It is best to check with your country's passport services agency for the answer to this question, as it varies by country. In the United Kingdom, for example, if there is any time left on your old passport, the Identity and Passport Service (IPS) will add it to your new passport, up to maximum of nine months. In the United States, on the other hand, your passport is always only valid for 10 years (if you are over 16 years old) or 5 years (if you are 15 years old or younger) from the date it is issued, regardless of the date on your old passport. The same is true for Australia. To find out the situation in your country please visit your Government's website.
One thing to consider when traveling: some countries will not admit visitors with passports expiring within six months or will not put visas them. So, check with your destination and renew accordingly.
Link for U.S. passport questions: http://travel.state.gov/passport/fri/faq/faq_1741.html.
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