Compare Mortgage Payments at Two Different Interest Rates (Formula)

Compare Mortgage Payments at Two Different Interest Rates (Formula)

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.

150.3075679

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..

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