### Video Transcript

Rachel bought her house for 120000
dollars. Her house increased in value by two
percent per year and the value of her house after đť‘› years can be calculated using
the equation đť‘‰ equals 120000 times đť‘Ą to the đť‘› power. Rachelâ€™s friend Hunter, who lives
in a different area, also bought his house for 120000 dollars, but his house
increased in value by three percent per year. After 10 years, the value of
Hunterâ€™s house is đť‘ź percent greater than the value of Rachelâ€™s house. Calculate đť‘ź to one decimal
place.

In order to calculate the percent
đť‘ź, weâ€™ll first need to think about the equation that will help us calculate the
increased value of the home. Rachelâ€™s home increases in value at
the rate 120000 times đť‘Ą to the đť‘› power. First, weâ€™ll need to know what đť‘Ą
is. For this đť‘Ą-value, we should
substitute 1.02 because the house increases by two percent every year. The whole number one represents the
starting value, and the 0.02 represents the two percent increase. To calculate the value of Rachelâ€™s
House after đť‘› years, you would multiply 120000 by 1.02 to the đť‘› power.

We can use the same strategy to
calculate the value of Hunterâ€™s house. In this case, we would substitute
1.03 because Hunterâ€™s house increases at a rate of three percent every year. We calculate the value of Hunterâ€™s
house after đť‘› years by multiplying 120000 times 1.03 to the đť‘› power. Hunterâ€™s house is đť‘ź percent
greater than Rachelâ€™s house. This đť‘ź percent greater represents
a percent increase. And we can calculate the percent
increase of something by taking a larger value, subtracting the smaller value,
dividing that by the smaller value, and then multiplying by 100.

In our case, it would be Hunterâ€™s
house value minus Rachelâ€™s house value, divided by Rachelâ€™s house value times
100. Letâ€™s plug in what we know. At this point, we can replace our
đť‘› values with the number 10. Weâ€™re interested in the percent
increase after 10 years. Now, at this point, something
really cool happens. Because both terms in the numerator
have a factor of 120000 and the denominator also has a factor of 120000, these all
cancel out. And weâ€™re left with 1.03 to the
10th power minus 1.02 to the 10th power divided by 1.02 to the 10th power times 100
percent.

Be careful here. 1.02 to the 10th power over 1.02 to
the 10th power canâ€™t be cancelled out because in the numerator weâ€™re
subtracting. At this point, itâ€™s probably safe
to go ahead and put this whole value into your calculator, being careful to put
brackets around 1.03 to the 10th power minus 1.02 to the 10th power. Make sure you put brackets around
the numerator. Whatever that value equals, youâ€™ll
then multiply by 100. And when you do that, you get
10.2479 continuing.

We need to round this to one
decimal place. Thereâ€™s a two in the tenths place,
the first decimal place. And to the right of that, we have a
four, which means weâ€™ll round down to 10.2. We can say that the value of
Hunterâ€™s house after 10 years is 10.2 percent higher than Rachelâ€™s house.