Solving Word Problems using Proportions 1

[ Silence ]
>> This is Part One of Solving Word Problems Using Proportions.
So there are a lot of word problems that you could use
by setting up a proportion.
And the idea is you want to keep things in proportion.
For instance, here's an example.
A recipe uses 5 cups of flour for every 2 cups of sugar.
If I want to make a recipe using 8 cups of flour,
how much sugar should I use?
We want to keep things in proportion
so the recipe tastes the same.
So one thing you can do is write down the proportion of flour
to sugar or sugar to flour,
and be clear how you're going to set that up.
So let's say I want to put the flour in the numerator,
how much flour I'm going to use,
and how much sugar in the denominator.
So I have two kinds of recipes.
Right. Recipe One, which is the way it is in the directions,
and then what I'm actually going to use.
I'll call that Recipe Two.
And we're going to set up a proportion,
which is an equation, where we've got the ratio of each.
So in the recipe book, I'm going to use 5 cups of flour.
Now remember flour goes up on top.
So I'm using 5 cups of flour and 2 cups of sugar.
Now I want to make a recipe using 8 cups of flour.
All right.
So flour goes in the numerator,
and what I don't know is the sugar.
So how about we let the amount of sugar I need be X. Okay.
So X is the amount of sugar that I want.
We set up a proportion, and now we can solve that proportion
to figure out how much sugar to use.
All right.
So remember how we could set up to do a proportion?
We could do the cross products,
five times X equals the other cross product,
which is 2 times 8, which is 16.
You can do that in your head.
And so we'll just divide both sides by five;
so our answer will be sixteen-fifths, right,
which is three and one-fifth.
Okay. So how much sugar do I need?
Use three and one-fifth cups of sugar.
You want to make sure you answer the question they ask you,
which is how much sugar you want to use.
If it was a word problem, you don't want
to just write X equals three and one-fifth or 3.2,
which also is correct, you could write 3.2.
You want to answer the question in words
because it was a word problem.
Now let's go back and see if that makes sense.
Three and a fifth cups of sugar.
So it's just a little over 3 cups of sugar.
Right. Let's see if that kind of makes sense.
You've got 5 cups of flour, you're using 8;
so you're not really doubling the recipe.
You're only using 8 cups of flour.
So if you doubled the recipe, you'd have 4 cups
of sugar but it's not doubled.
It's less than that.
Right. So three and one-fifth seems
like a reasonable amount to use.
So you should go back and to see if it seems reasonable.
[ Demonstration ]
>> Here's another problem.
A syrup is made by dissolving 2 cups of sugar
and two-thirds cups of boiling water.
How many cups of sugar should be used
for 2 cups of boiling water?
Okay. So we need to decide what, you know,
things we're talking about here.
We're talking about sugar and boiling water.
So all right, how about if I think of putting the sugar
in the numerator, and how much water is in the denominator
and I want to set up a proportion.
All right.
So see if you could set up the portion.
Put it on pause.
You're going to set up a proportion.
And you're looking for how much sugar could be used.
So how about letting X be how much sugar you're going to use.
All right.
So the original, quote, recipe uses 2 cups of sugar
for two-thirds cup of boiling water.
So in the numerator is the sugar
and the denominator's the water, two thirds.
Okay. How many cups of sugar should be used
for 2 cups of boiling water?
So the second part.
How much sugar?
I don't know, it's called that X,
and the denominator, the water is 2.
So that's what your proportion should look like, if you did it
with sugar over water.
If you did water over sugar,
the fractions would all be the reciprocals.
Okay. So now we can do our cross products.
So we could do two-thirds times X --
[ Demonstration ]
>> -- equals two times -- oops -- that's a mistake.
Sorry. -- 2 times 2, which is 4,
and now we can solve this equation.
There's different ways to solve this equation.
You can multiply both sides by three to eliminate fractions,
or what I would do is just multiply by the reciprocal,
three-halves, to isolate X, that, of course, there's more
than one way to solve this problem.
Okay. So then we have the 2 goes into the 4 twice,
2 times 3 is 6; so X is 6,
and we let X stand for how much sugar.
Right. For the sugar, we're going to use --
you set it up using the variable X.
So the question is how many cups of sugar should be used?
So you use 6 cups of sugar.
And now we want to make sure that answer makes sense.
All right.
So if X was 6, does that make sense?
Well, you're going from two-thirds cups to 2 cups,
and you're going from 2 cups of sugar to 6 cups of sugar, right,
and it ends up being 6, and that seems to make sense
because each is three times as much, right,
two-thirds times 3 is 2 and 2 times 3 is 6;
so that seems to make sense.
It's sort of like you're tripling your recipe
in this case.
[ Demonstration]
>> All right.
See if you could set this one up.
The school buys 8 gallons of juice for 100 kids.
How many gallons do they need for 175 kids?
See if you could write what the ratio looks like,
the original fraction, and then write a proportion.
Okay. We're talking about gallons of juice and kids.
So I'm just going to put juice over kids.
You could do it differently.
And we're going to make a proportion.
So I use 8 gallons for 100 kids.
Right. How many gallons are needed for 175 kids?
So I don't know.
I'm going to put X for what -- that's what I don't know,
how many gallons for 175 kids.
All right.
So there you go.
That's one way of writing your proportion.
Now if I went ahead and did my cross products here,
I have 100 times X --
[ Demonstration ]
>> -- equals the other cross product, 8 times 175.
[ Demonstration ]
>> Okay. Now you could go ahead and multiply 8 times 175, but --
and if you have a calculator, it's pretty easy to do.
The way I do this is by reducing things.
So I'm just going to go ahead and divide both sides by 100
and use canceling, because I never have my calculator around.
I like to use my mind.
All right.
So let's see, 175 and 100, you can cancel,
25 go into both of those numbers.
I think of quarters.
How many quarters are in a $1.75.
so I get 7 and 4.
Okay. And I could also cancel the 4 into the 8.
[ Demonstration ]
>> So my arithmetic ends up just being 2 times 7.
Now you would have gotten the same answer
if you would have done 8 times 175 and got 1400
and then divided by 100 and got 14.
So how many gallons of juice do they need?
Use 14 gallons of juice.
Does that seem reasonable?
Well, you're not doubling the number of kids.
Right. You're going from 100 kids to 175 kids.
If you doubled the recipe of how much juice you need,
you need 16 gallons and this is a little bit under.
So that answer seems reasonable.
Always look back and see if your answer seems reasonable.
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