SOLUTION: The annual income of Ana in 2011 was Php 450,000. From 2011 to 2015, her income increased by 30%. What was Ana's total income from 2011 to 2015?

Question 1168785: The annual income of Ana in 2011 was Php 450,000. From 2011 to 2015, her income increased by 30%. What was Ana's total income from 2011 to 2015?

Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
if it grew by the same percent increase each year, then the average amount that it grew each year would be 1.3 ^ (1/(2015-2011) = 1.3 ^ (1/4) = by a factor of 1.067789972.
take 1 away and multiply by 100 to get 6.7789972 percent increase each year.
rounded to the nearest dollar, her income each year from 2011 to 2015 would be:
2011 = 450,000
2012 = 480,505
2013 = 513,078
2014 = 547,860
2015 = 585,000

if it grew by the amount each year, then the average amount that it grew each year would (585,000 - 450,000) / 4 = 33,750 each year.
her income each year from 2011 to 2015 would be:
2011 = 450,000
2012 = 483,750
2013 = 517,500
2014 = 551,250
2015 = 585,000

the first growth was exponential growth.
it was the same percent increase each year.

the second growth was straight line growth rate.
it was the same amount each year.

the exponential growth multiplies by the same growth factor each year.
the amount of growth each year increases, but the percent growth remains the same.

the straight line growth adds the same growth each factor.
the percent growth decreases each year, but the amount of growth remains the same.

the percent growth equation is y = a * b^x
in your problem, a was 450,000 and b was 1.3 for every 4 years and b was 1.067789972 for each year.

to find the yearly growth rate, you had to take the 4th root of the 4 year growth rate.
1.3 ^ (1/4) = 1.067789972.

the straight line growth used the equation of y = m * x + b
m was the slope, or the average increase each year.
b was the y-intercept, which is the value of y when the value of x is 0.
in your problem the equation was y = 33,750 * x + 450,000
the slope was 33,750 and the y-intercept was 450,000.

the end points of the straight line growth and the exponential growth were the same, at least for the initial 4 year period.

after that, the exponential growth takes off and the straight line growth remains the same.

not sure which one you needed, so i gave you both.

let me know if you have any questions about this.
theo

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