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A budget deficit occurs when a government's expenditures exceed its revenues, causing it to borrow money and increase its debt. A budget surplus, on the other hand, occurs when a government's revenues exceed its expenditures, allowing it to reduce debt or use the surplus for other purposes. The debt/GDP ratio is a measure of a country's public debt relative to its GDP, which indicates its ability to pay back its debt.

It is possible for a nation to run budget deficits and still have its debt/GDP ratio fall if: 1. Its nominal GDP growth rate (which includes inflation) is greater than the rate at which the deficit is increasing. 2. The deficit is used to make productive investments in the economy that boost GDP growth in the long-term, thereby decreasing the debt/GDP ratio. In such a scenario, the country's economy is growing at a faster pace than its debt, leading to a decline in the debt/GDP ratio.

It is also possible for a nation to run budget surpluses and still have its debt/GDP ratio rise in the following cases: 1. There is a sudden increase in interest rates, making debt servicing costs higher. Even if the government is running a surplus, the increase in interest payments due to higher interest rates can push the debt/GDP ratio higher. 2. The GDP growth rate is sluggish, or the economy experiences a recession. In this case, even with a budget surplus, the debt/GDP ratio could increase because the economy is not growing enough to offset the government's debt. In these situations, running a budget surplus might not guarantee a decline in debt/GDP ratio due to adverse economic conditions or higher interest payments. In conclusion, it is possible for a nation to run budget deficits and have a falling debt/GDP ratio, as well as to run budget surpluses and have a rising debt/GDP ratio, depending on the circumstances like economic growth, inflation, and interest rates.