Each of the cucumbers in 100 pounds of cucumbers is composed of 99% water, by weight. After some of the water evaporates, the cucumbers are now 98% water by weight. What is the new weight of the cucumbers, in pounds?
Responding to a pm:
There are two ways of getting to the answer here.
Let's go forward to back. Say we have 98% mixture (2% pure cucumber extract and 98% water). Say, we also have plain water (100% water). If we mix these two, we get 100 pounds of 99% mixture (1% pure cucumber extract and 99% water). We want to find the amount of pure water that was mixed. (Usually, we are given two components and we have to mix them. This question does the opposite. It separates out the components of the mixture but overall it doesn't matter to us at all.)
w1/w2 = (A2 - Aavg)/(Aavg - A1) = (100 - 99)/(99 - 98) = 1/1
The 98% mixture and the pure water must have been mixed in 1:1 ratio. Since total mixture is 100 pounds, the amount of pure water must be 50 pounds and the 98% mixture must have been 50 pounds.
Another way to think about this problem: The amount of pure cucumber extract stays the same. Only the amount of water reduces. The amount of pure cucumber extract was 1 pound in the 99% mixture and it stayed 1 pound in the 98% mixture.
In 100 pounds of 99% cucumber mix, this pure cucumber extract was 1% and in 98% cucumber mix, this same pure cucumber extract was 2% of the cucumber mix.
Say, the total weight of 98% cucumber mix is x pounds.
1 pound = (2/100)*x
x = 50 pounds
The answer is a little un-intuitive since you don't expect so much water to disappear to just make it go down from 99% to 98% but instead, think that you need to make the amount of pure cucumber extract to go up from 1% to 2% i.e. you need it to double in proportion. For the same amount (1 pound) to be double in proportion, you need to reduce the total weight by half.
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