Is there an explanation for the reasoning?

prayforme

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[FONT=&quot][FONT=&quot]A valued painting appreciated by 13% every 10 years. If it’s value in 1990 was $5480 what will its value be in 2020?[/FONT][/FONT]
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[FONT=&quot][FONT=&quot]$5480 x 1.3 x 1.3 x 1.3 = $7907.08[/FONT][/FONT]
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[FONT=&quot][FONT=&quot]What is the 1.3? Or where is this value coming from? Why three times? Is it for each decade? Does the 1.3 correspond to that 13%? Sorry I’m just trying to understand the process and reasoning for future problems. Thanks![/FONT][/FONT]
 
Word Problem Process

[FONT=&quot][FONT=&quot]A valued painting appreciated by 13% every 10 years. If it’s value in 1990 was $5480 what will its value be in 2020?[/FONT][/FONT]
[FONT=&quot][FONT=&quot][/FONT]
[/FONT]

[FONT=&quot][FONT=&quot]$5480 x 1.3 x 1.3 x 1.3 = $7907.08[/FONT][/FONT]
[FONT=&quot][FONT=&quot][/FONT]
[/FONT]

[FONT=&quot][FONT=&quot]What is the 1.3? Or where is this value coming from? Why three times? Is it for each decade? Does the 1.3 correspond to that 13%? Sorry I’m just trying to understand the process and reasoning for future problems. I didn’t see my problem post under my profile or in the forum so sorry if this comes up twice. Thanks.[/FONT][/FONT]
 
A valued painting appreciated by 13% every 10 years. If it’s value in 1990 was $5480 what will its value be in 2020?


$5480 x 1.3 x 1.3 x 1.3 = $7907.08


What is the 1.3? Or where is this value coming from? Why three times? Is it for each decade? Does the 1.3 correspond to that 13%? Sorry I’m just trying to understand the process and reasoning for future problems. Thanks!
The correct answer is

\(\displaystyle 5480 * (1.13)^3 = 5480 * 1.13 * 1.13 * 1.13\)

if the rate of appreciation is 13% per decade.

If on the other hand the rate of appreciation is 30% per decade, then the correct answer is

\(\displaystyle 5480 * (1.3)^3 = 5480 * 1.3 * 1.3 * 1.3.\)
 
You're sort of on the right track, with the thought that the 1.3 term appears three times, once for each decade. However, the fact that it's 1.3 at all is a mistake and the answer is wrong. To see why that is, let's investigate a simpler version of the problem. Suppose the problem text had been:

A painting was valued at $1,000 in 1990. In 2000, its value had increased by 13%. What is its new value?

Now, if we think about this, we can see that by the exact same logic used in the original problem, the answer should be \(\displaystyle $1000 \times 1.3 = $1300\). Let's go a step further and analyze what this answer says to see why it's not correct. Can you see why we can also write the answer as \(\displaystyle $1000 \times (1 + 0.3) = $1000 \times 1 + $1000 \times 0.3 = $1300\)? Now, given that the word percent literally means "per hundred," we can write 1 as 100% and 0.3 as 30%. So what does our answer say in words? It says that the new value of the painting is "100% of the old value, plus 30% of the old value equals 130% of the original value" and... oops! That's not right at all!

I'll leave the rest of the solution to you, to think about what went wrong here and how we can fix it. Specifically keep in mind that we want to calculate 113% of the old value, not 130%. Then once you've correctly answered the simpler problem, you ought to be easily able to scale it up and solve the original problem.
 
f = p*(1 + r)^n
where:
p = present value (5480)
r = rate (.13)
n = number of periods (3)
f = future value (?)

f = 5040*(1.13)^3

Tattoo that on your wrist...
 
f = p*(1 + r)^n
where:
p = present value (5480)
r = rate (.13)
n = number of periods (3)
f = future value (?)

f = 5040*(1.13)^3

Tattoo that on your wrist...
Sorry, but I disagree Denis. Not with your maths but with you suggesting to memorise a rule. Prayforme wants to understand what's going on, not just use a rule to get the right answer. If you understand what's going on here, you don't need a rule or you can create it yourself.
 
Thanks Harry.
My intent was simply to provide the formula...
which can easily be found on-line.
 
A valued painting appreciated by 13% every 10 years. If it’s value in 1990 was $5480 what will its value be in 2020?


$5480 x 1.3 x 1.3 x 1.3 = $7907.08


What is the 1.3? Or where is this value coming from? Why three times? Is it for each decade? Does the 1.3 correspond to that 13%? Sorry I’m just trying to understand the process and reasoning for future problems. Thanks!
Firstly, as others have said, if the painting appreciates by 13% each decade then the correct answer will be 5480 x 1.13 x 1.13 x 1.13 = 7907.08 (rounded)

Here's why:
At the end of the first decade, the painting is worth
5480 + 13% of 5480 = 100% of 5480 + 13% of 5480 = 113% of 5480 = 1.13 x 5480 = 6192.40

At the end of the second decade the painting is worth
6192.40 + 13% of 6190 = 1.13 x 6190 = 1.13 x (1.3 x 5480) = (1.13)^2 x 5480 = 6997.412

At the end of the third decade the painting is worth
8923.20 + 13% of 8923.20 = 1.13 x 8923.20 = 1.13 x [(1.13)^2 x 5480] = (1.13)^3 x 5480 = 7907.08

(I've added in the bold steps just to show how each answer relates back to the original value.

(Note: If it increase by 30% each decade, then the multiplier would be 1.3 rather than 1.13.)

You can see now what's behind the formula that Denis B has given you.

\(\displaystyle F = P*(1+ i)^n\) where i = interest rate per time period (as a decimal) and n = number of time periods
 
Works same as (and easier to "see") a savings account at your Bank,
with opening deposit of $5480, earning 13% interest annually for 3 years:
Code:
year    interest   balance
  0                5480.00
  1      712.40    6192.40  : 5480.00 * .13 = 712.40
  2      805.02    6997.42  : 6192.40 * .13 = 805.02
  3      909.66    7907.08  : 6997.42 * .13 = 909.66
Clear?
 
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