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An old woman had the following assets:

(a) Rs. 70 lakh in bank deposits

(b) 1 house worth Rs. 50 lakh

(c) 3 flats, each worth Rs. 30 lakh

(d) Certain number of gold coins, each worth Rs. 1 lakh

She wanted to distribute her assets among her three children; Neeta, Seeta and Geeta.

The house, any of the flats or any of the coins were not to be split. That is, the house went entirely to one child; a flat went to one child and similarly, a gold coin went to one child.

The value of the assets distributed among Neeta, Seeta and Geeta was in the ratio of 1:2:3, while the gold coins were distributed among them in the ratio of 2:3:4. One child got all three flats and she did not get the house. One child, other than Geeta, got Rs. 30 lakh in bank deposits.

How many gold coins did the old woman have?

The answer is

90. Neeta got 20 coins and 30 in bank depostis, Seeta got 30 coins, a house and 20 in deposits, and Geeta got 40 coins, the flats and 20 in deposits.

Solution:

Neeta (N) got $2x$ coins and $i$ other goods, S got $3x+j$ and G got $4x+k$.

From the overall distribution, $3x+j = 2*(2x+i)$, therefore $j = 2i+x$. Similar to this, $k = 3i+2x$.
$i+j+k = 210$ (total worth of non-coin goods)

From previous equations, $6i+3x=210$.

Divide by 3: $j=2i+x=70$
$i$ is less than $j/2$, so, only G could get the flats and then, only S could get the house.


So, S got the house and 20 in deposits. Therefore, N got 30 in deposits, and G got the flats and 20 remaining deposits.
$i=30$; $70=j=2i+x$, so $x=10$. Total amount of coins is $2x+3x+4x=90$.

So my answer is the same as Verence with:

90 gold coins

But my method is different:

With Gold alone, the ratio of total assets would be 2:3:4 for N:S:G respectively. If we assume that they have have 2 gold, 3 gold, and 4 gold each, then with the addition of the other assets, their lowest ratio of total wealth would then become 3:6:9.


This is a different of +1:+3:+5. Increasing the values within this ratio gives us 4:8:12 and 5:10:15, each with an increase from 2:3:4 of +2:+5:+8 and +3:+7:+11 respectively.


As the rest of the given values deal with multiples of 10, I then wrote these out as a start of 20:30:40 gold, increasing by +10:+30:+50 etc other assets.

The next step was to find the total worth of the non-gold assets, to see of that matched the total of any of the increases I'd written out. The total increase in value was 70+50+90 = 210. I noticed that the 30:70:110 ratio also totalled 210.


I was then prepared to increase these values by factors that would allow for the other restrictions, but soon noticed that it was already viable to give Neeta the 30 bank deposit, give the 50 house to Seeta along with 20 in bank deposits to total 70, and give the 3 flats and the last 20 in bank deposits to Geeta to total 110.


This meant that the gold was given out as 20:30:40 (a ratio of 2:3:4), totalling 90, to make a total asset value of 50:100:150 (a ratio of 1:2:3).