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THE SHORTEST SHORT-CUT
A System of Mental Arithmetic

by Oluf Nielsen

The purpose of this pamphlet is to promote an alert mind. As training for the mind, it is helpful in all studies, not only mathematics.

It does not interfere with the current prescribed courses of study and is no criticism of present instruction. It is a helpful addition to a student's mathematical ability.

If a student will memorize the squares of the numbers between 10 and 20 (only nine!), and read this pamphlet, it will give him the power to multiply large numbers such as 175 x 175.

175 x 175 = (170 x 180) + (5 x 5)
= 30,625
275 x 275 = (270 x 280) + (5 x 5)
= 75,625

The reason for 280 instead of 270 is due to changing the problem to an equation such as:

75 = (70 x 70) + (5 x 70) + (5 x 70) + (5 x 5)
= 4900 + 350 + 350 + 25
= 5625

By adding 10 to the second 70, the two center sections of 350 are eliminated and it becomes:

75 = (70 x 80) + (5 x 5)
= 5625

All numbers ending in 5 can be squared in this manner.

65 = (60 x 70) + (5 x 5)
= 4225

It is fast and easy!

The student will need to memorize the squares of the numbers from 10 to 20.

10 = 10 x 10 = 100
11 = 11 x 11 = 121
12 = 12 x 12 = 144
13 = 13 x 13 = 169
14 = 14 x 14 = 196
15 = 15 x 15 = 225
16 = 16 x 16 = 256
17 = 17 x 17 = 289
18 = 18 x 18 = 324
19 = 19 x 19 = 361
20 = 20 x 20 = 400

How to Find the Square of All Numbers to 100

Examples:

  82 = 80 x 80 + (80 + 82) (2)
= 6400 + 324
= 6724
  81 = 80 x 80 + (80 + 81) (1)
= 6400 + 161
= 6561
Center 80 = 6400
  79 = 80 x 80 - (80 + 79) (1)
= 6400 - 159
= 6241
  78 = 80 x 80 - (80 + 78) (2)
= 6400 - 316
= 6084

Continuing this process:

  77 = 75 x 75 + (75 + 77) (2)
= 5625 + 304
= 5929
  76 = 75 x 75 + (75 + 76) (1)
= 5625 + 151
= 5776
Center 75 = 5625
  74 = 75 x 75 - (75 + 74) (1)
= 5625 - 149
= 5476
  73 = 75 x 75 - (75 + 73) (2)
= 5625 - 296
= 5329

How to Multiply Any Two Numbers in Any Ten Group

For example (60 to 70):

If the units added = 10:

61 x 69 = (6 x 7) (100) + (1 x 9)
= 4200 + 9
= 4209
62 x 68 = (6 x 7) (100) + (2 x 8)
= 4200 + 16
= 4216
63 x 67 = (6 x 7) (100) + (3 x 7)
= 4200 + 21
= 4221
64 x 66 = (6 x 7) (100) + (4 x 6)
= 4200 + 24
= 4224

If the units added do not = 10, adjust to use units which = 10 and add or subtract the difference.

63 x 69 = 61 x 69 + (2) (69)
= 4209 + 138
= 4347
71 x 77 = 73 x 77 - (2) (77)
= 5621 - 154
= 5467

Multiplying by Comparative Measure Called the Use of the Bar

The bar of any number is the nearest ten to it. For example, the bar of 38 is 40 and the bar of 78 is 80.

If both numbers are under their bars, subtract a number which is equal to multiplier plus bar of multiplicand times number under the bar from product of the two bars.

For example, 78 is 2 under its bar of 80.

78 (multiplicand) x 38 (multiplier)
78 x 38 = 40 x 80 - (80 + 38) (2)
= 3200 - 236
= 2964

If both numbers are over their bars, add a number which is equal to multiplier plus bar of multiplicand times number over the bar to product of the two bars.

