William recently inherited a large sum of money; he wants to use a portion of this money to set up a trust fund for his

two children. The trust fund has two investment options: (1) a bond fund and (2) a stock fund. The projected returns over the life
of the investments are 3% for the bond fund and 7% for the stock fund. Whatever portion of the inheritance he finally decides to
commit to the trust fund, he wants to invest at least 20% of that amount in the bond fund. In addition, he wants to select a mix that
will enable him to obtain a total return of at least 5.5%.

forget the "at least" part for now. If we invest exactly 20% in bonds, that gives

.03(.20) + .07(.80) = .006+.056 = .062 = 6.2%

So, that will do the trick. In order to get exactly 5.5%, he'd have to do

.03x + .07(1-x) = .055
x = .375

So, as long as he invests less than 37.5% in bonds, his return will be at least 5.5%

To determine the investment mix for William's trust fund, we need to find the minimum amount he should allocate to the bond fund and the stock fund to achieve at least a 5.5% total return.

Let's denote the amount William decides to commit to the trust fund as X.

According to the requirements, William wants to invest at least 20% of X in the bond fund. This means that the amount he invests in the bond fund will be at least 0.2X.

To calculate the remaining amount that he should invest in the stock fund, we subtract the bond fund investment from the total investment:
Amount invested in stock fund = X - 0.2X = 0.8X

The projected return on the bond fund is 3% and on the stock fund is 7%. To calculate the overall return, we need to consider the weight of each fund in the total investment. The weight of the bond fund is 0.2X / X = 0.2, or 20%, and the weight of the stock fund is 0.8X / X = 0.8, or 80%.

The total return can be calculated as the weighted average of the individual returns:
Total return = (Weight of bond fund * Return of bond fund) + (Weight of stock fund * Return of stock fund)
Total return = (0.2 * 0.03) + (0.8 * 0.07)
Total return = 0.006 + 0.056
Total return = 0.062, or 6.2%

To meet William's requirement of a total return of at least 5.5%, we see that the calculated total return is higher than the minimum requirement. Hence, William can invest any amount greater than 0 in the trust fund, and he will achieve a total return of at least 5.5% as long as he applies the minimum allocation of 20% to the bond fund and 80% to the stock fund.