“When the Final digits add up to 10” extensions

In our previous posts we discussed short methods of multiplying number pairs having identical starting digits and their “final digits add up to 10”. We first applied it to squaring numbers ending in 5 (5 + 5 = 10) and then we used it in multiplying pairs such as 74 x 76 and 62 x 68 (4 + 6 = 2 + 8 = 10).

We can extend this technique of using the Sutra, “By one More Than The One Before” whenever the final figures make up a “whole”. This “whole” can be a power of ten or even a whole foot!

It is easy to see that since 74 x 76 = 5624, then 7.4 x 7.6 = 56.24. We note here that 0.4 + 0.6 = 1.0.  

Less obvious is its application to 796 x 704. While it is true that 6 + 4 = 10, 79 is not equal to 70. However, we can consider the last two digits, 96 and 4, will add up to 100 and the initial digit is 7. So 796 x 704 = 7 x 8 | 96 x 4 = 56 | 0384 = 560,384.

We can also apply the procedure in finding the area of a rectangular room which measures 8’ 5” wide by 8’ 7 ”long. Since 5” + 7” will complete a foot, we have A = 8’ x 9’ | 5” x 7” = 72 sq. ft and 35 sq. inches.

The most time saving application is in the multiplication of mixed numbers like our example: 11 4/13 x 11 9/13.

Normally we have to convert the mixed numbers into improper fractions first, then use the algorithm for multiplication of fractions and convert the product back in the form of mixed number.

We can easily convert the multiplicands to improper fractions as 147/13 x 152/13. But, multiplying 147 by 152 and later dividing the product by 169 will certainly tempt us to find a calculator. And then again the answer we will get will have a decimal component not a mixed number.

A closer look at the multiplicands will show that their whole number part is identical while their fractional parts 4/13 and 9/13 add up to 1. Applying “By One More Than The One Before” 11 4/13 x 11 9/13 = 11 x 12 | 4/13 x 9/13 = 132 36/169.

Try these:

1. 9.2 x 9.8 =
2. 9 1/5 x 9 4/5 =
3. 4’ 3” x 4’ 9” =
4. 298 x 202 =
5. 12.3 x 12.7 =
6. 8 2/9 x 8 7/9 =
7.  6’ 1”x 6’ 11” =
8. 592 x 598 =
9. 1294 x 1206 =
10. 1989 x 1911

Answers to previous exercises:

1. 22 x 28 = 616
2. 49 x 41 =2009
3. 53 x 57 = 3021
4. 82 x 88 = 7216
5. 91 x 99 = 9009
6. 102 x 108 = 11016
7. 194 x 196 = 38024
8. 206 x 204 = 42024
9. 292 x 298 = 87016
10. 404 x 406 =164,024