A Tangled Tale · Chapter
§2: The Grurmstipths explained
This chapter presents a mathematical problem involving omnibuses and a traveller, exploring the relative speeds and times of meeting and overtaking.
What happens
This chapter presents a mathematical problem involving omnibuses and a traveller, exploring the relative speeds and times of meeting and overtaking. The solution is derived using algebraic reasoning, and several submitted answers are critiqued for their errors.
Themes in this chapter
Mathematical Reasoning
The chapter centers on solving a relative speed and distance problem using algebra and logical deduction.
Competition and Achievement
The class list and critique of answers highlight a competitive, evaluative approach to problem-solving.
Characters to notice
- Dinah Mite
Submitted a partially correct answer but miscalculated the overtaking time.
- Nolens Volens
Attempted a solution using the 'Achilles and the Tortoise' method but made an arithmetic error.
- Balbus
Listed as one of the correct solvers in the class list.
- Delta
Listed as one of the correct solvers in the class list.
Key passages
“Since the omnibus met is due at the starting-point in 2½ minutes, it goes in that time as far as the traveller walks in 12½; i.e. it goes 5 times as fast.”
The omnibus that the traveller meets will reach the starting point in 2.5 minutes, covering the same distance the traveller walks in 12.5 minutes, meaning the omnibus is five times faster.
Establishes the speed ratio between omnibus and traveller.
“Hence a + x = 5 x ; i.e. 4 x = a , and x = a 4 .”
The distance the overtaking omnibus travels (a + x) is five times the distance the traveller walks (x), leading to x being one quarter of a.
Key algebraic step in solving the overtaking time.