Cathy and Sara
Cathy and Sara have been friends ever since they can remember. At 17, they were still friends even though they have their own boyfriends. They often sat and just chatted.
Cathy: Dave just got his driver’s license. We are going to celebrate by going to Langley Park. We will just drive there and back.
Sara: That’s great. Which route will you take?
Cathy: Dave showed me this scenic route through the county side. There is very little traffic. He says that in order to enjoy the scenery we will go slowly, at 40 kilometers per hour, but we will come back at 60. Even though the speed limit is 50, they don’t give you a speeding ticket at 60. Do you and your boyfriend Johnny want to join us?
Sara: Let me ask Johnny. When are you going?
Cathy: Tomorrow.
Sara called Johnny and then told Cathy: We want to go but we will leave you lovebirds alone.
Cathy: What does that mean?
Separate cars
Sara: Johnny wants to take his own car, I will go with him, and you guys go in Dave’s car. We will take the same route. Also, he doesn’t like slowing the traffic while going and speeding while coming back. He may not get a ticket but he still wants to go at 50 both ways.
Cathy: Suit yourself but you are welcome to join us.
Sara: We can hang around after the trip. Since we will return 10 minutes before you, we will grab some coffee for you guys too.
Cathy cannot figure out how Sara would be back 10 minutes earlier. We are going at 40, coming back at 60 and they are going by the same road at 50 both ways. She must be nuts.
Well, the next day they went for the trip. Cathy and Dave did return 10 minutes later as Sara had mentioned. Cathy still cannot make out how this could happen. She thinks that they cheated. Sara swears that they stuck to 50 both ways.
What do you think?
How far is this Langley Park anyway?
Sara’s Joke
Two politicians were arguing on TV. Politician A said that there was a 90% chance that they would win the election. B said that the chance for A to win was 0%. The perplexed moderator decided to settle it by saying that the actual number might be some kind of a mean of the two values. B countered, ”I bet you it’s not the arithmetic mean – may be geometric or harmonic mean.”
Challenge
Let us say that you have two positive real numbers x and y.
Prove that the geometric mean ≤ arithmetic mean.
Prove that the harmonic mean ≤ arithmetic mean.
Remember: Arithmetic mean = (x+y)/2
Geometric mean = √xy
Harmonic mean = 2/(1/x + 1/y).
Explanations and solution to the challenge
Story explanation: First, let us consider this as a simple algebraic word problem.
Say Langley Park is d kilometers away.
For Cathy and Dave, at a speed of 40 km/h, it will take d/40 hours to go, and at a speed of 60 km/h, it will take them d/60 hours to come back. So the total time will be
d/40 + d/60 = 3/d120 + 2d/120 = 5d/120 = d/24 h.
Since there are 60 min in 1 hour, we can multiply the result by this to get
60d/24 min = 5d/2 min = 25d/10 min.
For Sara and Johnny, at a speed of 50 km/h in to go and to come back it will take
d/50 + d/50 h =2d/50 h =120d/50 min = 24d/10 min.
24d/10 min < 25d/10 min. So Sara is right that they will come back first.
The time difference between their journey will be
25d/10 – 24d/10 = 1d/10 min.
Sara says that they will come back 10 min earlier, that means
1d/10 = 10 min which gives d =100 km.
To be sure, we can check this answer. Langley Park is 100 km away, Sara’s group will take exactly 100/50 h to go and 100/50 h to return which is a total of 4 h or 240 min. Cathy and Dave will take 100/40 h to go and 100/60 h on their way back, totaling to
100/40 + 100/60 = 300/120 + 200/120 = 500/120 or 25/6 h which is 250 min.
Because Sara and Johnny returned in 240 min, they returned 10 min earlier than the other group. So the answer is verified.
You might ask why the time wouldn’t be the same since the average of 60 and 40 is 50, which is the average speed of the other group. The answer is that average of the two speeds is not the average speed, the average speed is known as the “harmonic” mean of the two speeds.
For Cathy and Dave, the total time will be (d/40+d/60) h. Because, they travelled the distance of 2d km in this time, the average speed will be 2d/(d/40+d/60) = 2/(1/40+1/60) = 48 km/h. This is the harmonic mean of 40 and 60 km/h.
For Sara and Johnny, the harmonic mean will be 2/(1/50+1/50) = 50 km/h.
Challenge
Prove that the geometric mean ≤ the arithmetic mean.
Prove that the harmonic mean ≤ the arithmetic mean.
Arithmetic mean = (x+y)/2
Geometric mean = √xy
Harmonic mean = 2/(1/x + 1/y)
Proof for geometric mean ≤ arithmetic mean
If x and y are both positive real numbers,
When x = y, (x-y)2 = 0.
When x ≠ y, (x – y)2 > 0 because the square of a positive or a negative real number is always positive.
Therefore, for all conditions (x – y)2 ≥ 0.
This means that x2 + y2 – 2xy ≥ 0.
Adding 4xy to each side and then dividing both sides by 4
x2 + y2 – 2xy + 4xy ≥ 4xy, or (x+y)2 ≥ 4xy or (x+y)2/4 ≥ xy or ((x+y)/2)2 ≥ xy.
Taking square root of both sides (x + y)/2 ≥ √ xy or √ xy ≤ (x + y)/2, the geometric mean (√ xy) or geometric mean ≤ arithmetic mean ((x+y)/2))
Prove that the harmonic mean ≤ arithmetic mean.
If x and y are both positive real numbers,
When x = y, (x – y)2 = 0.
When x ≠ y, (x – y)2 > 0 because the square of a real number is always positive,
Therefore, for all conditions (x – y)2 ≥ 0.
This means that x2 + y2 – 2xy ≥ 0.
Adding 4xy to each side, x2 + y2 – 2xy + 4xy ≥ 4xy, or (x + y)2 ≥ 4xy
Dividing both sides by 2(x + y), (x + y)/2 ≥ 2xy/(x + y).
Dividing the numerator and the denominator of the right hand side by xy, this becomes
(x+y)/2 ≥ 2/(1/y+1/x) or 2/(1/x+1/y) ≤ (x+y)/2
With the definitions of the harmonic mean (2/(1/x+1/y)) the arithmetic mean ((x+y)/2), it shows that the harmonic mean ≤ the arithmetic mean.
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