Real World Division


#1

We haven’t talked about division lately, so let’s revisit.

The other day; I was buying flowers and small talking the seller about the flowers.
He was telling me that for him the roses where 1.85 euros to buy and that he actually needs to sell them for 6 euros (ouch btw), but because he thought that was too expensive he sold them for 5 euros.

I was curious about the margin he calculates with, so I needed to calculate 6/1.85.
As you know if you read my previous posts, I usually start with a ballpark figure.
6/2 = 3, so 6/1.85 is more than 3.
6/1.5 = 4, so 6/1.85 is less than 4.

Then I start to think how to simplify this calculation.
First I double both sides, then I get rid of the decimal point, so both sides are integers:
6/1.85 = 12/3.7 = 120/37

In previous posts I wrote how I do this; a two step process.
37 is almost 40, so first I do 120/40 = 3.
Then I correct the remainder. 40-37 = 3. 3*3=9.
So 120/37 = 3 with a remainder of 9.

If this is abracadabra for you, think about dividing 120 apples over 40 baskets.
It is easy to see that each basket gets 3 apples.
In our case we need to divide 120 apples over 37 baskets instead.
By dividing over 40 baskets we over reach; we have done too much.
Since we only need to have 37 baskets filled, we take the 3 apples in each of the 3 superfluous baskets and thus we get a remainder of 9 apples.

Next digit. Starting from a remainder of 9, the next division is 90/37.
Again start with 40:
90/40 = 2r10.
Again we need only 37 baskets instead of 40 so we have 3 baskets too many with 2 apples in each. So add those 6 apples to the remainder of 10 to get 16 remaining apples.
90/37 then is 2r(10+6) = 2r16.

Btw, the normal way this is taught is to make a guess (2). Then, calculate 2*37 = 74 and last subtract 74 from 90: 90-74 = 16.
This leads to the same remainder of 16. However; I think my two step process is easier, faster and less prone to mistakes.

Third digit. 160/37. Again use 40:
160/40 = 4r0.
Instead of 40 baskets, we only need 37. We have three baskets with 4 apples in them that we don’t need, so 12 apples.
So: 160/37 = 4r12.

Take a step back now.
We have remainder of 12, so the next step is calculating 120/37.
This is our starting point(!), so from here you can just rattle the digits: 324 324 324 324 etc.

Final answer: 3.24324324…

Btw, repeating digits happen a lot in division. Always be looking for repeating digits. In this case a repetition of 3 digits is short and here this means we are done quickly.