Successive growth rates follow the same concept as Compound Interest. Growth rate of an item is applied on the value of the item at the end of the prior period.
Example 13
The price of potatoes was Rs. 80 per kg in 2011. Inflation over the next three years was 20%, 25% and 3331% respectively. What is the price of potatoes in 2014?
Solution
Price per kg in 2012=80×(1+10020)
Price per kg in 2013=80×(1+10020)×(1+10025)
Price per kg in 2014=80×(1+10020)×(1+10025)×(1+300100)
=80×(1+51)×(1+41)×(1+31)
=80×56×45×34= Rs. 160
Answer: Rs. 160 per kg
In the case of depreciation, the rate has to be subtracted instead of added.
Example 14
The price of petrol in 2028 was Rs. 200 per litre. It fell by 20% a year for 3 successive years. What was the price of petrol in 2031?
Solution
p=200 ; r=−20% ; n = 3
Price of a litre of petrol in 2031=p(1+100r)n=200(1−10020)3
=200×0.83=200×0.512=102.4
Answer: Rs. 102.4
4.1 Compound Annual Growth Rate
Where a certain initial investment has grown by different rates at the end of every year, Compound Annual Growth Rate (CAGR) is the geometric mean of the actual annual growth rates.
In other words, CAGR provides a constant rate at which the initial investment could have been compounded over n years to yield its final value. The Compound Interest formula applies.
A=p(1+100r)n
When a certain amount p has grown to an amount A over n years, then Compound Annual Growth Rate (CAGR) is the variable r.
Example 15
An investment grew from Rs. 1,000 to Rs. 2,500 over 5 years. What was the Compound Annual Growth Rate?
(1) 20%
(2) 25%
(3) 30%
(4) 35%
Solution
p=1,000;n=5; A=2,500
A=p(1+100r)n
⇒ 2500=1000(1+100r)5
⇒ (1+100r)5=2.5
Using the calculator in the CAT exam and substituting the options, we find that option (1) satisfies.
(1+10020)5=(1.2)5=2.488∼2.5
∴ r=20%
Answer: 20%
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