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Rational number word problem: ice

Word problems force us to put concepts to work using real-world applications. In this example, determine the volume of frozen water and express the answer as a fraction. Created by Sal Khan.

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Video transcript

Most liquids, when cooled, will simply shrink. Water, on the other hand, actually expands when it is frozen. Its volume will increase by about 9%. Suppose you have 1/3 of a gallon of water that gets frozen. What is the volume of the ice that you now have? So you're starting with 1/3 of a gallon of water. They tell us that when it gets frozen, when it turns into ice, its volume is going to expand by 9%. So the new volume is going to be your existing volume. So this is the original volume, 1/3 of a gallon, and it's going to expand by 9%. So your frozen volume is going to be your original volume plus 9% of your original volume. So you could say it's 9% times 1/3. So this right over here is going to be the expanded volume. Now, there's a bunch of ways we can figure it out. We could turn things to decimals or whatever else, but they tell us to express your answer as a fraction. So let's make sure that everything here is a fraction, and then we'll just try to simplify. So the one thing that's sitting here that is not a fraction is our 9%. Well, what does 9% actually represent? Well, 9% literally means 9 per 100. So we could rewrite this as-- so this is going to be equal to 1/3 plus, instead of writing 9%, I'll write that as 9 per 100, and then once again times 1/3. And we can simplify this expression right over here. We have a 9 in the numerator, a 3 in the denominator. If we divide both of them by 3, we get a 3 and a 1. And so we're left with 1/3 plus 300 times 1/1. Well, that's just going to be 3/100. So this is just going to be equal to 1/3 plus-- I'll write this in orange still, or maybe I'll do it in a new color-- plus 3/100. And now we have to add something, or two numbers that have different denominators. So let's find a common denominator. So this is going to be equal to, well, the least common multiple of 3 and 100. And they share no common factor, so it's really just going to be the product of 3 and 100-- the least common multiple is 300. So it's going to be something over 300 plus something over 300. Now to go from 3 to 300, in the denominator you multiply by 100, so you have to multiply the numerator by 100 as well. So 1/3 is the same thing as 100/300. And to go from 100 to 300, we have to multiply by 3 in the denominator, so we have to multiply by 3 in the numerator as well. So 3/100 is the same thing as 9/300. And now we're ready to add. This is going to be 100 plus 9/300, which is 109/300. So this is the volume of ice that I now have expressed as a fraction.