Maths problem

Mental Maths
New answer on Oct 23, 2020
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How do you calculate compound growth without a calulator? Question below:

Innovation Capital is 13.8 million and Human Capital 3.3 million

If total Innovation Capital in Exhibit 1 were to grow by 5% per year in the future, which of the following would be the MINIMUM required annual growth in Human Capital that would see it represent more than half of total Innovation Capital in 10 years? A. 10% B. 15% C. 20% D. 25%

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Hi there,

For many of the questions with compound rates you can apply the rule of 72.

According to it, the time to double, t, is equal to 72/r, where r is the compound growth rate:

t=72/r

So Innovation Capital will double in:

72/5 ≈14

So in 14 years will be around 28M. Growing at 5%, in 10 years will around 20% less (using now simple growth rate for simplicity), or approximately 22M

You now have to see the level of Human Capital to become 11M. This means that in 10 years it has to do a bit less than 4x.

Applying again rule of 72 (twice in this case), you find that to do 4x:

• With 10% you need around 14 years
• With 15% you need around 10 years
• With 20% you need around 7 years
• With 25% you need around 6 years

So the minimum is 15%.

Hope this helps,
Francesco

Hey there,

The trick to do such questions is not by formula but by laying things out in a table quickly and rounding things up/down slightly. Crazy to calculate 10th or 5th root without calculator :). But below method can be slightly time consuming for sure.

Human capital in year 1 stands at ~21% of Innovation Capital (3/14 *100). Remember this is underestimating due to rounding. So just common sensically we can say that human capital needs to grow near this 21% rate or will lag behind. Make sense?

Now by hit and trial, start with 15% growth rate year on year and do the maths for Innovation and Human Capital. E.g. in Year 2, Innovation capital will be (14+14/20) and so on.

Then try 20%. I think the correct answer will be 15% and you dont have to go all the way upto 10 years, my thinking is by 6th/7th year of calcs you can guess by elimination. Let me know how you get on.

This is my way but am sure there will be other tips/tricks :).

(edited)