Tsuru-kame-zan is a method of calculation based on figuring the number of cranes and tortoises from the totals of their legs. Let’s try it.
It is introduced in the rakugo Sanshi Katsura created. In ancient time, it was appeared in China as a method to calculate rabbits and sheep, I heard.
Example
There are cranes and tortoises. The number of them is 34, and the number of legs is 78. Now, how many cranes are there? And how about tortoises?
Answer
There are 29 cranes and 5 tortoises.
Solution
It’s important to think about difference.
Both crane and tortoise have 2 legs at least. So, there are ( 34 times 2 ) legs, at least.
[ 34 times 2 = 68 ( textrm{legs} ) ]Now, there are 78 legs and difference from 68 is 10.
Why does the difference 10 appear? It’s because tortoise have 4 legs. All of the difference 10 is tortoises’ legs.
One tortoise generate 2 legs difference, so
[ 10 div 2 = 5 ( textrm{tortoises}) . ]It’s the number of tortoises. Then cranes are ( 34 – 5 = 29 ) .
Practice
Question 1
There are two kind of books. One has 230 pages and the other has 180 pages. Now, there are 9 books and 1920 pages. Calculate the number of each kind of books.
Question 2
There are two kind of bags. One contains 3 kg of wheat, the other contains 2 kg of sugar. Now, there are 45 bags and weighs 113 kg. Calculate the weight of all wheat.
Answer
69 kg
Solve with Equation
Now, I’ll introduce solution by equation. There are cranes and tortoises. The number of them is 34, and the number of legs is 78.
Solution
It’s okey to use ( x ) , ( y ) and create multi equation, but solve with simple equation.
Denote the number of tortoises as ( x ). The number of tortoises and cranes is 34, then the number of cranes is ( left( 34 – x right) ) .
The number of legs are 78, then
[ 4 x + 2 left( 34 – x right) = 78 . ]So,
[ x = 5 . ]Thus, there are 5 tortoises and ( 34 – 5 = 29 ) cranes.