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### Course: 7th grade > Unit 5

Lesson 6: Rational number word problems- Rational number word problem: school report
- Rational number word problem: cosmetics
- Rational number word problem: cab
- Rational number word problem: ice
- Rational number word problem: computers
- Rational number word problem: stock
- Rational number word problem: checking account
- Rational number word problems

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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.

## Want to join the conversation?

- 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?(24 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).(25 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(14 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!(11 votes)

- is it important that denominator should be positive, while comparing rational numbers(5 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.(9 votes)

- Why do you have to multiply 1/3 into 1/3+9%(6 votes)
- I'm confused.... how did he went from 1/3+3/100 to 100/300+9/300?(4 votes)
- It is easier to find the LCM of the denominators and then add the numerators while keeping the denominator the same. The LCM of 3 and 100 is 300, so the denominator becomes 300, and you multiply both numerators by the number it takes to get the denominator to 300.(2 votes)

- 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)

- Sal divided the numerator of 9/100 and the denominator of 1/3 by a number, instead of dividing both numerators or both denomenators. *Why?*(1 vote)
- It's called cross cancelling. When multiplying fractions, you can divide out common factors before multiplying, or after multiplying.

If you multiply 1st: 9/100 * 1/3 = (9*1)/(100*3) = 9/300. Reduce the fraction by a common factor of 3 and you get 3/100

Sal is essentially doing the same thing, but he is dividing out the common factor, then multiplying.

See examples in these videos:

1) https://www.khanacademy.org/math/cc-fifth-grade-math/5th-multiply-fractions/imp-multiplying-fractions/v/multiplying-fractions

2) https://www.khanacademy.org/math/cc-fifth-grade-math/5th-multiply-fractions/imp-multiplying-mixed-numbers/v/multiplying-mixed-numbers

Hope this helps.(4 votes)

- Im confused why did he add? I thought he was going to multiply(1 vote)
- 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.(4 votes)

- what is the definition of ratio(1 vote)
- 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?(2 votes)

## 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.