Last week I travelled from London to Leeds, a distance I measured as 174 miles.
I started at 9.15am and completed the journey with an average speed of 40 miles per hour.
On the way back, in the evening, I travelled exactly the same route, starting at 5.15pm. The traffic was light, and I completed the journey with an average speed of 60 miles per hour.
What was the overall average speed for the round trip?
Hint
Remember that Time = Distance ÷ Speed, and Speed = Distance ÷ Time.
Answer
48 miles per hour.
Reasoning
Using:
Time = Distance ÷ Speed
Speed = Distance ÷ Time
The outward journey took 174 miles ÷ 40 mph = 4.35 hours.
The return journey took 174 miles ÷ 60 mph = 2.9 hours.
So I travelled a total of 2 x 174 = 348 miles. I did this in a total of 4.35 + 2.9 = 7.25 hours.
My overall average speed was therefore 348 miles ÷ 7.25 hours = 48 miles per hour.
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Puzzle 10
Can you find a five-digit number that has no zeros, and no repeated digits, where:
The first digit is a prime number. The second digit is the fifth digit minus the first digit. The third digit is twice the first digit. The fourth digit is the third digit plus three. The fifth digit is the difference between the first digit and the fourth digit.
Don't forget that 1 isn't prime, the prime numbers start with 2, 3, 5, …
Hint
Start with the possible answers where the first digit is a prime number, and then look at the third digit.
Answer
23,475.
Reasoning
By (1), the first digit is prime:
2----
3----
5----
7----
By (3), the third digit is twice the first digit, so we can eliminate 5---- and 7----:
2-4--
3-6--
By (4), the fourth digit is the third digit plus three:
2-47-
3-69-
By (5) the fifth digit is the difference between the first digit and the fourth digit:
2-475
3-696
We know from the introduction that no digit is repeated, so we can eliminate 3-696. And, by (2) the second digit is the fifth digit minus the first digit:
23475
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Puzzle 11
My local bus company has recently expanded and no longer has enough room inside its garage for all of its buses.
Twelve of their buses have to be stored outside.
If they decide to increase their garage space by 40% they will have enough room for all of their current buses, plus enough room to store another twelve in the future.
Hint
How many students study chemistry and physics, but not biology?
Answer
500 students.
Reasoning
The answer is more easily seen if three intersecting circles are drawn, and the numbers inside each section are worked out.
We know that 73 students study all three disciplines, which allows us to work out the numbers.
97 students study chemistry and physics, so 97 – 73 = 24 study just chemistry and physics.
138 students study physics and biology, so 138 – 73 = 65 study just physics and biology.
152 students study chemistry and biology, so 152 – 73 = 79 study just chemistry and biology.
280 students study chemistry, so 280 – 73 – 24 – 79 = 104 study just chemistry.
254 students study physics, so 254 – 73 – 24 – 65 = 92 study just physics.
280 students study biology, 280 – 73 – 65 – 79 = 63 study just biology.
For a total of: 104 + 92 + 63 + 24 + 65 + 79 + 73 = 500 students.
