Short Answer Question Show work please

Question

Billy walks around a square city block. On the first side his speed is 1 mph, on the second side his speed is 2 mph, on the third side his speed is 3 mph, and on the fourth side his speed is 4 mph. Find Billy's average speed. (Your answer must be a simplified fraction in mph) (Note: mph = miles per hour)

Answer 1

x = vt so t = x/v

Let

x be the distance of side of the block (in mi)

t be the time (in hr) it takes to travel that side

and v be the speed in mi/hr

To walk the first side at a speed of 1 mi/hr it takes (x mi)(1 mi/hr) = t hrs

To walk the second side at a speed of 2 mi/hr it takes (x mi)(2 mi/hr) = ½t hrs

To walk the second side at a speed of 2 mi/hr it takes (x mi)(3 mi/hr) = ⅓t hrs

To walk the second side at a speed of 2 mi/hr it takes (x mi)(4 mi/hr) = ¼t hrs

The total time it takes to walk the whole block is (1 + ½ + ⅓ + ¼)t =

(12/12 + 6/12 + 4/12 + 3/12)t = (25/12)t hrs

The total distance walked is 4x mi

v = x/t = 1 mi/hr for the first side

v_avg = (4x)/[(25/12)t] = 4/(25/12) × x/t = 4/(25/12) = 48/25

is the average speed for the whole block.

Answer

Billy's average speed is 48/25 mi/hr

Note

Expressed as a decimal this is 1.92 mi/hr

which is less than the number you arrive at

by taking the average of the four speeds.

This incorrect number is (1 + 2 + 3 + 4)/4 = 2.5 mi/hr