Mathematical Puzzles 

Workings

Hint
Try working backwards.

Answer
7,340 metres.

Reasoning
Stage 5's clue tells us that 150m was left. So the distance left at the start of Stage 4 must have been:
Dist4 = Dist4 + 75 + 150
        —————
          2

Which simplifies to give Dist4 = 450m.

Stage 3's clue tells us that 450m was one quarter of the remaining distance, so Dist3 = 1,800m.

Stage 2's clue tells us that the distance at the start of Stage 2 was:
Dist2 = Dist2 + 35 + 1800
        —————
          2

Which simplifies to give Dist2 = 3,670m.

Stage 1's clue tells us that 3,670m was half the overall distance, which means the entire ride was 7,340m.

Double-Checking
Starting with 7,340m:
Stage 1: cycled 3,670m, leaving 3,670m

Stage 2: cycled 1,870m, leaving 1,800m

Stage 3: cycled 1,350m, leaving   450m

Stage 4: cycled   300m, leaving   150m

Stage 5: cycled   150m, leaving     0m

Workings

Hint
In order for the clocks to show the same time (e.g. 2 o'clock), what must the total time gained by one, and lost by the other, total?

Answers
Midnight, 10 days later, when they will both show 4 o'clock.
Midnight, 30 days later, when they will both show 12 o'clock.
Reasoning #1
In order for the clocks to show the same time, the total time gained by one, and lost by the other, must be 12 hours.

For example, if the first clock were to show 2 o'clock, it would have gained 2 hours. In order for the second clock to also show 2 o'clock, it would have had to have lost 10 hours. This is a total of 12 hours gained and lost.

We know that for every hour that passes, the first clock gains one minute, and the second clock loses 2 minutes, for a total time gained and lost of 1 + 2 = 3 minutes.

The total time gained and lost will equal 12 hours when 12 x 60 ÷ 3 = 240 hours have passed. 240 hours = 10 days.

The first clock will have gained 240 x 1 minutes = 240 minutes = 4 hours.

The second clock will have lost 240 x 2 minutes = 480 minutes = 8 hours.

So, they will both show 4 o'clock, 10 days later.

Reasoning #2
In the first answer, we can see that 10 days later, the clocks both show 4 o'clock.

If we move forward another 10 days, both clocks would show 8 o'clock.

If we move forward another 10 days, both clocks would show 12 o'clock.

This will now be the correct time.

So, they will both show 12 o'clock, 30 days later.

Workings

Hint
You can either fix the distance rowed and see what numbers work, or you can fix the number of hours.

Answer
12 miles.

In the morning, rowing at 2 miles per hour, they rowed for 6 hours. In the afternoon, rowing at 4 miles per hour, they rowed for 3 hours.

There are a number of ways of working this out, and here are two of them:

Method 1
If we assume the distance the rowed upstream to be D miles, we know the morning took (D ÷ 2) hours, and the afternoon took (D ÷ 4) hours, with a difference of 3 hours. So:
 D   D
 - – - = 3
 2   4

Multiplying throughout by 4 gives:

 2D – D = 12

So:

 D = 12 miles

They rowed 12 miles upstream.

Method 2
If we assume they rowed for H hours upstream, we know they travelled H x 2 miles in the morning. In the afternoon, they rowed for (H – 3) hours, and travelled (H – 3) x 4 miles. We know these distances are the same, so:

 2H = (H – 3) x 4

Giving:

 2H = 4H – 12

Rearranging gives:

 12 = 2H

So:

 H = 6 hours

They rowed for 6 hours upstream at 2 miles per hour, which is a total of 12 miles.

Workings

Hint
Start by going up.

Answer
38.