Suppose you have a $\$ $100 debt at 10% monthly interest, and you pay $\$ $15 a month. The amortization for this payment is:
Month Principal Interest Payment Applied to Principal 1 100 10 15 5 2 95 9.5 15 5.50 3 89.50 8.95 15 6.05 4 83.45 8.35 15 6.65 5 76.80 7.68 15 7.32 6 69.48 6.95 15 8.05 7 62.43 6.24 15 8.76 8 53.67 5.37 15 9.63 9 44.04 4.40 15 10.60 10 33.44 3.34 15 11.66 11 21.78 2.18 15 12.82 12 8.96 0.90 9.86
So you pay it off in one year.
Now, I'm not clear from your description if your brother is planning to pay off double what the original amortization table would indicate, or double what that month's principal would have been (given that he already paid off more than he was expected to), so let me do both.
Suppose first that he simply pays pays his original 15, plus the column "applied to principal" from the original amortization: so the first month, instead of paying 15 with 5 towards principal, he pays 20 (10 towards principal). Next month, instead of 15, he pays 20.50 (so, whatever it will be towards principal, plus an extra 5.50 corresponding to the original amortization table "applied to principal" column; etc.) We have:
Month Principal Interest Payment Applied to Principal 1 100 10 15+5 10 2 90 9 15+5.50 11.50 3 78.5 7.85 15+6.05 13.20 4 65.3 6.53 15+6.65 15.12 5 50.18 5.02 15+7.32 17.30 6 32.88 3.29 15+8.05 19.76 7 13.12 1.31 14.43
so it takes 7 months, more than half. Here, what I add each month to the payment is what the "Applied to Principal" column indicated for that month in the original amortization table.
However, if each month he pays each again the amount that would now apply to the principal, we have:
Month Principal Interest Payment Applied to Principal 1 100 10 15+5 10 2 90 9 15+6 12 3 78 7.80 15+7.20 14.40 4 63.60 6.36 15+8.64 17.38 5 45.62 4.56 15+10.44 20.88 6 24.74 2.47 15+9.74 24.74
so you pay it off in six month (the last month because 15-2.47 = 12.53, and 15+12.53 is more than what is owed). Here, what I add to the payment each month is equal to the difference between the basic payment of $\$ $15 and the interest that is being paid off. So for example, in month 4, you have to pay down $\$ $4.56 interest; that means your $\$ $15 payment will pay down 15-4.56=10.44 principal, so you add another $\$ $10.44 to the payment.
Added. And then there seems to be a third option; from what youo describe, ti seems to me that he would look at the "Applied to Principal" line in the previous month, and add that amount to his payment fo the next month. If we do that, we get the following table:
Month Principal Interest Payment Applied to Principal 1 100 10 15 5 2 95 9.50 15+5 10.50 3 84.50 8.45 15+10.50 12.05 4 72.45 7.25 15+12.05 19.80 5 52.65 5.27 15+19.80 29.53 6 23.12 2.31 15+10.24 25.24
so it's 6 months this way, with a slightly larger final payment than the previous version, because he is not really doubling the amount of principal he would have paid down that particular month, but rather something slightly smaller. Here, what I add to the payment is the previous month's "Applied to Principal" column; so, since in month 4 there was $\$ $19.80 applied to principal, that's how much is added to the payment of Month 5.
So, in the first scenario (double what he would have paid towards principal in the original amortization), it takes him more than half the time. In the second, where he pays again over what he would have paid off that month in principal, it takes him about half the time. In the final scenario, where he pays again over what he paid last time towards principal, it's a bit more than half the time, but by very little compared to the first method.
These are just examples, of course, but they give you an indication of how things go over 30 years. It would seem he is correct, and it will take him just a bit over 15 years.
This agrees with the results reported by Ross using Excel: my "first scenario" is what Ross reports as taking 20 years; the second what he reports as taking 15 years. My third scenario is very close to the second, but he's a bit off from "doubling" because he is applying the previous amount that was used to pay down principal, not the current one.