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### Course: Operations and Algebraic Thinking 229+ > Unit 8

Lesson 2: Exponential expressions- Exponential expressions word problems (numerical)
- Exponential expressions word problems (numerical)
- Initial value & common ratio of exponential functions
- Exponential expressions word problems (algebraic)
- Exponential expressions word problems (algebraic)
- Interpreting exponential expression word problem
- Interpret exponential expressions word problems

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# Exponential expressions word problems (numerical)

Given the description of a real-world context, we write a calculation of a certain measure. The expression is exponential because it involves repeated multiplication.

## Want to join the conversation?

- From1:40to2:20how did Sal do that? I get that He is factoring but how did He do it?

What do those arrows represent? Where did the "1" come from?(22 votes)- Yeah, I had difficulty with this also, but look at it like this:

Instead of:`3800 + (3800 * 0.018)`

Think of it like this:`1(3800 + (3800 * 0.018))`

or`1(a + ac)`

then un-distribute the 3800 (or a)`1(3800 + (3800 * 0.018))`

(3800 + 1) * (1 + 0.018)

3800 * (1.018)

3800(1.018)

Watch the videos here to learn more about the distributive property over multiplication: https://www.khanacademy.org/math/pre-algebra/pre-algebra-arith-prop/pre-algebra-ditributive-property/v/the-distributive-property(23 votes)

- I don't understand how he keeps converting the fractions into decimals. (1:28is one example.) Can someone please explain it? I know it has something to do with the fact that 1.8% is 1.8/100, but I feel as if I didn't get a great percentage education. :-( Is it specific to the number you're trying to find the percentage of (3800 in this case) or universal? Thanks! :-)(17 votes)
- Hi Sophie Sunshine,

I don't quite understand your question, but I understood that you are confused about converting fractions into decimals. Watch this rewriting decimals as fractions playlist on KhanAcademy.

https://www.khanacademy.org/math/arithmetic-home/arith-review-decimals#decimals-to-fractions

Hope that helps!

- JK(10 votes)

- at2:00where did he get the one(8 votes)
- Simply put, it is his added deposit and interest put into the same equation to save time and space. If he were to multiply it by just 0.018 (his interest), his money in the bank suddenly gets much smaller; so the 1 is basically saying that it is the sum of the interest added to the sum of his deposit.

Really hope this helps! 😊(19 votes)

- I don't get why it says multiply by 105% three times instead of just multiply 5% three times in this problem:

Benjamin invests $400 dollar sign, 400 in a savings account that earns %5, percent interest each year.

Which expression does not give the correct balance in the account after 3 years?

1. 400(1.05)³

2. 400(1-0.05)³

3. 400+400(0.05)+420(0.05)+441(0.05)

4. 400(1+0.05)(1+0.05)(1+0.05)

Why 105% ?...(2 votes)- If you do 400 * 5%, the result you get is equal to just the interest earned for the 1st year. The problem is asking you to find the
**balance**in the account after 3 years.

-- At the end of year 1, the account balance = the original amount (400) + interest (400 * 0.05). This can be simplified into 400 * 1.05

-- At the end of year 2, the account balance = the balance from year 1 (400 * 1.05) + interest calculated on the year 1 balance (400 * 1.05 * 0.05). Or, in simplified form: 400 * 1.05 * 1.05 = 400 * 1.05^2

-- At the end of year 3, the account balance = the balance from year 2 + interest calculated on the year 2 balance. This ends up being 400 * 1.05^3

Hope this helps.(12 votes)

- How did he change 1.8% -> 0.018?

Shouldn't he just multiply 1.8 * 3,800?(3 votes)- Percentages are (by definition) fractions/ratios with a denominator of 100. If you use 1.8, you aren't using the correct value.

1.8% = 1.8/100 as a fraction or 0.018 as a decimal.

When ever you do math using a percent, you must convert it to fraction or decimal form to get the correct answer.

Hope this helps.(7 votes)

- Im worried i dont get the difference between Geometric Sequences and Exponential Functions.

A few lessons ago we talking about 15 + 3(n-1) as a Arithmentic Sequence. then 15 x 3(n-1) as a geometric sequence which seems exponential in nature to me.

now we are talking about exponential functions which i though we already talked about in the form of geometric sequences. Yet now we are talking about exponential functions in the form of y = 3 to the x power(4 votes)- Good observation - They both do use exponents. But, they are somewhat different...

In a geometric sequence, "n" is a counting number like 1, 2, 3, 4, etc. This is because "n" represent which term in the sequence you want to find. If you want the 10th term, then n=10

With an exponential function, the value of "x" can be any real number. It is not limited to counting numbers.

Hope this helps.(4 votes)

- Can somebody tell me why he changed 0.018 into 1.018(4 votes)
- The reason we do this is because each year we add 1.8% to the money in the account. Now, if we just convert 1.8% to a decimal and multiply by the interest we get the number 68.4. That is the amount of money you made in interest. To find out how much money you have in your account you'd have to add that number back into the principal giving us $3868.4. Doing this required two steps. So doing this in one step all we do is add 1 to the interest rate.

