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The quote currency is the second currency in a forex pair. It tells you how much of that currency is needed to buy one unit of the base currency (the first currency). Also called the counter currency or terms currency. In EUR/USD, the US dollar is the quote currency; in USD/JPY, the Japanese yen is the quote currency.
In USD/JPY at 150.00, JPY is the quote currency. The price means 1 US dollar costs 150 Japanese yen. If the price rises to 151.00, the quote currency (yen) has weakened — it now takes more yen to buy one dollar. Prices in this example are illustrative.