Rational number word problem: ice (video)

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7th grade

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• Rational number word problem: ice
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• Rational number word problems

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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• I’m confused. Why don’t you just multiply 9 percent by 1/3? Why do you have to add 1/3 to the product of 1/3 times 9? That is what confuses me. Is it because when you times 9 by 1/3 that only tells you how much the water expands? Then you have to add that amount to 1/3 to get the total amount of volume of water?(22 votes)
  • Yes, that’s exactly it. Multiplying 1/3 by 9% will give you the amount of additioal volume that will be added when the water freezes, but to get the total volume of the ice, you’ll have to add that addtional amount to the amount before freezing. The other way to do this is multiplying 1/3 by 109% (that’s the same as multiplying it by 1.09).(22 votes)
• Hi In this example sal took 9% of 1/3 and then added with original 1/3 = 109/300 = 0.3633
  What would have happened if we evaluated 1/3 = 0.33 and then taken 9% of 0.33 = 0.0297
  Hence final answer would have been 0.33 + 0.0297 = 0.3597.
  So can we say that taking % of fraction is more precise than converting it into decimal ?
  Thanks(13 votes)
  • Yes, usually working with fractions is more accurate than working with decimals, because when we use decimals we quite often create errors by rounding the numbers. You’ve given a good example, by saying that 1/3 = 0.33. It’s not: 1/3 = 0.33333333… and so on forever. Try taking 9% of that instead — you’ll see that you get 0.03 and not 0.0297, so the final answer becomes 0.36333333…, and again so on forever. By contrast, the fraction 109/300 is totally accurate!(9 votes)
• is it important that denominator should be positive, while comparing rational numbers(4 votes)
  • Good question. It makes it simpler to do the comparison if we don’t have to worry about negative signs in the numerator and denominator. We can still compare them, but it’s harder to do and we’re more likely to make mistakes.
    Making sure that the denominator is positive is one of those maths rules that doesn’t actually change the value of anything, it just makes it easier for us to use. It’s like using capital letters at the beginning of sentences – that doesn’t change the meaning of a sentence, it just helps us when we’re reading by making the beginning of the sentence more obvious.(8 votes)
• Why do you have to multiply 1/3 into 1/3+9%(5 votes)
• I’m confused…. how did he went from 1/3+3/100 to 100/300+9/300?(4 votes)
  • Adding fractions become easier when the denominator is the same. Thats what Sal did!(1 vote)
• i wonder, does water actually enlarge by 1/3 when frozen or is this bogus?(3 votes)
  • I believe so. My brother once put a can in the freezer and it exploded, because liquid expands when frozen.(2 votes)
• what is the definition of ratio(2 votes)
  • the quantitative relation between two amounts showing the number of times one value contains or is contained within the other.(3 votes)
• is there a practice session for this?(3 votes)
• is thie sal khan (the owner) is he talking(0 votes)
  • Sal Khan is the founder and CEO of Khan Academy.(7 votes)
• Hello I had a question why does he add 1/3 plus 1/3 plus 9 percent wouldn’t 1/3 plus 1/3 be 2/3 making the answer 2/3 plus 9 %(3 votes)
  • Thats another way to do it, he did it in another order :)(1 vote)

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.