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228 Rh Hence x = {x}{12} + 33, {11}{12}x = 33; x = 36.

Thus the time is 36 minutes past 4. If the question be asked as follows : "At what times between 4 and 5 o'clock will there be 13 minutes between the two hands? " we must also take into consideration the case when the minute-hand is 13 divisions behind the hour-hand. In this case the minute-hand gains 20 — 13, or 7 divisions. Hence x = {x}{12} + 7, which gives x = 7 {7}{12}. Therefore the times are 7{7'}{12} past 4, and 36' past 4.

Ex. 3. Two persons A and B start simultaneously from two places, c miles apart, and walk in the same direction. A travels at the rate of p miles an hour, and B at the rate of q miles ; how far will A have walked before he overtakes B ?

Suppose A has walked x miles, then B has walked x — c miles. A, walking at the rate of p miles an hour, will travel x miles in {x}{p} hours ; and B will travel x — c miles in {x -c}{q} hours : these two times being equal, we have ^ {x}{p} = {x -c}{q} qx = px — pc ; whence x = {pc }{p - q }

Therefore A has travelled {pc }{p - q } miles.

Ex. 4. A train travelled a certain distance at a uniform rate. Had the speed been 6 miles an hour more, the journey would have occupied 4 hours less ; and had the speed been 6 miles an hour less, the journey would have occupied 6 hours more. Find the distance.

Let the speed of the train be x miles per hour, and let the time occupied be y hours ; then the distance traversed will be represented by xy miles.