Equations with Decimals

>> Professor Perez: Hey!
This is Professor Perez from Saddleback College, again.
Today, we're going to work with equations that have decimals in them.
And, let's get started, let's get Charlie out.
He better be ready to go!
Hey, Charlie, you ready to go?
>> Charlie: Yeah!
>> Professor Perez: Okay, today we're going to set up equations
that have decimals in them, right?
Now, don't get scared.
Now Charlie, what number must be subtracted from 0.34 to obtain 6.46
So I'm going to help you out Charlie.
Pay attention.
What number, x, must be subtracted from 0.34 to get 6.46 right?
Okay, to solve this equation for x, we're going to subtract 0.34 from both sides.
Hopefully you don't need a calculator for this.
0.34 subtract 0.34, those cancel and that's 0, right?
And on the right hand side, we're left with what, Charlie?
>> Charlie: 6.12.
>> Professor Perez: 6.12.
Very nice.
Notice, it's a negative x on the left hand side because we have that subtract x
which we're treating as a negative 1 times x, remember that.
Okay, now, remember, we're trying to find a 1 x,
not a negative 1 x. And so what do we do, Charlie?
>> Charlie: Divide by negative 1.
>> Professor Perez: Very nice!
And so our answer is x equals a negative 6.12 That is correct and you can check your answer
by plugging it into the original equation that we wrote down, right?
Anyway, let's move on Charlie.
Don't get scared!
A rental car company, Charlie, charges $22.50 a day and 15 cents a mile.
If you rented a car for 2 days and you drove 237 miles, how much will you be charged?
That's what we're going to figure out.
Well, our total cost, or our charge, is going to be 2 times $22.50 because you rented it
for 2 days and it's $22.50 per day.
So obviously 2 times $22.50 will give you the amount
that they are going to charge you for the 2 days.
Now also, we must add to this, Charlie, the charge for the mileage.
Well, how many miles did we drive?
>> Charlie: 237.
>> Professor Perez: 237.
That's right.
And each mile costs us how much Charlie?
>> Charlie: 15 cents.
>> Professor Perez: 15 cents.
Now, 15 cents as a decimal is 0.15 because it's 15 hundreths, right?
15 hundredths of a dollar is 0.15 and that is what 15 cents is.
We have to keep the dollar amounts the same.
Notice we have $22.50 which means 22 dollars and 50 cents, right?
The .50 is 50 cents.
Therefore the 15 cents must be written as a decimal because that's 15 hundredths.
All right, now, Order of Operations says we have to do what first, Charlie?
>> Charlie: Multiplication.
>> Professor Perez: Multiplication.
2 times $22.50, it's not too bad, is $45, right?
But notice we put $45.00 because we're talking about dollar amounts, right?
$45.00 means 45 dollars exact, right?
Plus, now, 237 times 15 cents gives us how much, Charlie?
>> Charlie: $35.55
>> Professor Perez: $35.55 which is 35 dollars and 55 cents.
Notice, both of our decimals are to the nearest hundredths
because we're talking about dollars, right?
All right, Charlie, now, what's 45 dollars plus 35 dollars and 55 cents?
>> Charlie: 80 dollars and 55 cents.
>> Professor Perez: 80 dollars and 55 cents.
That's your total amount.
That's how much you're going to be charged.
All right, Charlie, you got that one.
Let's do another one.
Here we go, Charlie.
Now, Candice, has 5 dollars and 45 cents in quarters and dimes.
Don't get scared.
She has 5 less dimes than she has quarters.
How many quarters and dimes does she have?
>> Charlie: I don't know!
>> Professor Perez: Well that's what we're going to figure out, Charlie.
Anyway, let's move on.
Let's let x equal the number of quarters.
And since there are 5 less dimes, the number of dimes must be x take away 5
because there's 5 less dimes than there are quarters, right?
All right, now, Charlie, if you had 4 quarters, how much money do you have?
>> Charlie: 1 dollar.
>> Professor Perez: A dollar because you can think of 25 cents times 4 as 1.00 Watch,
so if x represents the number of quarters,
then 0.25 25 cents times x represents the amount from the quarters.
And now, we have to also add the amount of the dimes.
Now, a dime is what decimal, Charlie?
>> Charlie: 0.10
>> Professor Perez: It's 0.10 because a dime is 10 cents of a dollar right?
It's 10 one hundredths of a dollar which is the 0.10 And now,
how many dimes do we have, Charlie?
>> Charlie: x subtract 5.
>> Professor Perez: x subtract 5.
So, the 0.10 times x subtract 5 represents the amount from the dimes.
So what you're seeing here is we're summing the amount from the quarters
and the amount from the dimes, right?
Those two amounts, when you add them together, gives you how much, Charlie?
>> Charlie: $5.45
>> Professor Perez: 5 dollars and 45 cents.
So it's like here's the quarters over here, here's the dimes,
this value plus this value must equal the 5 dollars and 45 cents.
All right, now, there's our equation.
Okay, so 0.25 times x, what do we have to do here, Charlie?
The distributive property, that's right.
0.10 times x is 0.10x, 0.10 times 5 is what, Charlie?
>> Charlie: 0.50
>> Professor Perez: 0.50, that's right.
And all of this has to equal 5.45, the 5 dollars and 45 cents.
Now, here we go, Charlie.
What do we do next?
>> Charlie: Combine like terms.
>> Professor Perez: That's right.
And 0.25x plus 0.10x is how much, Charlie?
>> Charlie: 0.35x
>> Professor Perez: Very nice there Charlie!
0.35x Bring down our work.
And now, what do we do next to solve for x, Charlie?
>> Charlie: Add 0.50 to both sides.
>> Professor Perez: That's right.
So let's go up there.
Let's add 0.50 to both sides, they cancel out there.
The left hand side is 0.35x, right hand side is what, Charlie?
>> Charlie: 5.95
>> Professor Perez: Very nice there.
Now, what do we do to solve for x?
Remember, we want 1 x.
>> Charlie: Divide both sides by 0.35
>> Professor Perez: Very nice there!
And we get that 1 x equals what, Charlie?
>> Charlie: 17.
>> Professor Perez: Very good!
That's some good calculator work.
All right.
So if x equals 17, remember, x minus 5 is what, Charlie?
>> Charlie: 12.
>> Professor Perez: 12.
x represents the number of quarters and x subtract 5 equals the number of dimes, right?
So there we have it.
17 quarters and 12 dimes.
If you take 17 times 0.25 and 12 times 0.10, add those two numbers up,
obviously you get 5.45 which is 5 dollars and 45 cents.
Very nice there Charlie!
Now, those coin problems are tough!
It's the first time you saw that, but we're going to do a lot more here in pre-algebra
and you're going to do a whole lot more in Beginning Algebra.
That's next semester!
Anyway, we'll see you all again soon!