82 x 42 = 80 x 40 + (42 + 80) (2)
= 3200 + 244
= 3444
79 x 39 = 80 x 40 - (39 + 80) (1)
= 3200 - 119
= 3081
77 x 37 = 80 x 40 - (37 + 80) (3)
= 3200 - 351
= 2849

If the numbers are both over or under the bar but not in equal amounts, adjust to make them equal and add or subtract the difference.

78 x 37 = 77 x 37 + 37
= 2849 + 37
= 2886

If the multiplicand is under the bar and the multiplier is over the bar, then add the difference between the multiplier and the bar of the multiplicand.

78 x 42 = 80 x 40 + (80 - 42) (2)
= 3200 + 76
= 3276

If the multiplicand is over the bar and the multiplier is under the bar, subtract the difference between the multiplier and the bar of the multiplicand.

82 x 38 = 80 x 40 - (80 - 38) (2)
= 3200 - 84
= 3116

A split bar is when one number is over the bar and the other is under.

Examples:

88 x 62 = 90 x 60 + (90 - 62) (2)
= 5400 + 56
= 5456
Split Bar
92 x 58 = 90 x 60 - (90 - 58) (2)
= 5400 - 64
= 5336
Split Bar
148 x 92 = 150 x 90 + (150 - 92) (2)
= 13500 + 116
= 13616
Split Bar
148 x 88 = 150 x 90 - (150 + 88) (2)
= 13500 - 476
= 13024
Both Under
152 x 92 = 150 x 90 + (150 + 92) (2)
= 13500 + 484
= 13984
Both Over

Note: Always select the bar closest to the numbers to be multiplied.

How to Multiply Any Number by 33 1/3 or 25

To multiply by 33 1/3, as a short cut, multiply by 100 and divide by 3.

33 1/3 x 84 = 8400 / 3 = 2800

33 1/3 x 87 = 8700 / 3 = 2900

But if you multiply by 33 only, you subtract 1/100 of the answer or 28 or 29 respectively.

33 x 84 = 8400 / 3 - 28
= 2800 - 28
= 2772
33 x 87 = 8700 / 3 - 29
= 2900 - 29
= 2871
33 x 86 = 8600 / 3 - 28
= 2866 - 28
= 2838

Note: Here remainders of 2 from 8600 / 3 were lost and were not needed.

To multiply by 25, multiply by 100 and divide by 4.

25 x 54 = 5400 / 4 = 1350

Mulitiplication by Formula

In high school algebra:

(4x) (2x) = 8x

In college algebra:

The answer equals half the sum squared minus half the difference squared.

(4x) (2x) = (6x / 2) - (2x / 2)
= 9x - x
= 8x
93 x 57 = (150 / 2) - (36 / 2)
= 75 - 18
= 5625 - 324
= 5301
766 x 534 = (1300 / 2) - (232 / 2)
= 650 - 116
= 422,500 - 13,456
= 409,044
75,500 = (75 x 76 x 1,000,000) + (5 x 5 x 10,000)
= 5,700,000,000 + 250,000
= 5,700,250,000
75,500 x 80,500 = (156,000 / 2) - (5,000 / 2)
= 78,0000 - 2,500
= 6,084,000,0000 - 6,250,000
= 6,077,750,000

When the student becomes inquisitive enough he will wonder what put the planets like our earth in orbit at 93,000,000 miles from the sun. What keeps it from being drawn into the sun?

If the north pole was not pointing towards the north star, would we have the four seasons?

At what speed do we travel to make the orbit in 365 1/4 days?

What would happen if the earth did not roll over 1,000 miles per hour establishing day and night divided in the 24 hours?

Would it work if it rolled at 500 miles per hour? Or would the nights be too cold and the days too hot for vegetation?

Wishing to know is the start of thinking. Only the drive to know will start the process. Out of more than 6,000,000 brain cells, some must be inactive. Let us call it cold storage.

Maybe with this mental arithmetic, we can get some of those cells out of cold storage!

  Oluf Nielsen, Author (1891–1967)
Council Bluffs, Iowa
Circa 1965
All copyrights reserved

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