# I'm stuck with this compound percentage problem

#### JAMPS

##### New member
Lets suppose that in 2014, in Peru, health expenditures were 75347.2 million, and each year that ammount grows by 5% compared to the previous year. (so it adds to the previous ammount)

If i start with 75347.2 (Year 0) then in the first year the expenditures will be 79114.56

what i did was:

find 5% (0,05) of 75347,2 by:

75347.2 x0,05=3767.36

and then add it to the first number

3767.36+75347.2=79114.56 in expenditures the first year.

Now my starting number will be 79114.56 (Year 1)
so:
Year 0: 75347.2
Year 1: 79114.56
Year 2: 83070.228
and so...

How many years will it take to reach 300000 in expenditures?
This is the question i don't know how to answer.
Year X: 300000

Doing one year at a time will take forever.

#### Harry_the_cat

##### Senior Member
Look at a shorter way of doing what you did:

75347.2 x 0.05+ 75347.2 = 75347.2 x 0.05+ 75347.2 x 1 = 75347.2 ( 0.05+ 1 ) = 75347.2 x 1.05 (don't multiply it out just yet)

So rather than finding 5% of your amount and then adding to your amount, you could just find 105% of your amount.

What do you need to do to 75347.2 x 1.05 to get the next years expenditure?

#### Subhotosh Khan

##### Super Moderator
Staff member
Lets suppose that in 2014, in Peru, health expenditures were 75347.2 million, and each year that ammount grows by 5% compared to the previous year. (so it adds to the previous ammount)

If i start with 75347.2 (Year 0) then in the first year the expenditures will be 79114.56

what i did was:

find 5% (0,05) of 75347,2 by:

75347.2 x0,05=3767.36

and then add it to the first number

3767.36+75347.2=79114.56 in expenditures the first year.

Now my starting number will be 79114.56 (Year 1)
so:
Year 0: 75347.2
Year 1: 79114.56
Year 2: 83070.228
and so...

How many years will it take to reach 300000 in expenditures?
This is the question i don't know how to answer.
Year X: 300000

Doing one year at a time will take forever.
What subject (class/Grade) has assigned this problem?

This will require knowledge of Geometric Series. Do you know the equation (formula) for summation of a geometric series?

#### Dr.Peterson

##### Elite Member
Lets suppose that in 2014, in Peru, health expenditures were 75347.2 million, and each year that ammount grows by 5% compared to the previous year. (so it adds to the previous ammount)

How many years will it take to reach 300000 [million?] in expenditures?
This is the question i don't know how to answer.
Year X: 300000

Doing one year at a time will take forever.
Does the question refer to the expenditure in a given year (which is what you are calculating), or to the total accumulated expenditures (which is what would require a geometric series)? I could interpret your "it adds to the previous amount" either way (adding 5%, or adding to the total).

#### JAMPS

##### New member
Does the question refer to the expenditure in a given year (which is what you are calculating), or to the total accumulated expenditures (which is what would require a geometric series)? I could interpret your "it adds to the previous amount" either way (adding 5%, or adding to the total).
to the total accumulated expenditures

#### JAMPS

##### New member
What subject (class/Grade) has assigned this problem?

This will require knowledge of Geometric Series. Do you know the equation (formula) for summation of a geometric series?
It's math class 8th grade. My teacher just came, gave me a copy with exercises and said solve them without teaching anything :/ this one is the only one i couldn't do.

#### JAMPS

##### New member
Look at a shorter way of doing what you did:

75347.2 x 0.05+ 75347.2 = 75347.2 x 0.05+ 75347.2 x 1 = 75347.2 ( 0.05+ 1 ) = 75347.2 x 1.05 (don't multiply it out just yet)

So rather than finding 5% of your amount and then adding to your amount, you could just find 105% of your amount.

What do you need to do to 75347.2 x 1.05 to get the next years expenditure?
Yes
75347.2 x 1.05
That to the power of X and equal to the ammount i want has to give me the year.
300000 = 75347.2 * 1.05^X
300000/75347.2 = 1.05^X
logarithm of both sides
log (300000/75347.2) = log 1.05^X = X log 1.05
X= log (300000/75347.2) / log 1.05
X= 28.3187346
75347.2 * 1.05^28.3187346= 300000

This in time would be 28 years and 116.338129 days
0.3187346 x 365 for days.
Since the 5% gets added at the end of the year it would take 29 years to go pass and reach 300000, right?

#### Subhotosh Khan

##### Super Moderator
Staff member
That is not correct if you need answer "to the total accumulated expenditures". You'll need summation of

75347.2 * [(1.05)^0 + (1.05)^1 + (1.05)^2 + (1.05)^3 ...... (1.05)^n] = 3 * 10^5

Now comes the summation of geometric series. Do you know the equation?

#### JAMPS

##### New member
That is not correct if you need answer "to the total accumulated expenditures". You'll need summation of

75347.2 * [(1.05)^0 + (1.05)^1 + (1.05)^2 + (1.05)^3 ...... (1.05)^n] = 3 * 10^5

Now comes the summation of geometric series. Do you know the equation?
Oh i understand. No, it is not asking for the total accumulated expenditures, just when will the starting expenditure of X year reach 300 000 (and the word used in the exercise should be pass, because it gives the idea that an exact natural number will be equal to 300 000. like 28 years = 300 000.

And no i don't know any equation for summation of geometrics.

This table has values for other years i used to check that 75347.2 * 1.05^X was right.
If for example i plug 2 in the power of X it gives: 83 070.288

And the way the B) question is written in spanish lends to the misunderstanding you mentioned. If it's asking for the accumulated expenditures that reach 300 000 or not.
I know the teacher just wants me to say the year that the dedicated expenditures reached 300 000 for that year only
so the question should be B) How many years will it take for the dedicated health expenditures of a year to reach/reach over/pass 300 000?
or something like that. I'm just trying to let you guys understand since english is not my first language.

attatched is an image of the table and question.

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#### Subhotosh Khan

##### Super Moderator
Staff member
Oh i understand. No, it is not asking for the total accumulated expenditures, just when will the starting expenditure of X year reach 300 000 (and the word used in the exercise should be pass, because it gives the idea that an exact natural number will be equal to 300 000. like 28 years = 300 000.

And no i don't know any equation for summation of geometrics.

This table has values for other years i used to check that 75347.2 * 1.05^X was right.
If for example i plug 2 in the power of X it gives: 83 070.288

And the way the B) question is written in spanish lends to the misunderstanding you mentioned. If it's asking for the accumulated expenditures that reach 300 000 or not.
I know the teacher just wants me to say the year that the dedicated expenditures reached 300 000 for that year only
so the question should be B) How many years will it take for the dedicated health expenditures of a year to reach/reach over/pass 300 000?
or something like that. I'm just trying to let you guys understand since english is not my first language.

attatched is an image of the table and question.
In that case, response #7 is correct.

#### JAMPS

##### New member
I'm truly grateful to everybody for your help!

#### fezziwig

##### New member
Thank's guys for helping this guy x)