Grades 9-10 Video Solutions 2021-video 2021 9-10/16

video 2021 9-10/16

Video Transcription

Problem number 16 states, in a team competition there are 5 teams waiting to start. Each team consists of either only boys or only girls.  The number of team members are 9, 15, 17, 19 and 21. After all members of the first team have started, the number of girls who have not started yet  is 3 times the number of boys who have not started yet. How many members are on the team that had already started?  So we know that the first team had to have either 9, 15, 17, 19 or 21 kids on it. If we want to know the total number of kids we can just add up all of these values.  The sums of the numbers of kids on each individual team will be the total number of kids. Adding that up we get 81 students.  So let's say that X of them are on team 1, the team that went first. That means that 81 minus X of them are still at the starting line after team 1 leaves.  We can call this variable Y. The problem then tells us that 3 fourths of the total kids left are boys and 1 fourth are girls.  Mathematically speaking we can represent this as 3 fourths Y and 1 fourth Y. This means we know that 3 fourths Y and 1 fourth Y must be integers as fractional boys  and girls doesn't make any sense. But more importantly we know that 1 fourth Y and 3 fourths Y must be a valid sum of team members.  We know this because there are only boys or girls on a team so 1 fourth Y might be one big team of girls or two smaller teams of girls.  So let's calculate these values and see how many kids are on the team that have already started, X. So these are the possible values of X. Now let's compute 81 minus X.  So we simply take 81 and subtract it from every possible value of X.  This is how many possible kids there are remaining at the start line after team 1 leaves. Now let's take 1 fourth of these values and disregard any that are not integers and  any that are not the sum of the team members. So in our case 16 and a half and 15 and a half are not integers and 18 and 16 cannot be made as a sum of the yellow numbers but 15 can.  And this would mean that there were 60 students after the first team left and that the first team had 21 people on it. So that's our answer.  So the question asked us, how many members are on the team that has already started? The answer is 21, letter E.

Video Summary

The problem involves five teams with either only boys or girls, having 9, 15, 17, 19, and 21 members each, totaling 81 members. After the first team starts, the remaining girls are three times the number of remaining boys. This condition is expressed as 1/4 of the remaining members being boys and 3/4 being girls. Calculating possible team start sizes and verifying which matches the given condition, it’s determined that the first team, which already started, had 21 members. Hence, the team that went first consisted of 21 members.