7/27/2019 Tips for Math Calculation
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(1) Multiplication by 5
It's often more convenient instead of multiplying by 5 to
multiply first by 10 and then divide by 2. For
example,1375=1370/2=685. (2) Division by 5
Similarly, it's often more convenient instead to multiply first
by 2 and then divide by 10. For example,1375/5=2750/10=275.(3)
Division/multiplication by 4
Replace either with a repeated operation by 2. For example
124/4=62/2=31. Also, 1244=2482=496. (4)Division/multiplication by
25
Use operations with 4 instead. For example,
3725=3700/4=1850/2=925. (5) Division/multiplication by 8Replace
either with a repeated operation by 2. For example
1248=2484=4962=992. (6) Division/multiplication by
125
Use operations with 8 instead. For example,
37125=37000/8=18500/4=9250/2=4625(7) Squaring two digit
numbers.
You should memorize the first 25 squares:
1 2 3 4 5 6 7 8 9 10 11 12 13 14
1 2 9 16 25 36 49 64 81 100 121 144 169 196
15 16 17 18 19 20 21 22 23 24 25
225 256 289 324 361 400 441 484 529 576 625
Squares of numbers from 26 through 50.
Let A be such a number. Subtract 25 from A to get x. Subtract x
from 25 to get, say, a. Then A2=a2+100x. Forexample, if A=26, then
x=1 and a=24. Hence 262=242+100=676. Similarly, if A=37, then
x=37-25=12, and
a=25-12=13. Therefore, 372=132+10012=1200+169=1369. Why does
this work?
(25+x)2-(25-x)2=[(25+x)+(25-x)][(25+x)-(25-x)]=502x=100x.
Squares of numbers from 51 through 99.
The idea is the same as above. (50+x)2-(50-x)2=1002x=200x. For
example, 632=372+20013=
1369+2600=3969.
Squares of numbers from 51 through 99, second approach (this one
was communicated to me by my
father Moisey Bogomolny).
We are looking to compute A2, where A=50+a. Instead compute
100(25+a) and add a2. Example: 572. a=57-
50=7. 25+7=32. Append 49=72. Answer: 572=3249.
In general, a2 = (a + b)(a - b) + b2. Let a be 57 and, again, we
wish to compute 572. Let b = 3. Then 572 = (57
+ 3)(57 - 3) + 32, or 572 = 6054 + 9 = 3240 + 9 = 3249.
(8) Squares of numbers that end with 5.
Let A=10a+5. Then A2=(10a+5)2=100a2+210a5+25=100a(a+1)+25. For
example, to compute 1152, where a=11, firstcompute
11(11+1)=1112=132 (since 3=1+2). Next, append 25 to the right of
132 to get 13225! Another example, tocompute 2452, let a=24. Then
24(24+1)=242+24=576+24=600. Therefore 2452=60025. Here is another
way to compute
2425: 2425=2400/4=1200/2=600. The rule naturally applies to
2-digit numbers as well. 752=5625 (since 78=56).
(9) Product of two one-digit numbers greater than 5.
This is a rule that helps remember a big part of the
multiplication table. Assume you forgot the product 79. Do
this.
First find the access of each of the multiples over 5: it's 2
for 7 (7 - 5 = 2) and 4 for 9 (9 - 5 = 4). Add them up to get 6
