Crane-and-Turtle (Tsurukame) Problems ① Basic Method
“Crane (2 legs)” and “turtle (4 legs)” total 10 animals, and when the total number of legs is 32, the problem of finding how many of each there are is called a crane-and-turtle (tsurukame) problem.
Method Using a Table
Consider how many legs there would be if they were all cranes, and use the difference from the actual number.
Example 1 (Basic)
There are 10 cranes and turtles in total. The total number of legs is 32. How many cranes and turtles are there?
Answer (Assuming they are all cranes):
If they were all cranes: 2×10＝20 legs
Difference from the actual number: 32－20＝12 legs
Changing one animal to a turtle increases the number of legs by 2
Number of turtles: 12÷2＝6
Number of cranes: 10－6＝4 (cranes), 6 (turtles)
There are 10 cranes and turtles in total. The total number of legs is 32. How many cranes and turtles are there?
Answer (Assuming they are all cranes):
If they were all cranes: 2×10＝20 legs
Difference from the actual number: 32－20＝12 legs
Changing one animal to a turtle increases the number of legs by 2
Number of turtles: 12÷2＝6
Number of cranes: 10－6＝4 (cranes), 6 (turtles)
Method Using a Table
It is also effective to make a table by changing the number of cranes to 0, 1, 2, … and calculating the number of legs.
Key Points
- Make an assumption using “If they were all ___.”
- Divide the difference from the actual number by “the difference when one is changed.”
- If the answer is not an integer, suspect a calculation error.
Practice Problems
- You buy a total of 20 pencils at 10 yen each and ballpoint pens at 25 yen each, for a total of 330 yen. How many pencils and ballpoint pens did you buy?
- You buy a total of 15 apples at 80 yen each and mandarins at 50 yen each, for a total of 1,050 yen. How many apples and mandarins did you buy?
Answers and Explanations
- Answer: No integer solution satisfies the conditions
Explanation: Let the number of ballpoint pens be x, and the number of pencils be (20－x). Then, 25x＋10(20－x)＝330. Since 15x＝130, x＝26/3, but the number of items must be an integer. Therefore, there is no combination that satisfies the conditions in the problem. - Answer: 10 apples and 5 mandarins
Explanation: If they were all mandarins (50 yen): 50×15＝750 yen. Difference from the actual amount: 1,050－750＝300 yen. Changing one to an apple increases the price by: 80－50＝30 yen. Number of apples: 300÷30＝10. Number of mandarins: 15－10＝5. Check: 80×10＋50×5＝800＋250＝1,050 yen ✓