Maggie has a box of 300 colored blocks.The box has a equal number of green and blue blocks and an equal number of red and yellow blocks.If Maggie arranged all of the green blocks in stacks of 10 and all of the blue blocks in stacks of 7,how many yellow blocks are in the box?
clearly, if there are x stacks of green & blue, and y of red & yellow, then
20x+14y = 300
10x+7y = 150
y = (150-10x)/7
y = 10(15-x)/7
Since there must be an integer number of blocks, and 7 does not divide 10, it must divide 15-x
So, 15-x = 7 or 14
and y = 10 or 20
So, there may be 70 yellow and red, and 80 green & blue
or 140 yellow and red, and 10 green & blue
70
To find out the number of yellow blocks in the box, we need to determine the number of blocks in the box that do not belong to the green and blue colors. Given that there is an equal number of green and blue blocks in the box, we first need to find the number of blocks in each color.
Let's denote the number of green and blue blocks as G and B, respectively. We know that G = B and that the total number of blocks in the box is 300. So, G + B + R + Y = 300, where R represents the number of red blocks, and Y represents the number of yellow blocks.
Since there is an equal number of red and yellow blocks, R = Y. Substituting this in the equation, we have G + B + 2R = 300.
Now, we are given that Maggie arranged all of the green blocks in stacks of 10 and all of the blue blocks in stacks of 7. Therefore, the number of green blocks, G, must be a multiple of 10, and the number of blue blocks, B, must be a multiple of 7.
To find the possible values of G and B, we can start by assuming an arbitrary number of green blocks.
Let's assume G = 10. In this case, B = G = 10.
Now, substituting these values in the equation G + B + 2R = 300, we get:
10 + 10 + 2R = 300
20 + 2R = 300
2R = 280
R = 140
Since R = Y, we have Y = R = 140.
Therefore, when G = 10, B = 10, R = 140, and Y = 140.
So, there are 140 yellow blocks in the box.