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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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  • blobby green style avatar for user Harikesh
    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
    (12 votes)
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    • male robot hal style avatar for user Ben Willetts
      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!
      (0 votes)
  • male robot hal style avatar for user californiahotdogsnl
    Why do you have to multiply 1/3 into 1/3+9%
    (3 votes)
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  • hopper happy style avatar for user Rashel
    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?
    (2 votes)
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    • marcimus pink style avatar for user Gargarbl
      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).
      (2 votes)
  • hopper cool style avatar for user SV
    is there a practice session for this?
    (2 votes)
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  • aqualine sapling style avatar for user lee
    Why did he add 9%*1/3 to the original volume(1/3)? Is that how you find the volume...I forget. Sorry if this question is weird.
    (4 votes)
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    • leafers seedling style avatar for user Fieso Duck
      Hi Lee, what the video basically says in the beginning is that if you have 1/3 gallon of water and you freeze it, the volume of this 1/3 gallon will increase by 9%.

      So after freezing your 1/3 gallon of water it's volume will have increased by 9%, so you are left with 100%*1/3 gallon + 9%*1/3 gallon, which is 109%*1/3 gallon.

      Later in the video this result is worked out as a fraction.
      (0 votes)
  • female robot grace style avatar for user Allyson Whipple
    I know reducing and multiplying fractions have both been covered before, but the way the 9/100 * 1/3 expression was reduced was very confusing to me. Help?
    (1 vote)
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  • old spice man green style avatar for user harshvardhangalhotra
    is it important that denominator should be positive, while comparing rational numbers
    (0 votes)
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    • piceratops ultimate style avatar for user moomoosnake
      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.
      (4 votes)
  • marcimus pink style avatar for user late melissa
    Im confused why did he add? I thought he was going to multiply
    (1 vote)
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    • piceratops ultimate style avatar for user fr33d0g
      you have to add the original amount you started with, plus what it expanded to get the total amount. To do this you take the total volume you started with plus 9% of that volume. Which gives you your total volume.
      (2 votes)
  • male robot donald style avatar for user Aritra Das
    By what number should -33/8 be divided to get -11/2
    (1 vote)
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  • leaf orange style avatar for user Lori Doran
    why did he simplify the fraction first then add? and why did he add not multiply?
    (1 vote)
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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.