Crane-and-Turtle (Tsurukame) Problems ② Solving with Equations
Crane-and-turtle problems can be solved more logically using simultaneous equations. Master how to set up the equations.
Solving with Equations
Let the number of cranes be x and the number of turtles be y, and set up two equations.
Example Problem
There are 10 cranes and turtles in total, and they have 32 legs altogether. How many of each are there?
Solution:
Let the number of cranes be x and the number of turtles be y.
① x＋y＝10 (total number of animals)
② 2x＋4y＝32 (total number of legs)
①×2: 2x＋2y＝20
②－①×2: 2y＝12 → y＝6
x＝10－6＝4
4 cranes and 6 turtles
There are 10 cranes and turtles in total, and they have 32 legs altogether. How many of each are there?
Solution:
Let the number of cranes be x and the number of turtles be y.
① x＋y＝10 (total number of animals)
② 2x＋4y＝32 (total number of legs)
①×2: 2x＋2y＝20
②－①×2: 2y＝12 → y＝6
x＝10－6＝4
4 cranes and 6 turtles
Application Problems
Example Problem 2
There are 30 100-yen coins and 50-yen coins in total, worth 2,200 yen altogether. How many of each coin are there?
Solution:
Let the number of 100-yen coins be x and the number of 50-yen coins be y.
① x＋y＝30
② 100x＋50y＝2200
②÷50: 2x＋y＝44
(2x＋y)－(x＋y): x＝14
y＝30－14＝16
14 100-yen coins and 16 50-yen coins
There are 30 100-yen coins and 50-yen coins in total, worth 2,200 yen altogether. How many of each coin are there?
Solution:
Let the number of 100-yen coins be x and the number of 50-yen coins be y.
① x＋y＝30
② 100x＋50y＝2200
②÷50: 2x＋y＝44
(2x＋y)－(x＋y): x＝14
y＝30－14＝16
14 100-yen coins and 16 50-yen coins
Key Points
- Two unknowns → two equations
- Set up an equation for the number of items and an equation for the total amount (quantity or value).
- Multiply equation ① by a coefficient and subtract it to eliminate one of the unknowns.
Practice Problems
- There are 25 adults and children in total. The admission fee is 800 yen for adults and 400 yen for children, and the total was 14,800 yen. How many adults and children are there? (Solve using equations.)
- There are 20 five-point and three-point questions in total. If all the questions are answered correctly, the score is 80 points. How many five-point and three-point questions are there?
Answers and Explanations
- Answer: 12 adults and 13 children
Explanation: Let the number of adults be x, and the number of children be (25-x). 800x＋400(25-x)＝14800. 800x＋10000－400x＝14800. 400x＝4800. x＝12. There are 12 adults and 13 children. Check: 800×12＋400×13＝9600＋5200＝14800 yen ✓ - Answer: 10 five-point questions and 10 three-point questions
Explanation: Let the number of five-point questions be x, and the number of three-point questions be (20-x). 5x＋3(20-x)＝80. 5x＋60－3x＝80. 2x＝20. x＝10 questions. Three-point questions: 20－10＝10 questions. Check: 5×10＋3×10＝50＋30＝80 points ✓