7 trick
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7 trick
Everyone knows the 9 trick for multiplication:
- To multiply a one digit number by 9: subtract one from the number and write down the answer in the tens (3 x 9 : 3-1 = 2 : 2x ) then subtract you new number from 9 and write down the result in the ones (9 - 2 = 7 : 27) and you have your answer.
And it's friend the "9 test" for division:
- To test if 9 goes evenly into a number add up the digits: if the result is 9 you can divide by 9 and have no remainder (and you can get the right answer for 2 digit numbers just by adding one to the tens digit).
I have heard of a 5 test (must end in a 0 or 5) and a 3 test (digits must add up to 2 6 or 9).
But I have never heard of a 7 test. So I decided to see if I could some up with one.
Binary 7
In decimal (normal counting numbers from first grade) every digit must be 0, 1, 2, 3, 4, 5, 6, 7, 8 or 9 (ten possible values).
In binary you get the value of a number by adding up powers of two. Every digit must be either 0 or 1 (two possible values).
In decimal (normal counting numbers from first grade) the columns are (from right to left) ones, tens, and hundreds. Just keep multiplying by 10.
- 234 = (4 x 1) + (3 x 10) + (2 x 100) = 4 + 30 + 200 = 234
In binary the columns are (from right to left) ones, twos and fours. Just keep multiplying by 2.
- 101 = (1 x 1) + (0 x 2) + (1 x 4) = 1 + 0 + 4 = 5
So in binary
- 111 = (1 x 1) + (1 x 2) + (1 x 4) = 1 + 2 + 4 = 7
This suggests a 7 trick:
The NEW 7 Trick
- To multiply a number by 7, take the original number and write it down.
- 3 x 7 : 3
- Then double it and write down the answer under it
- 3
- 6
- Next, double the new number, and write it down
- 3
- 6
- 12
- Finally add up the three numbers:
- 3
- 6
- +12
- ---
- +21
The reason this works is as follows:
Since
- 7 = 1 + 2 + 4
we can show that
- N x 7 = N x ( 1 + 2 + 4 )
it follows that
- N x 7 = (N x 1) + (N x 2) + (N x 4)
the steps give you the original N (3 in our example) then the original N x 2 (6 in our example) then the last number doubled: N x 2 x 2 = N x 4 (12 in our example).
The NEW "7 test" trick
- Divide the number by 8 (ignore the remainder for now)
- Half the number three times: (easy for even numbers, for odd numbers just drop down one... we are ignoring the remainder)
- (first halving) 87 is odd, half of 86 is 43
- (second halving) 43 is odd, half of 42 is 21
- (third halving) 21 is odd, half of 20 is 10
- Jot this number down at the edge of the paper
- 10
- Multiply the new number by 7 (using any method you know... see above for a trick)
- 10 x 7 = 70
- Subtract the result from the original number
- 87 - 70 = 17
- Repeat the above steps for the new number
- (first halving) 17 is odd, half of 16 is 5+3 = 8
- (second halving) 8 is even, half of 8 is 4
- (third halving) 4 is even, half of 4 is 2
- Jot this number down at the edge of the paper below the other intermediate result
- 10
- 2
- Multiply the new number by 7 (using any method you know... see above for a trick)
- 2 x 7 = 14
- Subtract the result from the original number
- 17 - 14 = 3
- Keep repeating until the result is smaller than 7
- 3 is smaller than 7 so stop
- This left-over number is the remainder, circle it and add a letter "R"
- ( R 3 )
- You have completed the test... Remainder is not "0" to 87 is not evenly divisible by 7.
- Special Case.... Remainder is ... 7?
- Because we are dividing by 8 instead of 7 (for a low approximation of dividing by 7) the remainder may sometimes end up as 7
- This happens because 7 divided by 8 is 0 (8 doesn't go into 7).
- If this happens, the remainder is really 0, and the number is evenly divisible by 7.
Going Further... division by 7
- Add up the numbers at the edge of the paper to get the dividend
- 10
- + 2
- -----
- 12
- You now have your answer (and not a multiplication table in sight)
- 87 / 7 = 12 ( R 3 )
- Special Case .... Remainder is ... 7?
- if the remainder is 7 then add one to the dividend and change the remainder to "0".
- Final Check
- Use The 7 trick above to multiply the final dividend by 7 and subtract it from the original number. You should get the remainder.
- If not ... check you math.
- Use The 7 trick above to multiply the final dividend by 7 and subtract it from the original number. You should get the remainder.
Optional: If you want the answer in decimal
- If you want the answer in decimal, like money, decide how many places you want to get (say 2 or 3) to the right of the decimal point
- I want an answer with 2 places to the right of the decimal : 12.xx (accurate to 2 places)
- Add this many zeros to the right of the reminder
- ( 3 R ) -> 3 with 2 zeros after it = 300
- Repeat the process above for this new number
- half 300 = 100+50 = 150
- half 150 = 50+ 20+5 = 50 + 25 = 75
- odd 75 -> half 74 = 30+5 + 2 = 37
- Save
- 37
- Then
- 37 x 7 = 37 + 74 + 148 = 111 + 148 = 259
- 300 - 259 = 41
- Repeat
- odd 41 -> half 40 = 20
- half 20 = 10
- half 10 = 5
- Save
- 37
- 5
- Then
- 5 x 7 = 5 + 10 + 20 = 5 + 30 = 35
- 41 - 35 = 6
- 6 is smaller than 7 so stop
- ( R 6 )
- 37
- + 5
- ----
- 42 ( R 6 )
- take your new number and write it with a decimal point on it's RIGHT
- 42.
- we wanted accuracy to 2 places so wright in 2 zeros on the LEFT
- 0042.
- Now move the decimal point 2 places to the left
- 00.42
- Add the original dividend, place a decimal and 2 zeros on the RIGHT of it
- 12.00
- + 00.42
- -------
- 12.42
Optional: Rounding
- If you want your answer to be the closest it can be (to the true answer) with only 2 digits: you may round.
- If the final Remainder is 4 or more round up. If the final Remainder is 3 or less leave it alone (called rounding down).
- ( R 6 ) -> 6 is more than 4 -> round up
- To round up, just add one to the last digit on the right
- 12.42 -> 2 + 1 = 3 -> 12.43
- 87 / 7 = 12.43 (rounding to 2 places of accuracy)