Grades 7-8 Video Solutions 2022-2022_7-8

Video Transcription

Question 28. Several points were marked on a line. Renard then marked another point between each pair of adjacent points on the line. He repeated this process a further three times.  There are now 225 points marked on the line. How many points were marked on the line initially?  So let's start off drawing a line.  Let's start off by drawing two points on the line. If we imagine that this is the question, Renard first marks one point between each pair of adjacent points.  So in this instance, only one. He then repeats the process three times. So he would draw lines between each of these, like so. Then he repeats, and repeats, like so.  Now, in this instance, we started off with two points. He then added one less point than there were in total. So just one. And he got three.  Then, he added one less total than there were, so he added two. Then we had five. Again, he added one less than there were before. So he added four, which gives us nine.  And he added one less than that, which is eight. So we ended up with 17 points. To figure out this question, we simply have to follow this pattern for every other answer.  Let's start off with A, 10. If we start off with 10, in the first instance, he will draw one less point than 10. So he adds nine. This gives us 19.  He draws one less than that, adding 18, giving us 37. He draws one less than that, adding 36, giving us 73. And then he draws one less than that, adding 72, giving us 145.  So 10 could not have been the answer. Next, we try 12. Here, Renard would have added 11, since it is one less than 12, giving us 23.  Then, adding 22, since it is one less than 23, getting 45. Adding 45, getting 89. Adding 88, getting 177. Again, this is not the amount of points that we ended up with.  So this is not the answer. If Renard started with 15 points, he would have added 14, having 29. He then repeats this process, adding one less, getting 57.  Adding one less, getting 113. And then, one less, 112. 113 plus 112 gives us 225. So the answer will be C, 15.

Video Summary

Renard marked several points on a line and added a point between each pair of adjacent points, repeating this process four times. Now, there are 225 points. Initially, calculating with different starting points, we see that beginning with 15 points fits the conditions: <br /><br />1. Start with 15 points. <br />2. Add 14 (15-1), resulting in 29 points.<br />3. Add 28 (29-1), resulting in 57 points.<br />4. Add 56 (57-1), resulting in 113 points.<br />5. Add 112 (113-1), resulting in 225 points.<br /><br />Thus, the initial number of points was 15.

Keywords

Renard

points

line

sequence

calculation