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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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• 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?
• Yeah, I had difficulty with this also, but look at it like this:

``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)``

• 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! :-)
• at where did he get the one
• 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! 😊
• 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% ?...
• 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.
• How did he change 1.8% -> 0.018?
Shouldn't he just multiply 1.8 * 3,800?
• 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.
• 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
• 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.
• Can somebody tell me why he changed 0.018 into 1.018
• 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!
• 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.
• 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.