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

Video Transcription

This is problem 20 on the grade 11 and 12 2025 math kangaroo contest. When grandma started knitting woolen socks, she had a huge ball of yarn with a diameter of 30 centimeters.  After finishing 70 socks, she still has a ball of yarn with a diameter of 15 centimeters. How many more socks can grandma knit with the yarn that is left?  We have the answer choices of A, 70 more socks, B, 50 more socks, C, 30 more socks, D, 20 more socks, and E, 10 more socks. Take your time. Pause the video here. Try to solve it.  Okay. So there are a few like implicit assumptions that we're making. So assume that like the number of socks that grandma can produce is like directly proportional to volume.  And then we can also assume that like our grandma lives in three dimensions and that's working with a like three dimensional ball. Or you know, you may know this as a sphere.  So yeah, we have a sphere. And that the number of socks is proportional to the volume of said sphere.  So then we have that the old diameter, old diame, is equal to two times the new diame. And I'm just going to note these as like DN and DO. So DO equals two times DN.  So then if we think back to our like sphere volume result, we get that it's four over three pi times R cubed.  So then yeah, if D old equals two times D new, then we have that R old is equal to two R new. And like putting this back in, we get that the old volume is equal to eight times the  new volume. But because we're working with like the old, the same sphere, what we've done is we've taken like eight new volumes and we've sort of subtracted seven of them away.  So minus seven new to obtain like the one new volume that we're working with. So that means we've taken like seven of the eight or seven over eighths the volume to create 70 socks.  Then we only have one over eight the volume left. And this corresponds to just exactly 10 new socks because one eighth is one seventh of seven eighths.  So we just take one seventh of 70. Then we have 10 new socks that can be made.

Video Summary

Grandma started with a ball of yarn with a diameter of 30 cm and, after knitting 70 socks, she was left with a smaller ball with a diameter of 15 cm. Since the number of socks is proportional to the volume of the yarn, and assuming a spherical shape, initially the yarn volume was eight times that of the remaining yarn. Seven-eighths of the initial volume was used for the first 70 socks, leaving one-eighth of the volume, sufficient to knit 10 additional socks. Therefore, Grandma can knit 10 more socks, corresponding to option E.

Keywords

knitting

yarn volume

socks

spherical shape

proportional