For every red flower in a garden, there are three blue flowers. For every two blue flowers, there are five purple flowers. For every flower of these colors, there are six yellow flowers. If there are fifteen purple flowers, how many flowers are there in all?

*Solution to yesterday’s problem:*

To solve this problem we need to know that for any triangle, the sum of two sides *must* be greater than the length of the other side. So if we call *x* the length of the missing side, we can write these inequalities:

or or

If we know that x is between 6 and 26, and not including 6 and 26, we can add all these values together to find the answer. Here’s a quick way to do 7 + 8 + 9 + … + 24 + 25:

Notice that the sum of the first and last terms, 32, is the same as the sum of the second and second-last terms. Also notice that we are adding 19 terms overall (26-6 and minus one for including 26). So to get the sum of the terms, we multiply 32 by half of the number of terms: 32 x 9.5 = *304*.

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