When the price of bonds decreases, the interest rate rises. Explain?

Bond prices and interest rates have an inverse relationship, meaning that when one increases, the other decreases, and vice versa. To explain this, we will make use of the following example:
Company X issues new bonds that have a 5% annual coupon rate and $1000 par value. When interest rates go up to, say, 7%, buyers will be reluctant to invest in Company X’s 5% bonds. This is because at a 5% rate, a buyer will be expecting yearly interests of $50 only, compared to $70 that they would have gotten elsewhere for the same principal of $1000. Therefore, in order to make its bonds attractive to buyers, Company X settles on a lower price p, for which it will give a similar interest of $50 at current interest rates of 7%. That is:
p x 7% = $50 ; p = $50/0.07 = $714.30
Company X has to sell its bond at a discount price of $714.30.
When interest rates go down to, say, 3%, then the bonds become attractive to buyers. The increased demand pushes bond prices up. Company X must look for a price p that will yield the same interest of $50 at the current market interest rate of 3%. That is:
p x 3% = $50 ; p = $50/0.03 = $ 1666.70
Company X has to sell its bond at a premium price of $1666.70.
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