Subtraction by Steps

In last week’s issue of our MATH-Inic Newsletter, we discussed how the Sutra, All from 9 and the Last from 10, can be used to in a general method of subtraction which eliminates “borrowing”. We will now discuss the first of 3 methods found in Chapter 3 – “Subtraction without Borrowing”  of our book, 25 Math Short Cuts (pp. 7 – 10).

“Imagine subtraction to be going down from a higher number (minuend) to a lower number (subtrahend). The distance between them is their difference. On the way down, we can make a stopover on an intermediate value. The difference then is the sum of the distance from the higher number and the stopover and the distance from the stopover to the lower number.”

This method can be understood better if we will take a real-life example: if you went to a convenience store with P435 and then purchased goods worth P397. How much money will you have left?

Of course, the conventional paper and pencil solution is to place the minuend on top of the subtrahend and perform subtraction column by column from the right:

                        P 4 3 5

              -3 9 7

This would entail two regroupings or borrowings.

But we can compute this mentally using subtraction by steps:

1. First go down from P435 to a suitable number midway like P400; deducting P400 from P435 to get P35 should be very easy even for very young kids.
2. Next getting P397 from P400 is even easier, leaving P3
3. Finally add P35 and P3 to get P38, the amount left.

Some may complain that this procedure will only complicate computations, but this happens every day. When told that your total purchase is P397, you would only give the cashier P400 leaving the P35 with you. Then you add the P3 change she will give you to the P35 you still hold to get P38.

This technique is very useful when the numbers involved are near and in the opposite sides of a nice round number.

Let’s look at this question which was given in the 2^nd Philippine National Vedic Mathematics Olympiad in 2021: 4321 – 3877= ?

Using the conventional right to left method of subtraction would require 3 regroupings. But by using 4000 as our “stopover” and employing All from 9 and the Last from 10, we can easily get 321 + 123 = 444.

Let us practice using subtraction by steps:

1. 513 – 487 =
2. 726 – 668 =
3. 734 – 277 =
4. 842 – 594 =
5. 2222 – 1879 =
6. 3124 – 1666 =
7. 5432 – 3778 =
8. 20204 – 19678 =
9. 23232 – 18598 =
10. 747353 – 698765 =

Answers to last issue’s exercises:

1. 632 – 345 = 632 + 655 – 1000 = 1287 – 1000= 287
2. 745 – 268 = 745 + 732 – 1000 = 1477 – 1000 = 477
3. 813 – 444 = 813 + 556 – 1000 = 1369 – 1000 = 369
4. 5232 – 3475 = 5232 + 6525 – 10000 = 11757 – 10000 = 1757
5. 7524 – 3667 = 7524 + 6333 – 10000 = 13857 – 10000 = 3857
6. 8413 – 5578 = 8413 + 4422 – 10000 = 12835 – 10000 = 2835
7. 43,264 – 24,789 = 43,264 + 75,211 – 100,000 = 118,475 -100,000 = 18, 475
8. 75,243 – 36,564 = 75,243 + 63,436 – 100,000 = 138,679 – 100,000 – 38,679
9. 405,432 – 236,567 = 405,432 + 763,433 – 1,000,000 = 1,168,865 – 1,000,000 = 168,865
10. 823,526 – 545,687 = 823,526 + 454,313 – 1,000,000 = 1,277,839 – 1,000,000 = 277,839