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Workings

Hint
A little algebra might help.

Answer
Width=40, height=20, depth=18.

Reasoning
If we call the height H, the width W and the depth D we have:
   H x W = 800

       H = 800 ÷ W (1)

   W x D = 720

       W = 720 ÷ D (2)

   H x D = 360

       H = 360 ÷ D (3)

Using (1) and (3) we have:
   800   360
   ——— = ———
    W     D

   800 x D = 360 x W

Using (2):
   800 x D = 360 x 720
                   ———
                    D

   800 x D² = 259,200

         D² = 324

          D = 18

Using (2), W = 40.

Using (1), H = 20.

Workings

Hint
In the road race, where could Charlie have finished?

Answer
Cycling : Billie > Alex > Charlie > Drew

Road race : Drew > Charlie > Billie > Alex

Road Race
In the road race, where could Charlie have finished?

By (4), he wasn't last. By (5), he wasn't first.

By (2), if Charlie was third, then he would have won the cycling, but by (3) Alex beat Charlie in the cycling, so Charlie can't have won the cycling.

So Charlie must have finished in second place.

By (5), Drew was first.

By (1), Alex didn't win the cycling, so by (2) he didn't come third in the road race. Therefore, Alex came last.

Leaving Billie in third place.

Road race : Drew > Charlie > Billie > Alex

Cycling
By (2), Billie won.

By (4), Charlie wasn't last.

By (3), Alex beat Charlie. So Alex finished in second place, with Charlie in third place.

Leaving Drew in last place.

Cycling : Billie > Alex > Charlie > Drew

Workings

Hint
The items can be eaten.

Answer
Coconut, kumquat, peanut, and nectarine.

Workings

Hint
Adding the speeds won't work.

Answer
205 mph.

Reasoning
If we take the number of miles in a lap to be L miles and use Time = Distance ÷ Speed.

The first ²/₅ are covered at 123 mph. Therefore, the time taken was [(²/₅) x L] ÷ 123 hours.

The next ²/₅ are covered at 164 mph. Therefore, the time taken was [(²/₅) x L] ÷ 164 hours.

The last ¹/₅ is covered at y mph. Therefore, the time taken was [(¹/₅) x L] ÷ y hours.

The entire trip is covered at an average of 150 mph, and we can conclude that:
 2L      2L      L     L
————— + ————— + ——— = ———
5x123   5x164    5y   150

Rearranging and simplifying gives:
 1     1    2     2
——— = —— - ——— - ———
 y    30   123   164

That means that y = 1 ÷ (¹/₃₀ - ²/₁₂₃ - ²/₁₆₄)

Using a calculator gives y = 205.