Grades 9-10 Video Solutions 2014-Levels 9&10 Video Solutions 2014 problem10

Video Transcription

Question number 10. The big wheel of this bicycle here has a perimeter of 4.2 meters and the small  wheel has a perimeter of 0.9 meters. At a certain moment the valves of both wheels are at their  lowest point. The bicycle then rolls to the left. After how many meters will both valves first be  at their lowest point together again? So just to make sure we understand what's going on, let's  mark here a position on the wheels. Let's say that's a valve on the big wheel and at the same  position at the very bottom, so touching the ground, we have a valve on the small wheel and  then the bicycle here rolls along to the left like that. So let's say this is going in that  direction and so at some point the bicycle will again end up in this exact position after the big  wheel rolls around and the small wheel probably more than once and the valves will line up again.  Okay, so let's calculate how many revolutions we have per revolution of the big wheel of the small  wheel. So every time the big wheel revolves exactly once, we move 4.2 meters and at the same time  we see that the small wheel will revolve 4.2 meters divided by its perimeter so 0.9 meters  that gives us 42 over 9 which is 36 over 9 so 4 and 6 ninths which we can simplify as 4 and 2 thirds so every one revolution of the big wheel the small wheel revolves  four and two-thirds of the time and then we count the revolutions so we have the  big wheel and then the small we have one revolution here then the small wheel moves  around four and two-thirds of the time if we have two revolutions so that would be  four and two-thirds plus four and two-thirds which gives us eight and  four-thirds and then after one more revolution of the big wheel we have eight and four-thirds plus  four and two-thirds which gives us 12 plus six-thirds and that is equal to  14 revolutions so here this is what the problem was describing we have an integer number of  revolutions of the big wheel and an integer number of revolutions of the small wheel so that is when  they line up again like in the diagram and how far do we have to travel again well this is after  the big wheel which makes three revolutions at 4.2 meters and that is 12.6 meters after we multiply that out and that is answer c so the large  wheel has to move and the bicycle indeed has to move 12.6 meters for the valves to line up again

Video Summary

The problem involves a bicycle with two wheels of different perimeters: 4.2 meters for the large wheel and 0.9 meters for the small wheel. The task is to find out how far the bicycle must travel for both wheel valves to align at the lowest point simultaneously again. The large wheel revolves once every 4.2 meters, and during this, the small wheel makes 4 and 2/3 revolutions. After analyzing the revolutions needed for alignment, it is determined that after the large wheel completes 3 revolutions (12.6 meters), both valves will align. Thus, the answer is 12.6 meters.

Keywords

bicycle

wheel alignment

large wheel

small wheel

revolutions