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If you start with the notion that inflation is a period of prices rising faster than incomes, then rising interest rates will slow down the economic activity (which will slow down inflation).
When demand for a good exceeds supply, the price of the good will rise. Users or consumers will pay more for the good, driving price inflation. People who need to borrow money to purchase that good shall, until the cost of borrowed money goes up to a point that the purchase is less desirable. Think residential home purchases.
An increase in the cost of money decreases the real value of the house being purchased. While the typical limit is felt at the bank, when income has not increased, but the house price and the cost of money both have. Price inflation will start to slow, since there will be less buyers chasing that and other homes.
Central banks (the US Fed and others) have long used interest rate tools to manage inflation. The quantitative easing program in the US was designed to stimulate demand, until price inflation exceeded Fed targets.
If rates are raised, even a bit, starting tomorrow, stock market prices are likely to decline, as is demand for borrowed money (it will be more expensive). In this indirect way, interest rates affect inflation.
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