MHB How many glasses total are there?

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The problem involves calculating the total number of glasses at a restaurant, where 2/7 are green and the remaining are white, with 210 more white glasses than green. The equations derived show that the total number of glasses, T, can be expressed as T = W + G = 210 + 2G. By substituting G with 2/7 of T, the solution leads to T = 490. This method efficiently confirms the total number of glasses without extensive factor listing.
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the problem reads. 2/7 of the glasses at a restaurant are green. the rest are white. 210 more are white than green. How many glasses total are there.

I was only able to come up with 490 but that was by listing all the factors of 7 and just checking each one until I found that 490 works. any help on how to do this problem quickly and more efficiently would be great.

thanks,
 
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Hi amyjojoja and welcome to MHB! :D

Let $T$ be the total number of glasses, let $W$ be the number of white glasses and let $G$ be the number of green glasses:

$$T=W+G=210+2G$$

$$\frac27T=G\Rightarrow\frac27T=\frac{T-210}{2}\implies T=490$$
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...

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