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When the value of the US dollar drops, your travel expenses will, all other things being equal, rise. This is, of course, only true if the US dollar drops relative to the currency of the country where you are travelling. It is possible for the dollar to drop relative to the currency of one country while remaining strong against another country’s currency.
When you travel to a foreign country, you will typically need to pay for everything in the currency of that country. You will have to use your own dollars to buy the currency of the country to use for buying things. When the dollar depreciates, your dollars buy less of the other country’s currency. This means that you will have to use more of your dollars to buy each unit of the other country’s currency. The prices might stay constant in the other country’s currency, but they will cost more of your dollars.
Thus, a falling dollar tends to make your travel expenses rise. You will need to spend more dollars to buy the foreign currency that you need.
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