Ramanujam Maths Puzzle

I got this puzzle today: http://www.imsc.res.in/~rao/ramanujan/newnow/q289.htm

(sweat). Still working to find a formal answer to both the puzzles. Jesus. 

Elementary nested root problem, the author says. My foot.

Answer (inelegant) in green in this sheet: https://docs.google.com/open?id=0B1xQDlpMnLKxRUV6SExvdGRwWDQ

Formal solution here: http://www.imsc.res.in/~rao/ramanujan/newnow/sq289.htm and below.

Additional reading:

• http://paramanands.blogspot.in/2013/06/congruence-properties-of-partitions-part-1.html#.Vpp7X5p95dg
• http://www.had2know.com/academics/nested-radical-square-root-calculator.html
• http://en.wikipedia.org/wiki/Nested_radical

Pros And Cons Of Buying New Vs Old Flat

A friend of mine was planning to buy a new flat. He gave me a link which the builder had given him that stated the cost of the flat and amenities like 2 covered parking, 24 hours power backup and what not. The cost of the flat was Rs1.5 crore (approx $268,000).

It was a 1100 sq ft apartment which had a UDS (Undivided share) of 567 sq ft. The land cost in this area is Rs 4 crore per ground (per 2400 sq ft). I worked out that the UDS cost for 567 sq ft is about Rs 93 lacs (about $166,000) and the premium he was paying (excess of market price excluding tax / registration charge over UDS price) was about Rs 57 lacs ($100,000). I treat the UDS price as a book value. The ratio of market price 1.5 crore / UDS 93 lacs is like the price to book value (P/BV) ratio.

I felt it was not a very good deal to pay one third of the total price as premium, it should be far lesser. Could he not look for a flat which had a higher ratio of UDS price to market price. My friend's contention was that the if he only bought the UDS, it is like buying land and it will not fetch rent. Having a construction and roof over the land made it livable. And this construction essentially becomes the premium that one paid. My friend and I both agreed that the annual rent of a residential property across India was about 3.5% of the current market value. This figure of 3.5% is from my own data based on what I have seen in Chennai. I am sure it applies to other Indian cities as well. 

My friend pointed out that the rent is 3.5% of the total current value of the property, UDS + premium. Very true.

But the point is, while the value of UDS keeps increasing (say @ 10% CAGR), the premium depreciates. All your battery and woodwork and A/c are depreciating assets. At the end of 20 years they may lose all their worth. My data shows that the premium probably gets a negative worth after about 25 or 30 years. 

If this is so, the premium that you pay fetches you 3.5% annually on a declining value. Bank deposit gets you more than 7% where the capital is protected. What sense does it make to pay a high premium value? Is it not a better idea to buy a real estate with very low premium, for example very old flats? The entire property value would fetch 3.5% annual return PLUS there is little / no depreciation. Two flats, one old and one new are compared and the detailed workings are in the sheet below.

Buying a flat with low premium (example: a very old flat), fetches you 1 to 1.5% more CAGR.

This is another article which talks about the same subject.