How does a currency like USD get stronger against other currencies?

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What you're asking is a FOREX question and it is the basis of everything foreign exchange in the entire world. Basically speaking, currencies are susceptible to increments in their value with respect to other currencies. These increments are caused when a person, company, firm, etc buys a large amount of one currency while selling another. By dumping one currency and buying another, you increase the value of the currency that is bought and you decrease the value of the currency that is sold. That is how currencies like the US dollar strengthen against others. Let's take an example.

Let's say Toyota is about to be sold to an American man. Toyota is worth about 18 trillion yen. Converting that to the US dollar gives 236 billion dollars. So this American man is about to buy Toyota for this price. But since he's buying from a Japanese man, he has to pay in yen. So he converts 236billion dollars into yen to make the payment. By doing so, he dumps 236 billion dollars and buys 18 trillion yen. Thus reducing the value of the dollar incrementally and raising the value of yen at the same time. This is a very direct way of looking at it but there are other ways by which a country's currency strengthens.

It's not everyday giant companies are bought and sold between countries. But investments are made everyday. A large investment into a nation will cause it's currency to strengthen. That's why most countries are always looking for foreign investors. By investing heavily in a country, you strengthen their currency. So this is how currencies gain strength against each other