IMC 2017 Q21
21. Brachycephalus frogs are tiny − less than 1 cm long − and have three toes on each foot and two fingers on each ‘hand’, whereas the common frog has five toes on each foot and four fingers on each ‘hand’.
Some Brachycephalus and common frogs are in a bucket. Each frog has all its fingers and toes. Between them they have 122 toes and 92 fingers.
How many frogs are in the bucket?
- A 15
- B 17
- C 19
- D 21
- E 23
Intermediate Mathematical Challenge 2017, Question 21. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: A 15
Let the number of Brachycephalus frogs and common frogs in the bucket be \(b\) and \(c\) respectively. Note that each Brachycephalus frog has 6 toes and 4 fingers, while a common frog has 10 toes and 8 fingers.
Therefore, \(6b + 10c = 122\) (1); \(4b + 8c = 92\) (2). Subtracting (2) from (1) gives \(2b + 2c = 30\), so \(b + c = 15\).
Solution © UK Mathematics Trust, from the IMC 2017 solutions. Longer versions with investigations: Solutions and investigations (PDF).