Grades 7-8 Video Solutions 2025-2025_7-8

Video Transcription

Problem number 13. Edgar wants to write a number in each circle in the diagram. He wants each number to be equal to the sum of the numbers in the two adjacent  circles. He has already written two numbers as shown. What number should he write in the gray circle? To solve this problem, let's try to fill out the  numbers one by one and see how much information we can get. For example, in the diagram in the bottom left corner, you can see that I've marked one of the  circles with a question mark. We should already know what number should go here because we know that each circle should contain the sum of the numbers in the  adjacent circles. Since the circle with the question mark is next to 1 and 2, it must have the number 3. Now, let's look at the two circles now marked with a  question mark. Well, we don't exactly know what numbers are in these circles directly, but we know that the number 1, for example, is adjacent to a 3 and a  question mark, and similarly the 2 is adjacent to a 3 and a question mark. That is, we know that 3 plus the question mark has to equal 1 because those are the  circles next to 1. Similarly, 3 plus question mark has to equal 2 since those are the circles next to 2. And using those equations, we can solve for the  question mark. In the first case, the question mark is negative 2. In the second case, the question mark is negative 1. Now, we just have to fill out  the last circle, and we can see that this circle will be equal to the sum of the two numbers next to it, and that is negative 2 and negative 1. And we know  that for negative 2 plus negative 1, it was negative 3. That means that the number in the gray circle is D, negative 3.

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