Let the value of the index be 100. Then, the shares of R, H and F have the respective values: 7, 13 and 15. The remaining shares would have the value of 100 - (7 + 13 + 15) = 65.

Since the values of the shares rose by 9%, 10% and 4%, the new values of R, H, and F wil be 7.63, 14.3 and 15.60. The new value of the remaining shares then would be 68.47.

Hence, the required percentage increase in the value of the remaining shares equals (3.47/65)*100 = 5.34%.

[spoiler]Answer: (E)[/spoiler]

R = 7% of total investment
H = 13% of total investment
F = 15% of total investment
Let remaining shares = X
X = 100% - (13 + 7 + 15)%
X = 100% - 35%
X = 65%


Increase in R = 9%
Increase in H = 10%
Increase in F = 4%
Find an increase in X


Assuming that the whole investment index = $100
R = $7, H = $13, F = $15, X = $65
The whole index increased by 6% i.e it became => $100 + (6% of 100) = $106 out of which R = 7 + (9% of 7) = 7.63; H = 13 + (10% of 13) = 14.3; F =15 + (4% of 15) = 15.6
Total of the 3 = 7.63 + 14.3 + 15.6 = 37.53
X= 106 - 37.53 = 68.47


$$\%\ increase\ of\ X=\frac{increase}{original\ }\cdot\frac{100}{1}$$
$$=\frac{68.47-65}{65}\cdot\frac{100}{1}$$
$$=\frac{3.47}{65}\cdot\frac{100}{1}$$
$$=0.0534\cdot100$$
$$=5.43\%$$

$$Answer\ =\ E$$