# 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.**

**54 ^{2} = 25 + 4 | 4^{2 }= 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 **

**47 ^{2} = 25 – 3 | 3^{2} = 22 | 09 = 2209**

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

Proof:

**(50 ****±**** a) ^{2} = 50^{2 }**

**±**

**2(50)a + a**

^{2}** = 2500 ****± 100a + ****a ^{2}**

**= (25** **± a) 100 + ****a ^{2}**

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:

**504 ^{2} = 250 + 4 | 4^{2} = 254 | 016 = 254, 016**

**512 ^{2 }= 250 + 12 | 12^{2} = 262 | 144 = 262,144**

**486 ^{2 }= 250 – 14 | 14^{2} = 236 | 196 = 236,196**

**535 ^{2}= 250 + 35 | 35^{2 }= 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:

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

Answers to exercise P-3

- 108; 2) 10,094; 3) 9,894; 4) 9,996; 5) 9,991; 6) 9,765; 7) 7,725; 8) 9,968;

9) 1,002,892; 10) 1,119,375

From Prudente, Virgilio, *25 Math Short Cuts*, pp 73 -74.