Grades 11-12 Video Solutions 2025-2025_11-12

2025_11-12_24

Video Transcription

This is problem 24 on the grade 11 and 12 2025 math kangaroo. Julia and her little sister Paula  go out together for a bike ride. Both ride at a constant speed, Julia at 18 kilometers an hour  and Paula at 12 kilometers an hour, and they follow the same path. Julia feels tired after 20  minutes and decides to go back. When she meets Paula, Julia tells her to turn around and both  return home, each at her own speed. How many minutes later than Julia will Paula arrive home?  Our answer choices are A, four minutes, B, six minutes, C, eight minutes, D, 10 minutes, and E, 15 minutes. Take your time, pause the video and try and solve it yourself.  Okay. For me, it always helps to sort of draw this one out, especially on tests.  So they have their starting point here,  and I'll use blue for Julia and green for Paula.  So Julia bikes out for 20 minutes and she gets to right here.  In 20 minutes, that means she rode 20 over 60 times 18 kilometers,  which means that this distance is six kilometers. And during that time,  her sister Paula went at two-thirds of that pace and therefore got to four kilometers.  Julia then doubled back, and we can denote that down here. So she doubled back,  and during this time, you know, Paula was riding forward. So they met somewhere in the middle here, and where they met in the middle, that doesn't matter  for now. But what does matter is that these two distances sum to those missing two kilometers  from Paula's initial ride. So that means after they met, like at exactly the time they met,  they had both ridden 12 kilometers in total. So 12 km in total from both of them.  And because Paula is running at two-fifths of the speed of her sister, then that means that Paula,  at the time that she meets, has ridden 24 over 5, or two-fifths times 12 kilometers,  and then must bike home from that point. So that means that Paula has ridden 24 over 5 kilometers. They're both biking home 24 over 5 kilometers,  but Julia is going to do it a lot faster than Paula will.  Okay, so then, yeah, now we just need to determine exactly how much time they both take on this. So Paula, return to blue, Paula on her return home  is going 18, yeah, this is unchanging, so she's going 18 kilometers an hour.  And therefore takes 24 over 5 kilometers over 18 kilometers per hour.  And doing this division, we get that this turns into 4 over 15 hours, or, not hours, but hours, or 16 minutes. So it takes her 16 minutes to get home. And Paula goes at two-thirds  her pace, and therefore takes three halves her time. So I'll write that in green. So yeah, Paula is going at two-thirds the pace, and therefore takes three halves the time.  And then three halves of 16 is 24. So Paula takes 24 minutes.  Takes 24 minutes compared to the 16 minutes, so minus 16. That means that there's an eight-minute differential between the two of them, so the answer is eight.

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