If the people turning to Ramsey for financial advice were, by and large, analytical types, they probably wouldn’t need to be turning to Ramsey for financial advice.
Would it be better to push more money to a $7000 debt at 7% than a $1000 one at 5%? Analytically, yes.
Is it more likely that someone who had become nihilistic about their financial situation will actually get both of those paid off if they get to see that $1000 account “paid in full” in two or three months? That’s what Ramsey has bet on, and apparently is right about.
There’s a psychological reason too (already commented on by others here), but there is a math reason too (cash flow).
The math doesn't work out if you give up before paying off that high-interest debt.
Say you have a debt payment budget of $500 / month for payments of $80, $100 and $200. Also, for sake of the example assume that the amount owed on each is proportional to the payment size (smallest to largest).
You have a total of $380 with an extra $120 to apply to one of the payments. You’ll apply it to the $80 payment until it’s paid off so you’re now paying $200 / month towards it.
Once it’s paid off, you focus on the $100 / month but because the $80 has been paid off you now have an extra $200 / month so you can pay $300 / month today towards it.
Once that is paid off, you can now focus the entire $500 towards the $200 / month payment.
It would be interesting to see some models.
For messier situations, I wonder if a better short-term paydown strategy is to maximize liquidity.
E.g., suppose a person's debt is: (a) ($500/month, 3% interest, 3 months remaining) + (b) ($2000/month, 6% interest, 20 years remaining), and they're barely making ends meet each month. Prioritizing the paydown of (a) could eliminate a lot of psychological/financial stress by reducing the risk of a cash crunch.
Smallest debt first is not mathematically optimal but it's more tractable for people. And once you kick the little debts that's only costing you small monthly payments that gives you more financial stability making paying off the other debts easier.