Multiplying by 11

Although the Philippines uses the International System of Units(SI) many Filipinos prefer to express their heights in feet and inches ang their weight in pounds. But most weighing scales are in kilograms. So how can we quickly convert the kilogram reading into pounds?

The exact conversion is 1 kg. = 2.2046226218 but using 1 kg = 2.2 pound is 99.79% accurate. Now if we use 1 kg = 2.0 we can easily convert any kilogram by menatlly by using a topic which is discussed in Chapter 4 of 25 Math Short Cuts and also Chapter 4 of Algebra Made Easy as Arithmetic – multiplying by 11. Note that 2.2 = 2 x 1.1 or 0.2 x 11.

To multiply any two-digit number by 11, simply split the two digits and insert their sum in between them:

35 x 11 = 3 |(3 + 5)| 5 = 385.

A similar procedure can be applied to longer multiplicands:

32,453 x 11 = 3 | 3 + 2 | 2 + 4 | 4 + 5 | 5 + 3 | 3 = 356,983

If the sum of the adjacent digits is 10 or more, we will introduce a slight modification to our technique:

47 x 11 = 4 | 4 + 7 | 7 = 4 | 11 | 7 = 4 +1 | 1| 7 = 517

To check the result of multiplication by 11, we use the alternating sum check which is actually the divisibility rule for 11. If the result is zero or a multiple of 11, then the number is divisible by 11. This is done in two ways:

1. Alternately add and subtract the digits of the number. Let us check the results of our examples:

35 x 11 = 385 → +3 – 8 + 5 = 0  

                        32,453 x 11 = 356,983 → +3 – 5 + 6 – 9 + 8 – 3 = 0

                        47 x 11 = 517 = +5 – 1 + 7 = 11

• Take the difference of the sums of alternating digits:

35 x 11 = 385 → (3 + 5) – 8 = 0  

                        32,453 x 11 = 356,983 → (3 + 6 + 8) – (5 + 9 + 3) = 17 – 17 = 0

                        47 x 11 = 517 = (5 + 7) – 1 = 11

Now what is 35 kilos in pounds?

We can do it in a number of ways:

1. 35 x 0.2 x 11 = 7 x 11 = 77
2. 35 x 11 x 0.2 = 385 x 0.2 = 77
3. 35 x 2 x 11 x 0.1 = 70 x 11 x 0.1 = 770 x 0.1 = 77

How about 48Kg?

1. 48 x 1.1 x 2 = 52.8 x 2 = 105.6

Multiply the following numbers by 11 and check your answers:

1. 53
2. 72
3. 57
4. 85
5. 352
6. 462
7. 5216
8. 7429
9. 81354
10. 87659

Answers to last week’s exercises:

1. 2.22 – 0.222 = 1.998
2. 9.99 – 0.999 = 8.991
3. 13.33 – 3.333 = 9.997
4. 5432 – 2435 = 2997
5. 7884 – 5889 = 1995
6. 6777 – 4888 = 1889
7. 87878 – 77879 = 9999
8. 78756 – 59768 = 18988
9. 345678 – 135579 = 210099
10. 578987 – 278999 = 299988