TO T£IE RESOLUTION OF PROBLEMS. 91 
, „ » . 
Qax—a 2 = b; whence 2ax = b + a z , and x - f -g~ 
Cl *i 
±r 17 the same as before. 
PROBLEM XXX. 
- If one agent A, alone, can produce an effect e, in the 
time a, and another agent B, alone, in the time b; in how 
long time will they both together produce the same effect.- 
Let the time sought be denoted by x, and it will be, 
as a : x :: e : —-, the part of the effect produced by A : 
(Theor. 3. p. 72) also, as b : x e : -y, the part pro 
duced by B: therefore f ~ — e. Divide the 
whole by e, and you will have-- -f y = 1; and this,. 
reduced, gives x — ~ After the same manner, if 
there be three agents, A, B, and C, the time wherein 
they will altogether produce the given effect, will come 
abc . 
out — 7 ; r-. 
ah 1 ac f oc 
Example. Suppose A, alone, can perform a piece of 
work in 10 days; B, alone, in 12 days; and C, alone, 
in L6 days; then all three together will perform the 
same piece of work in 4jV days; for in this case, 
a being = 10, b — 12, c — 16, it js ( plain that 
abc 10/13/16 , _ 
ab f ac b be no x 12 l 10 x 16 1 12 x 16 — ijy ‘ 
PROBLEM XXXI. 
Two travellers, A and B, set out together from the same 
place, and travel both the same way; A goes 28 miles the 
first day, 26 the second, 24 the third, and so on, decreasing 
two miles every day; but B travels uniformly 20 miles 
every day; now it is required to find how many miles each 
person must travel before B comes up again with A f