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>> This is a coin problem.
Ed has 6 more quarters than dimes.
The total value of his coins is $6.05.
Find out how many of each coin Ed has.
Well of course you could guess right?
You can say well let's see what
if I had I've got you know 6 more quarters than dimes
so let's say I've got 10 dimes,
and let's say I have then 16 quarters,
and we could just see how much each of these is worth.
Well if you had 10 dimes then each dime is worth 10
cents right?
So you'd have to multiply each one by 10 cents
and I'm gonna put this in cents, that would be 100 cents,
and if we had 16 quarters, 16 and each one is worth 25 cents
and that ends up being 400 cents and adding together
that would mean you've got 500 cents or $5 but the total value
of his coins is $6.05
so for instance this is not the correct answer.
Well we want to do this algebraically as opposed
to just keep changing it like next I could try how
about 11 dimes and 17 quarters, or 12 dimes and you know 6 more
than that, 18 quarters etc. So let's do this algebraically
but keep in mind that how many
of each coin you have is different
from how much its worth and I'm writing the worth in cents
but you could have done it in dollars and cents as well.
So let's start again.
Ed has 6 more quarters than dimes.
I'm gonna make up a chart for us to keep track
of some information here
and we're gonna define our variables right in the chart
so we're talking about 2 kinds of coins right
so this is the type of coin, type of coin,
well we've got dimes and we've got quarters right now what I'm
gonna do next is write what is called the unit value,
in other words how much is a dime worth?
And I'm gonna do everything in cents
to avoid working with decimals.
We know that each dime is worth 10 cents
and we know each quarter is worth 25 cents so that's a given
from what you know about the worth of dimes and quarters.
Now what we don't know is how many of each
so we could write how many or you could write the word number,
whatever is easiest for you.
What do we know about the number of dimes
and the number of quarters?
Well we know that Ed has 6 more quarters than dimes
so if I knew how many dimes there were then I would add 6
to that number and that'll tell me how many quarters.
So the key here is that we want
to define our single variable as dimes.
How about D for dimes?
Now you can use an X or an N. I'm using D so I remember
when I'm done if I solve for D that'll be the number of dimes.
Alright so if I have D dimes and I have 6 more quarters
than dimes I've got D+6, that's how many.
So so far on my chart I'm just keeping track how much each dime
and quarter is worth and then how many of each I'm going
to use a variable and the trick is the value.
Alright so what this means is in money.
Well to get the value for 10 dimes you would take 10 cents
for each dime so you're gonna multiply the unit value times
how many you have will give you the value of all the dimes.
So in this first row 10 times D will be just 10 D. Now
for the quarters be careful here you've got D+6 quarters you've
gotta do 25, 25 cents for each of those quarters right?
So you've gotta do 25 times that whole quantity
so you're doing 25 times the entire amount which is D+6
so 25, you must put that in parenthesis D+6.
Now if you want you could make a note here that the total value,
sorry for that writing, the total value we know is $6.05
and I'm doing everything
in cents see how I did the unit value in cents so this is 605
and what is true then is the total value
of the dimes plus the total value
of the quarters is the total value of all the coins
and that is the equation we want.
Take the value of the dimes, 10D,
that's the dimes plus the value of the quarters
which is 25 times D+6= 605 cents and that's the equation.
Alright that's the hard part, getting the equation.
The rest of it will be easy
and then we will go back and check our answer.
So go ahead and put this on pause and solve it.
I'm gonna do it quickly here.
We have 10D+25D now when I do 25 times 6 remember
to distribute you're gonna get 150=605 and over
on the left here I can add the like terms.
10D+25D and I'm also gonna subtract 150 from both sides
at the same time so that I get 35 D = 455
and then I've gotta divide both sides by 35;
you could use a calculator or do this long hand.
So we get D=13.
Now remember what D stood for.
D stood for the number of dimes right
and we know we had 6 more quarters so D+6, that'll stand
for the number of quarters, so now we're gonna check our answer
by making a little chart and see if this all makes sense.
Alright so if we look at the chart again we now know
that D is 13 so I am going to put that in
for the number of quarters.
We know that I'm sorry for the number of dimes,
so we have 13 dimes, 10 cents each
and I'm gonna do this in real money.
This is a dollar and 30 cents right cause you do 10 times 13
so the real value in dollars
and cents would be the decimal points.
You could keep it in all in cents as well if you'd like.
Now how many quarters?
Remember there were 6 more quarters
so we've got 19 quarters so we also need to figure
out the value of the quarter so you do .25 times 19
which is $4.75 and then to check it, it says it was 605,
actually add it up and see what you get and you get 605
and that checks from the original problem cause the
original problem if we look back up says that we've got the value
of all the coins $6.05.
So here's the original problem Ed has 6 more quarters
than dimes, the total value is 605 and so here we are
when we checked it we have 6 more quarters than dimes right
and the total value is 605
so now we're ready to write the answer.
It asked how many of each: Ed has 13 dimes and 19 quarters.
If it just asked us for how many quarters we would just say oh he
has 19 quarters.
If it asks just for the number
of dimes you would say we have 13 dimes.
We have to remember we're gonna answer the question
from the original problem and so this is how to do a coin problem
and of course you could have kept this as just 1025
and not used an decimals and the check still works the same.
If you want you can also do your check a little bit differently
you could say I've got 13 quarters at 25 cents
and then just write 475 so you could say ok I'm checking
out the quarters, I'm checking out the dimes,
and then I had I'm sorry I had 19 quarters,
25 cents each was 475, and then I had 13 dimes,
10 cents each was 1.30.
Add it up so you don't have to put it back in a chart
but you somehow want to you know go back and figure out the value
of the dimes and the quarters from you know what you got
for your variable D and check it out.
So there's different ways to show your check at the end.