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The Race


A and B run a mile, and A wins by 80 yards.
A and C run over the same course and A wins by 20 seconds.
B and C run and B wins by 5 seconds.
In what time can A run a mile?

Solution:

Suggested Solution (Kurt Ottum & Logan O'Sliabh, WWU students)

In order to know the time for A to run a mile, we must consider rate. Therefore A's rate will be "A " and likewise with B and C. We are concerned with finding the time it takes for A to run a mile, call it "t." Since time is given in seconds in the problem, t will be in seconds. Therefore,

A*t = 1 (mile)

Since B is short 3*80 (or 249) feet of a mile (5280 feet) when A finishes the mile:

B*t = 1 - 240/5280 = 21/22 miles

Since C must take an extra 20 seconds at rate C to finish the mile after A:

C*t = A*t - 20*C or C*t = 1 - 20*C

Since C takes an extra 5 seconds after B and B takes [22/21]*t to finish:

C*[22/21]*t = B*[22/21]*t - 5*C and C*t = B*t - (21/22)*5*C

C*t = 21/22 - [21/22]*5*C

Using C*t identities:

1 - 20*C = 21/22 - (21/22)*5*C or 1/22 = [335/22]*C or C = 1/335

Since C*t = 1 - 20*C:

[1/335]*t = 1 - 20*[1/335] or t = 335 - 20 = 315

Thus, t = 315 seconds, or 5 min 15 seconds for A to finish the mile.

Source: Charles Pendlebury's Arithmetic (London: G. Bell & Son, 1918)