fractions tricks
Solving vulgar fractions ending in odd digits:
There are two methods of solving vulgar fractions, the multiplication method and the division method.
I’ll explain the multiplication method first using an example. For example the fraction we have is 1/19. The first digit of the denominator (1) will form the last decimal digit before the decimal places start repeating. Now we start calculating backwards. Add one to this last digit and we get 2. This will occupy the decimal place before the last digit. This is called a multiplier. It will be used to multiply each digit to obtain its predecessor decimal place. To determine the number of decimal places in the answer subtract the numerator from the denominator. To understand this better let me solve the example for you.
1/19
So in this case the number of decimal places in the answer will be 19-1=18.
The first digit of the denominator is 1. To it add 1 to get 2. Hence the last two digits will now be 21. Now since 2 is the multiplier hence multiple 2 by 2 to get the next digit. Hence now the digits are 421.
Now multiply 4 by 2
Therefore the last few decimal places now are 8421
Now when we go further we will have to multiply 8 by 2 but since the answer is a two digit number we carry the first digit forward as in the normal multiplication. And so on and so forth. And the final answer we get is 1/19=0.052631578947368421.
Now to solve the same sum using the division method, in this method also 2 would be the multiplier as we calculated using the earlier method. But we derive the answer in this case by going forward from the decimal point. Hence the first decimal place is calculated by dividing 1 by 2. The answer is 0 with a remainder of 1. Hence the first decimal place is 0.0. Now since 1 is the remainder for the next decimal place, divide 10 with 2 to get 5. Hence the second decimal place is 5. And so on and so forth to get the same answer as above.
We use the same formula to solve fractions with single digit denominators as well. For example the fraction is 1/ 7=7/49
Now the multiplier in this case is 4+1=5
Hence we can solve it as follows, the last digit will be 7 as in the original fraction and the multiplier will be 5. So the answer will be determined as follows,
7
The second last decimal place will be 7*5=35. Carry 3 forward
57
Now multiply 5*5=25+3(carried forward) =28 carry 2 forward
857
And so on and so forth till we get six decimal places (numerator-denominator= 7-1 =6). Hence we will stop after 6 decimal places.
Thus the answer obtained is 0.142857.