You want to buy a $222,000 home.
You plan to pay 10% as a down payment
and take out a 30 year loan for the rest.
Part A, how much is the loan amount going to be?
Because you are putting 10% down,
the loan amount it going to be 90% of $222,000,
since 100% minus 10% is 90%.
So we need to find 90% of $222,000.
To find the percent of a number,
we convert the perfect to a decimal and multiply.
90% of the decimal is 0.90 or just 0.9,
giving us 0.9 times 222,000.
And now let's go to the calculator.
0.9 times 222,000 is 199,800
and therefore, the loan amount is $199,800.
If you were asked to find the down payment,
we would find 10% of 222,000,
which would be 0.1 times 222,000,
which would be 22,200.
So another way to find the loan amount
would be to take 222,000 and subtract 22,200.
And now for part B, what will you monthly payments be
if the interest rate is 6%?
To answer this questions we will use
the loan formula shown below,
where Po is the loan amount,
which we now know is $199,800.
Giving us 199,800 equals, on the right side,
PMT is the loan payment which we are solving for,
so we have PMT, then in the numerator we have
the payment times the quantity one minus
and in parenthesis, we have one plus r divided by n,
where r is the annual interest rate as a decimal
and n is the number of compounds per year.
Which if not specifically given,
we use number of payments per year.
So for part B, our 6%, which is a decimal, is 0.06.
This is divided by n,
because you're making monthly payments and there's 12.
All this has raised the power of negative n times t,
which is negative 12 times t
as the length of the loan in years
for 30 year loan and therefore, t is 30.
Close parenthesis, all this is divided by r divided by n,
which gives us 0.06, divided by 12.
And then to solve for PMT,
we will evaluate this quotient here
which will give us PMT times n value.
And then we can solve for PMT
by dividing both sides by that value.
So going to the calculator,
we will evaluate this quotient here
on the right side of the equation.
So we have open parenthesis,
one minus open parenthesis one plus 0.06
divided by 12, close parenthesis,
it says raise to the power of
negative 12 times 30 which is negative 360.
Right arrow to exit the exponent, close parenthesis
and then divide it by the parenthesis we have,
0.06 divided by 12.
Which means on the right side of the equation,
we have PMT times approximately 166.7916144.
Let's go ahead and write this down.
The equation is now 199,800 equals PMT times,
again, 166.7916144.
And that'll solve for PMT the loan payment,
we divide both sides of the equation by 166.7916144.
Notice on the right side, this quotient is equal to one,
giving us PMT times one, of course, which is just PMT.
So the monthly payment is going to be this quotient here,
which we will round to the nearest cent.
So going back to the calculator,
we have 199,800 divided by 166.7916144, enter.
To the nearest cent, if the interest rate is 6%,
then the monthly payment will be $1,197.90.
And now for part C,
we're asked to determine the monthly payments
if the interest rate is 7% instead of 6%.
So to make this change,
we can go back up to this equation here
and simply change 0.06, which is 6% as a decimal to 0.07,
which is 7% as a decimal.
So again, we will now change 0.06 here and here to 0.07.
So we have 0.07 here and 0.07 here.
And now let's go back
and determine this quotient here again.
If we press second enter,
it brings up the previous entry which we can then edit.
So if you press second enter twice,
it brings us back up to this expression
where now we can just change 0.06 to 0.07
to save ourselves some time.
Press the left arrow
until we're over the six for 0.06 here,
change this to seven
and then change this 0.6 to 0.07 as well.
Then we press the right arrow until we're over the six
and change this to seven and then press enter.
And now this quotient is approximately 150.3075679
and therefore, the right side of the equation
can be written as PMT times this value here.
Let's go ahead and do that.
When the interest rate is 7%,
we have the equation 199,800
equals PMT times 150.3075679.
And that'll solve for PMT the monthly payment,
we divide both sides of the equation by 150.3075679.
Simplifying on the right, this quotient is one,
giving us PMT times one, which is PMT.
The monthly payment is equal to the quotient on the left,
which you will now evaluate on the calculator
and round to the nearest cent.
Now 199,800 divided by 150.3075679,
to the nearest cent, if the interest rate is 7%,
the monthly payment is now $1,329.27.
Looking at the monthly payments,
notice how the monthly payment goes up
by over $130 when the interest rate goes from 6% to 7%,
which is why the interest rate
of a mortgage loan is so important.
I hope you found this helpful.