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

Video Transcription

Question 17. The numbers 1 to 8 are placed, once each, in the circles shown. The numbers by the arrows show the products of the three numbers in the circles on that straight line.  What is the sum of the numbers in the three circles at the bottom of the figure?  Let's take a closer look at the diagram and list out all these numbers, which can only be used once each.  The first thing to notice is that there are two numbers that are divisible by 5, 30 and 105. So, we should put the 5 in the circle that is shared by those two numbers, like so.  Next, there are only two numbers that are divisible by 7. So we will put 7 here, at the 105 and 28. Next, we notice that 105 is only missing one number.  So the one number that we have to multiply by is 3, so we can take that out. Now the next step is finding out the numbers that will multiply up to 30.  Since we already have 5, we can do 30 divided by 5, and this gives us 6. Normally, we can break 6 down into 1 times 6 and 2 times 3.  However, we have already used our 3, so that will not be a valid option. We are left with 1 and 6 for the two circles at the top, so we can put them in, like so.  The reason that we are able to put the 1 in the first circle is because the number it has to multiply to is 28, which is possible when we have a 1.  So the 1 versus the other 1 has to multiply towards 144, so we need the 6 in the second circle. Next, it is just a matter of filling out the numbers that are missing.  We can put in the 4 here, since 1 times 4 times 7 is 28. Then we can put the 2 here, since 2 times 4 times 6 is 48, and we are left with the 8.  And all we are left to do is add together the three bottom numbers, 2, 7, and 8. And that will give us our answer, which is D, 17.

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