.

The digits in the units place are subtracted,
Then the digits in the tens place are subtracted.
Then the digits in the hundreds place are subtracted.

The difference is 142.

Consider the subtraction 156-11. The answer is not 46. The correct answer is 145. Note that the units are subtracted from units, and the tens are subtracted from the tens.

Consider the subtraction 156-88. The simplified procedure, subtraction by place-value with de-grouping, is given in the figure.

The digits in units place are subtracted. Since 6 minuend is smaller than 8 subtrahend, the one ten from tens place is borrowed as 10 units into units place. The borrowed 10 units and the units in the minuend 6 together make 16.

Subtraction at units place is 16-8=8.

The digits in the tens place are subtracted. Since one ten was borrowed into the units place, the tens place of minuend is revised to 4. The 4 is less than the subtrahend 8. So, one hundred from hundreds place is borrowed as 10 tens. The borrowed 10 tens and the 4 tens in the minuend together make 14.

Subtraction at the tens place is 14-8=6.

Subtraction in the hundreds place is 0-0=0.

The difference is 68.

Consider the subtraction 675-396. The simplified procedure is given in the figure. Note the subtraction of place values and the borrow are highlighted in color. The difference is 279.

Subtraction by Place-value with de-grouping -- Simplified Procedure : Two numbers are subtracted as follows:

•  the place-value positions are arranged units under units, 10s under 10s, etc.

•  the units are subtracted and if the minuend digit is smaller than the subtrahend digit, then one ten's place is borrowed to make the minuend at the position larger. Then subtraction is carried out.

•  continue to the higher place-value position. Note: The "borrow" is the simplification of taking 1 from a place value to 10 pieces of a lower place value.

Solved Exercise Problem:

What is the difference 2332-1297

• 1145
• 1035
• 1035

The answer is "1035"

Solved Exercise Problem:

What is the difference 122-99

• 23
• 23
• 33

The answer is "23"

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