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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.

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  • hopper happy style avatar for user False Memory
    From to how 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?
    (17 votes)
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  • aqualine ultimate style avatar for user Sophie Sunshine
    I don't understand how he keeps converting the fractions into decimals. ( is 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! :-)
    (13 votes)
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  • blobby green style avatar for user tysnickimouse
    at where did he get the one
    (7 votes)
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    • duskpin ultimate style avatar for user Cowboy
      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)
  • purple pi purple style avatar for user The first integral proponent
    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)
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    • stelly blue style avatar for user Kim Seidel
      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.
      (11 votes)
  • aqualine ultimate style avatar for user Priscilla
    How did he change 1.8% -> 0.018?
    Shouldn't he just multiply 1.8 * 3,800?
    (3 votes)
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    • stelly blue style avatar for user Kim Seidel
      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.
      (6 votes)
  • blobby green style avatar for user juan.suero
    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)
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    • stelly blue style avatar for user Kim Seidel
      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)
  • primosaur sapling style avatar for user Joseph Hagar
    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)
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    • aqualine sapling style avatar for user Beaniebopbunyip
      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)
  • aqualine ultimate style avatar for user RedBlackandBlue
    At how did he get the 1
    (3 votes)
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  • male robot johnny style avatar for user eshaanbilling
    I found the answer with the formula, A(t) = 3800 * e^(0.018 * t)
    Where t represents time. Is that correct?
    (2 votes)
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  • female robot ada style avatar for user Taylor Poff
    On the first year, how/why is he adding 3.8% to 3,800 then timing it again to 3,800?
    (1 vote)
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    • aqualine ultimate style avatar for user Hannah Alisse
      Because of the order of operations (parenthesis, exponents, multiplication, division, addition, subtraction), it doesn't matter which way he writes it. He's not adding 1.8%, he's adding 1.8% * 3800.

      3800 + 1.8% * 3800 is the same as 3800 + 3800 * 1.8 which is also the same as 3800 + (1.8 * 3800) and it's the same as 3800 + (3800*1.8)

      Hope this helps :)
      (3 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.