Doubling: Numbers with Big Digits.

Last week we showed how to double numbers when all the digits are small (0, 1, 2, 3, and 4). This week we will explain doubling when numbers contain big digits ( 5, 6, 7, 8 and 9).

As we have written in our book 25 Math Short Cuts we can also double each digit starting from the left but we need to anticipate regrouping. Before writing down or announcing the result, we look at the next digit of the multiplicand. If it is “big”, add 1 to the partial result. Then double the big digit and write only the last digit of the result.

For example, if we want to double 27 using the traditional right to left method, we start with 7 x 2 = 14. Then we write down 4 and “carry” 1. Next we double 2 to get 4 and add the carry figure 1 to get 5. So the answer is 54. 

If we do it from the left to right we start with  2 x 2 = 4 and then upon seeing that the next digit, 7,  is big, we immediately add 1 to 4 and announce “fifty…”. Now 7 x 2 is 14 but since we have earlier performed the carry operation, we only attach the last digit of the product, “four”.

This can be illustrated as

27 x 2 = 4 | 14 = 4+1 | 4 = 5|4 = 54

For 469 we have

469 x 2 = 8 | 12 | 18 = 8 + 1| 2 + 1| 8 = 938

For our featured example 264, 368 x 2, upon reading the multiplicand as “two hundred sixty-four thousand…” we can start doubling the number from the left and recite the answer knowing that it will be in the hundred thousands:

1. 2 times 2 is 4 but the next digit is big so we add 1 to get 5, “five hundred…”    
2. 6 times 2 is 12 but we have carried “1” so only 2 remains and the next digit, 4 is small so we continue, “twenty…”
3. 4 x 2 is 8 and the next digit is “small”, so it remains “eight thousand, …”
4. 3 x 2 is 6 but the next digit is “big”, so we anticipate the regrouping and we have “seven hundred…”  
5. 6 x 2 = is 12 but we have done the carry operation and the last digit is “big”, so we add 1 to get “thirty…”
6. Finally, we double the last digit 8 to get 16, and mention only the last digit, “six”.

As we have written in our previous post, doubling is a very important skill in Arithmetic. It is useful also in mentally multiplying by 4 and 8, dividing by 5, 25 and 125 and in factoring and expanding our multiplication table. Further explanation of doubling can be found in chapter 5 of our book “25 Math Short Cuts”.

 Practice by doubling these numbers:

1. 39
2. 68
3. 348
4. 459
5. 2, 436
6. 4, 829
7. 26, 467
8. 63, 476
9. 385, 498
10. 754, 849

Here are the answers to last week’s practice exercises:

1. 123 = 246
2. 402 = 804
3. 4, 123 = 8, 246
4. 4, 432 = 8, 864
5. 33, 243 = 66, 486
6. 21, 342 = 42, 684
7. 421, 231 = 842, 462
8. 342, 221 = 684, 442
9. 232,323 = 464, 646
10. 333,444 = 666, 888