A part of this question i don't get. Percentage increase

Question

Over 3 years, 5.5 cents is increased exponentially by the same percentage each year to 7.7 cents. Calculate percentage increase *each year*

 

And the solution is :

5.5 * X³
=7.7

so my answer was 1.119 but the answer in the mark scheme is 11.9. I asked a friend and she told me that we minus 1.119 by 1 (1.119 - 1)=0.119 then to find the percentage with multiply 0.119*100=11.9. What i don't understand is why we minus by 1

 

 

Answer 1

the exponential growth formula is y=a(1+r)^t where y=final amount, a=initial amount, r=rate of growth, and t=time

you used y=ab^x where b=(1+r) and x=t

7.7=5.5b^3

7.7/5.5=b^3

1.4=b^3

(1.4)^(1/3)=b

b=1.1186889

b=1.119

b replaced (1+r) in the original formula

1.119=1+r

r=1.119-1

r=0.119

r=11.9%

 

 

Answer 2

Kimia, the 7.7 is gained by multiplying the previous year by 100% plus the percentage, so you have to subtract 1 (100%) to get the annual percentage.

 

Let's look at your example. Each year, the 5.5 cents is increased by 11.9%, meaning the new total is 119% of the year before.

 

5.5(1.119)(1.119)(1.119) = 7.7

 

Just like if a shirt is on sale for $5 with 8% sales tax. The total price is $5(1.08) = $5.40, but the tax rate is 1.08 - 1 = 0.08 = 8%.

 

Hope this helps!