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Photo: David Goehring via Flickr.

If small money does not go out, big money will not come in.
--Chinese proverb

When many Americans think about how well they're preparing for retirement, they may end up despairing. Things are probably not quite as bad as they seem, though, because relatively small sums can go a long way, as some simple math demonstrates.

A common goal for retirement is a nest egg of $1 million. If you withdraw a relatively conservative 4% from it annually (adjusting for inflation in future years), you'll start off with an income of $40,000. Between that and Social Security and perhaps some pension or annuity income, a comfortable retirement can seem within grasp.

A million dollars is a lot of money, though, and here's the problem: According to the 2014 Retirement Confidence Survey, for example, 60% of American workers have less than $25,000 saved for retirement (excluding the value of their home), and 36% of American workers have less than $1,000 saved for retirement. Yikes!

There's hope
Depending on your age now and your earning potential, it may not be possible to accumulate $1 million by retirement. Still, there are lots of steps you can take to improve your retirement prospects. (And plenty of people can get by on less than $1 million, too.)

I'll run you through some scenarios to give you an idea of how you might accumulate $1 million, as well as how you might save smaller yet significant sums. Keep in mind that it's important to spend some time fiddling with online calculators to see what's realistic for you.

Imagine, for example, that you're 35 and hope to retire at 65, in 30 years. You sock away just $100 per month (only $1,440 per year) in a broad stock market index fund and earn an annual average of 8% -- less than the market's long-term average of about 10%. By retirement, you'll have accumulated $149,000. Unless you will have significant other sources of income, that's not likely to be enough, but it would put you far ahead of the many people who have $25,000 or less to their name. Want that million? You'd need to sock away $675 per month. A more reasonable sum for many people is $400 per month, which would get you nearly $600,000.

Another way to boost your nest egg is to work a little longer -- especially if the market dips shortly before your planned retirement age and your savings take a hit. Going back to our $400-per-month example, if you decided to work just two more years, retiring at 67, you could end up with $710,000, having added more than $113,000 to your nest egg in just two years, though you invested far less than that.

Bear in mind that growth rate matters a lot. An 8% return is not guaranteed, and the market doesn't go up in a straight line. If you managed average annualized returns of 10%, then your $400 per month would become $904,000 in 30 years, and in two additional years it would top $1.1 million. If you were unlucky and the market averaged an uncharacteristic 6% over those 30 years, you would end up with only $463,000.

Further, many people will shift some assets out of stocks and into bonds as retirement approaches, which, though it will lower their risk, will likely throttle their returns. There's no guarantee that you'll get to a million dollars, but by crunching the numbers, you can get some idea of what it will take.

Tweak those numbers
Of course, odds are you're not exactly 35. Let's say you're actually 50 and looking to retire in 15 years. Let's imagine that you save aggressively and are able to put away $1,000 per month, or $12,000 per year. (You might put $5,000 in a Roth IRA and $7,000 in a 401(k) plan at work, for example.) If you average annual gains of 8%, you'll accumulate $346,000. Want more? Retire at 67 and you'll have nearly $432,000, and by age 70 you'll be close to $600,000. It's not a million, but it's a lot.

Bear in mind that the ride to a cushy retirement portfolio will not be smooth. There will be good years and bad years. However, the stock market has proven again and again to be an unbeatable investment over periods of decades. That's why it's important to invest for the long term and stick to your well-laid plans.

Small sums become big sums
The more you can sock away, the better, and it's important to appreciate the power of small sums. Every single dollar that is invested for 30 years and averages 10% annual growth will turn into $10.83 that you can spend in retirement. Time can make small sums quite powerful.

You'll find that you can plan for whatever kind of nest egg you need. (My colleague Chuck Saletta has detailed how you can become a millionaire on minimum wage.) It's important to run the numbers and see how much you need to be investing. Clearly, the sooner you start, the smaller your regular contributions to your retirement accounts will have to be. No excuses!

Longtime Fool specialist Selena Maranjian, whom you can follow on Twitter, has no position in any stocks mentioned. The Motley Fool has no position in any of the stocks mentioned. Try any of our Foolish newsletter services free for 30 days. We Fools may not all hold the same opinions, but we all believe that considering a diverse range of insights makes us better investors. The Motley Fool has a disclosure policy.