3800(1 + 0.018) = 3800 + 0.018*3800

We see that mathematically this does the same as the two steps we took.

So, when we talk about compound interest we add 1 to the interest rate so that it will add the interest to the principal.

I hope this helps!

FreeRadical(3 votes)

- Can someone explain to me how I should interpret '15 years'. I tried to do the question before Sal showed how, and I used the formula I learnt from geometric sequence:

3800(0.018)^(15-1).

So I put 14 as the power because I thought year 1 was 3800? I always get stumped in the wording of the question, if someone can give me some tips that would be very helpful, thanks.(3 votes)- With a geometric sequence, the 15 usually would represent that you want the 15th term in the sequence and the 3800 is your 1st terms. This is why, you would have to use 15-1 because you want to apply the common factor 14 times, not 15.

With exponential growth / decay problems, 3800 it the initial amount (before the clock starts, not the value after 1 year). By doing 15-1, you are interpreting the 3800 as the value at the end of year 1. This is why it is different than what you see in a geometric sequence.

Hope this helps.(2 votes)

- When you plug the equation into the calculator you get back a decimal. Is it ok to round to the nearest dollar in this problem?(2 votes)
- If the question says that it’s okay to round, you should do so. If it’s an equation where you can do it by hand and get a fractional answer, you probably should do that, or convert your decimal to a fraction. If it’s an exponential problem (or logarithmic, or involving square roots, pi, e, etc.) then you should round a few decimal places. Hope this helps!(3 votes)

- So, the geometric sequence is an exponential function with domain of integers, did I get is right?(1 vote)
- Almost... the domain is usually natural numbers (1, 2, 3, 4, etc.) rather than integers.(5 votes)

## Video transcript

- [Instructor] You put
$3,800 in a savings account. The bank will provide
1.8% interest on the money in the account every year. Another way of saying that, the money in the savings account
will grow by 1.8% per year. Write an expression that
describes how much money will be in the account in 15 years. So let's just think
about this a little bit. Let's just think about
the starting amount. So in the start, we're
just gonna put $3,800. We could view that as year zero. Actually, let me write it that way. So the start is the
same thing as year zero and we're gonna start with $3,800. Now let's think about year one. How much money will we
have after one year? Well, we would have the original
amount that we put, $3,800, and then we're gonna get the
amount that we get an interest and they say that the bank
will provide 1.8% interest on the money in the account so it'll be plus 1.8% times $3,800 and we could also write this as a decimal. This is equal to 3,800
plus and I'll just write, I'll switch the order
of multiplication here, plus 3,800 times 0.018. 1.8% is the same thing
as 18,000ths or 1.800ths depending on how you want to pronounce it. And so here you might say, "Well, there's an interesting
potential simplification "mathematically here." I could factor 3,800 out
of each of these terms. I have a 3,800 here, I have a 3,800 here so why don't I factor it out? Essentially undistributed. So this is going to be 3,800 times, when you factor it out here, you get a one plus, when you factor it out here, you get 0.018 and so I could just rewrite
this as 3,800 times 1.018. So this is an interesting time to pause. We're not at the full answer yet, how much will we have in 15 years, but we have an interesting expression for how much we have after one year. Notice that if the
money is growing by 1.8% or another way it was growing by 0.018, that's equivalent to multiplying the amount that we started the year with by one plus the amount that
it's growing by or 1.018. And once again, why does
this make intuitive sense? Because at the end of the year, you're going to have the
original amount that you put, that's what that one really represents, and then plus you're gonna have
the amount that you grew by so you multiply both the sum here times the original amount you put and that's how much you'll
have at the end of year one. What about year two? So year two. Well, we know what we're going
to start with in year two. We're gonna start with whatever
we finished year one with. So we're gonna start
with 3,800 times 1.018, but then it's gonna grow
by 1.8% or grow by 0.018. And we already said, if you're
gonna grow by that amount, that's equivalent to
multiplying it by 1.018. Well, this is the same
thing as 3,800 times 1.018 to the second power. I think you see where this is going. Every time we grow by 1.8%, we're gonna multiply by 1.018. And if we're thinking about
15 years in the future, we're gonna do that 15 times. So one year in the future, your exponent here is essentially one. Two years, your exponent is two. So year 15, I can just
cut to the chase here, so year 15, well that's just going to be, we're going to have the
original amount that we invested and we are going to grow 1.018 15 times so we're gonna multiply
by this amount 15 times to get the final amount. And one of the fun things, this is actually called compound growth where every year you
grow on top of the amount that you had before. You'll see if you type
this into a calculator that even though 1.8% per
year does not seem like a lot, over 15 years it actually would amount to a reasonable amount, but this is the expression. They're not asking us to calculate it. They just want us to know
an expression that describes how much money will be in
the account in 15 years.