67.—­AVERAGE SPEED.

The average speed is twelve miles an hour, not twelve and a half, as most people will hastily declare.  Take any distance you like, say sixty miles.  This would have taken six hours going and four hours returning.  The double journey of 120 miles would thus take ten hours, and the average speed is clearly twelve miles an hour.

68.—­THE TWO TRAINS.

One train was running just twice as fast as the other.

69.—­THE THREE VILLAGES.

Calling the three villages by their initial letters, it is clear that the three roads form a triangle, A, B, C, with a perpendicular, measuring twelve miles, dropped from C to the base A, B. This divides our triangle into two right-angled triangles with a twelve-mile side in common.  It is then found that the distance from A to C is 15 miles, from C to B 20 miles, and from A to B 25 (that is 9 and 16) miles.  These figures are easily proved, for the square of 12 added to the square of 9 equals the square of 15, and the square of 12 added to the square of 16 equals the square of 20.

70.—­DRAWING HER PENSION.

The distance must be 63/4 miles.

71.—­SIR EDWYN DE TUDOR.

The distance must have been sixty miles.  If Sir Edwyn left at noon and rode 15 miles an hour, he would arrive at four o’clock—­an hour too soon.  If he rode 10 miles an hour, he would arrive at six o’clock—­an hour too late.  But if he went at 12 miles an hour, he would reach the castle of the wicked baron exactly at five o’clock—­the time appointed.

72.—­THE HYDROPLANE QUESTION.

The machine must have gone at the rate of seven-twenty-fourths of a mile per minute and the wind travelled five-twenty-fourths of a mile per minute.  Thus, going, the wind would help, and the machine would do twelve-twenty-fourths, or half a mile a minute, and returning only two-twenty-fourths, or one-twelfth of a mile per minute, the wind being against it.  The machine without any wind could therefore do the ten miles in thirty-four and two-sevenths minutes, since it could do seven miles in twenty-four minutes.

73.—­DONKEY RIDING.

The complete mile was run in nine minutes.  From the facts stated we cannot determine the time taken over the first and second quarter-miles separately, but together they, of course, took four and a half minutes.  The last two quarters were run in two and a quarter minutes each.

74.—­THE BASKET OF POTATOES.

Multiply together the number of potatoes, the number less one, and twice the number less one, then divide by 3.  Thus 50, 49, and 99 multiplied together make 242,550, which, divided by 3, gives us 80,850 yards as the correct answer.  The boy would thus have to travel 45 miles and fifteen-sixteenths—­a nice little recreation after a day’s work.

75.—­THE PASSENGER’S FARE.

Mr. Tompkins should have paid fifteen shillings as his correct share of the motor-car fare.  He only shared half the distance travelled for L3, and therefore should pay half of thirty shillings, or fifteen shillings.