Puzzle 114
Can you find a five-digit number that …
… if you place a 9 at the beginning is four times larger than if you place a 9 at the end instead?
Puzzle Copyright © Kevin Stone
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Hint
What's a mathematical way to describe placing a 9 at the beginning?
Answer
23,076.
Reasoning
Placing the 9 at the beginning is equivalent to adding 900,000 (for example, 12,345 + 900,000 becomes 912,345).
Placing the 9 at the end is equivalent to multiplying by 10 and adding 9 (for example, 12,345 x 10 + 9 becomes 123,459).
Let's call the unknown number k, and we know that one of these numbers is 4 times the other, so:
k + 900,000 = 4 x (10k + 9)
k + 900,000 = 40k + 36
900,000 – 36 = 40k – k
899,964 = 39k
899,964 ÷ 39 = k
Therefore, k = 23,076.
Double-Checking
923,076 = 4 x 230,769.
Puzzle 115
At the local racetrack, Alex was testing the latest car.
During the eight test laps, Alex managed to average the first two laps at 100mph, the next two laps at 102mph, the next two laps at 140mph, and the last two laps at 150mph.
What was the average speed for the entire 8 laps?
Puzzle Copyright © Kevin Stone
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Hint
Remember that Time = Distance ÷ Speed, and try setting the track length to be a random length.
Answer
119mph.
Reasoning
It doesn't matter how long a lap is, so let's set the track length to be a random number, say 5 miles.
Using:
Time = Distance ÷ Speed
The first two laps took:
10/100 hours
The second two laps took:
10/102 hours
The next two laps took:
10/140 hours
The final two laps took:
10/150 hours
Therefore, the total time for the 8 laps was:
10/100 + 10/102 + 10/140 + 10/150 hours
The 8 laps were 40 miles, and using Speed = Distance ÷ Time:
average speed = 40 ÷ (10/100 + 10/102 + 10/140 + 10/150)
= 119 mph
Puzzle 116
A farmer buys chicken feed for 47c per tonne, pig feed for $1.23 per tonne and cattle feed for $1.44 per tonne. The feed can only be purchased by the tonne, and part tonnes aren't sold.
How much of each would they have if they bought 100 tonnes for $100?
Puzzle Copyright © Kevin Stone
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Hint
One of the answers is 28 tonnes.
Answer
38 tonnes of chicken feed, 34 tonnes of pig feed and 28 tonnes of cattle feed.
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