Video: US-SAT04S4-Q38-202127891526

Rachel bought her house for $120000. Her house increased in value by 2 percent per year and the value of her house after 𝑛 years can be calculated using the equation 𝑉 = 120000 (𝑥)^𝑛. Rachel’s friend Hunter, who lives in a different area, also bought his house for $120000, but his house increased in value by 3 percent per year. After 10 years, the value of Hunter’s house is 𝑟 percent greater than the value of Rachel’s house. Calculate 𝑟 to one decimal place.

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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 𝑛 years can be calculated using the equation 𝑉 equals 120000 times 𝑥 to the 𝑛 power. Rachel’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 Hunter’s house is 𝑟 percent greater than the value of Rachel’s house. Calculate 𝑟 to one decimal place.

In order to calculate the percent 𝑟, we’ll first need to think about the equation that will help us calculate the increased value of the home. Rachel’s home increases in value at the rate 120000 times 𝑥 to the 𝑛 power. First, we’ll need to know what 𝑥 is. For this 𝑥-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 Rachel’s House after 𝑛 years, you would multiply 120000 by 1.02 to the 𝑛 power.

We can use the same strategy to calculate the value of Hunter’s house. In this case, we would substitute 1.03 because Hunter’s house increases at a rate of three percent every year. We calculate the value of Hunter’s house after 𝑛 years by multiplying 120000 times 1.03 to the 𝑛 power. Hunter’s house is 𝑟 percent greater than Rachel’s house. This 𝑟 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 Hunter’s house value minus Rachel’s house value, divided by Rachel’s house value times 100. Let’s plug in what we know. At this point, we can replace our 𝑛 values with the number 10. We’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 we’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 can’t be cancelled out because in the numerator we’re subtracting. At this point, it’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, you’ll then multiply by 100. And when you do that, you get 10.2479 continuing.

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

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