 # VMO J-3: Squaring numbers near 50.

Numbers near 50 can also be squared easily. This time, however, we add the excess or subtract the deficiency from 25 and not from the number. The second part is still equal to the square of the excess or deficiency.

To square a number just above 50, add the excess of the number from 50 to 25 and set-up the square of the excess.

542 = 25 + 4 | 42 = 29 | 16 = 2916

To square a number just below 50, deduct the deficiency of the number from 50 from 25 and set-up the square of the deficiency

472 = 25 – 3 | 32 = 22 | 09 = 2209

The second part must occupy two places so 9 is written as 09.

Proof:

(50 ± a)2        = 502 ± 2(50)a + a2

= 2500 ± 100a + a2

= (25 ± a) 100 + a2

To square numbers near 500, just add(deduct) the number’s excess(deficiency) from 500 to 250 to get the first part of the answer. The second part, the sqjuare of the excess(deficiency) must occupy the last three places of the answer.

Examples:

5042 = 250 + 4 | 42 = 254 | 016 = 254, 016

5122 = 250 + 12 | 122 = 262 | 144 = 262,144

4862 = 250 – 14 | 142 = 236 | 196 = 236,196

5352= 250 + 35 | 352 = 285 | 1225 = 286,225

In the last example, the second part has 4 digits so the 1 must be carried to the next column on the left.

Exercise J-3: Find the square of the following numbers:

1. 51; 2) 57; 3) 61; 4) 49; 5) 42; 6) 39; 7) 506; 8) 521; 9) 494; 10) 465