Dana has $5.15 in quarters and nickels in her car.
The number of nickels is seven more
than the number of quarters.
How many of each type of coin does she have?
This is kind of a silly question,
because I would just tell Dana
to count her quarters and nickels,
but we'll go ahead and solve this mathematically anyway.
We need to begin by writing expressions
for the number of nickels and the number of quarters.
And because we're told the number of nickels
is seven more than the number of quarters,
we will begin by letting x equal the number of quarters.
And again from here, because the number of nickels
is seven more than the number of quarters,
if x is the number of quarters,
then we can use the expression x plus seven
for the number of nickels.
And now from here we can write our equation,
which will be the value of a quarter
times the number of quarters,
plus the value of a nickel times the number of nickels,
must equal the total value which in this case is $5.15.
However, to avoid decimals in our equation,
we will use a units of cents rather than dollars.
So the value of a quarter is 25 cents,
and since x is the number of quarters,
the value of the quarters is 25 cents times x,
or 25x for our equation,
plus the value of the nickels is going to be five cents
times the number of nickels.
And since the number of nickels
is the quantity x plus seven,
the value of the nickels is five cents
times the quantity x plus seven,
or for our equation, five times the quantity x plus seven,
and this must equal the total value of $5.15.
But we want this value in cents.
$5.15 is equal to 515 cents.
And now we need to solve the equation for x.
The first step is to clear the parentheses on the left
by distributing five.
We have 25x, and now distributing five,
five times x is five x giving us plus five x.
And then five times seven is 35
giving us plus 35 is equal to 515.
The next step is to combine like terms on the left side,
25x plus five x is 30x.
Giving us 30x plus 35 equals 515.
The next step is to add or subtract
to isolate the variable term of 30x.
We need to undo plus 35 by subtracting 35
on both sides of the equation.
Simplifying, 35 minus 35 is zero,
the equation simplifies to 30x equals 515
minus 35 is equal to 480.
The last step is to multiply or divide to solve for x.
And since 30x means 30 times x,
to solve for x we divide both sides by 30.
Simplifying 30 divided by 30 simplifies to one.
One times x is x.
We have x is equal to 480 divided by 30
which is equal to 16.
Now that we know x is equal to 16,
we can determine the number of quarters
and the number of nickels.
And let's do this on the next slide.
And again we just found x is equal to 16.
So now we know there are 16 quarters.
The number of nickels is x plus seven.
And since x is 16,
that gives us 16 plus seven which is equal to 23.
Dana has 16 quarters and 23 nickels in her car.
Before we go let's verify our value equation below,
which is 25 cents times the number of quarters which is 16
plus five cents times the number of nickels which is 23
must equal the total value of the coins
which is in cents 515 cents.
And let's verify this on the calculator.
And 25 times 16 plus five time 23 does equal 515
verifying our solution is correct.
I hope you found this helpful.