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   Alison Kiddle (alisonkiddle@mathstodon.xyz)'s status on Saturday, 21-Sep-2024 18:20:09 JST Alison Kiddle

   Here's a question that I'm interested to know how people will answer:

   I'm travelling by car. If I drive for half the journey at 60mph and half the journey at 40mph, what will be my average speed for the whole journey? How did you work it out?

   • GreenSkyOverMe (Monika) repeated this.
   • Embed this notice
     GreenSkyOverMe (Monika) (greenskyoverme@ohai.social)'s status on Saturday, 21-Sep-2024 18:20:06 JST GreenSkyOverMe (Monika)

     in reply to

     @alisonkiddle Half the journey by time or half the journey by distance?

   • Embed this notice
     Annika Backstrom (annika@xoxo.zone)'s status on Saturday, 21-Sep-2024 18:29:31 JST Annika Backstrom

     in reply to

     @alisonkiddle If you drive for one hour at 60mph, then one hour at 40mph, that's 100 miles in 2 hours, an average of 50mph. The average is the same for any length of travel (30 minutes per leg = 30 miles + 20 miles = 50 miles in 1 hour)

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     GreenSkyOverMe (Monika) (greenskyoverme@ohai.social)'s status on Saturday, 21-Sep-2024 18:44:48 JST GreenSkyOverMe (Monika)

     in reply to

     @alisonkiddle
     If it’s by time: 50mph.

     If it’s by distance: t1 hours * 40 mph = t2 hours * 60 mph (so t2 = 2/3 t1)
     We are looking for total distance / total time = (t1*40mi/h+t2*60mi/h)/(t1+t2)
     = 2*t1*40mi/h / (5/3 t1)
     = 80 mi/h * 3/5
     = 48 mi/h

     (I don’t know if the sign for miles is actually mi.)

   • Embed this notice
     Alison Kiddle (alisonkiddle@mathstodon.xyz)'s status on Saturday, 21-Sep-2024 18:44:49 JST Alison Kiddle

     in reply to

     • GreenSkyOverMe (Monika)

     @GreenSkyOverMe I left it deliberately ambiguous to see whether/what people would assume!

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     Annika Backstrom (annika@xoxo.zone)'s status on Saturday, 21-Sep-2024 19:48:06 JST Annika Backstrom

     in reply to

     @alisonkiddle common north america thing to measure travel by time rather than distance :D

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     Alison Kiddle (alisonkiddle@mathstodon.xyz)'s status on Saturday, 21-Sep-2024 19:48:07 JST Alison Kiddle

     in reply to

     • Annika Backstrom

     @annika nice, thank you for sharing. I deliberately didn't specify whether it was half the journey by time or distance, because I wanted to see which of those would be the natural assumption

   • Embed this notice
     Karen Campe (karencampe@mathstodon.xyz)'s status on Sunday, 22-Sep-2024 04:10:53 JST Karen Campe

     in reply to

     @alisonkiddle
     Second method!!
     Notice that the harmonic mean of 60 and 40 is 5/120

     \[ \frac{1}{60} +\frac{1}{40}= \frac{5}{120} \]

     The answer to the average rate question is the reciprocal of half the harmonic mean.

     \[ \frac{1}{2(\frac{1}{60}+\frac{1}{40}) }\]

     \[ \frac{1}{\frac{5}{240 }}\]

     Harmonic mean is a way to average rates.

     When this was shown to me, I was amazed! It isn't usually taught to HS Ss in US Alg 2 when they see problems with rate of work.

   • Embed this notice
     Karen Campe (karencampe@mathstodon.xyz)'s status on Sunday, 22-Sep-2024 04:10:55 JST Karen Campe

     in reply to

     @alisonkiddle love this (and yes very confusing for many students)

     Must find out total distance divided by total time to get average speed, not just average the speeds.
     Two methods:

     Assume whole distance = 120 for ease of calculation, then each half of journey = 60
     Distance = Rate • Time or D = RT
     60 = 60 • T
     1st part takes 1 hour
     60 = 40 • T
     2nd part takes 1.5 hours

     Total distance = 120
     Total time = 2.4 hours
     Ave speed = 120/(5/2) = 240/5 = 48 mi/hr or 48 km/hr

     GreenSkyOverMe (Monika) repeated this.
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     Karen Campe (karencampe@mathstodon.xyz)'s status on Sunday, 22-Sep-2024 04:11:02 JST Karen Campe

     in reply to

     @alisonkiddle yes me too!! Been meaning to write a post along the lines of "what does the mean mean?"

   • Embed this notice
     Alison Kiddle (alisonkiddle@mathstodon.xyz)'s status on Sunday, 22-Sep-2024 04:11:04 JST Alison Kiddle

     in reply to

     • Karen Campe

     @KarenCampe I love the harmonic mean!

     GreenSkyOverMe (Monika) repeated this.