Since the days of the Puritans, many Americans have believed in a simple equation: Hard work equals success.

The recent sale of the start-up company WhatsApp for $19 billion is the latest eye-popping deal to raise obvious questions about whether that equation is still reliably true.

Certainly the founders and employees worked hard to create a company that Facebook thinks is worth that much in cash and stock. But did they work 1,900 times harder than the owner of a ­company that managed to find a buyer for $10 million? Or 19,000 times harder than a small-business owner who struggled to reach a net worth of $1 million?

There have always been ­winners and losers in a dynamic market economy, of course, but deals like this make the game look more and more like a giant lottery. The more often that big-dollar deals appear to land like some sort of random financial lightning strike, and a disproportionate share of the proceeds go to an ever-shrinking group, the harder it will be to sell the idea of the equality of opportunity to younger entrepreneurs.

That basic premise of American life, that there is equality of opportunity, is increasingly something only older Americans believe. They continue to buy the story that hard work pays off and effort is far more important than where you started in life.

Younger Americans are far more likely to say that the wealthy in the United States got there mainly because “they know the right people or were born into wealthy families,” according to a 2011 survey by the Pew Research Center. A majority of those 26 to 34 years old thought these kinds of things were far more important than skill, ­education and effort.

To some extent these are ­attitudes that are catching up with the reality of the job market, particularly as the middle class loses ground, but part of what drives entrepreneurs is their perception that ­business remains a meritocracy.

In the past two years, in conversations with entrepreneurs in coffee shops or at co-working sites, they told their stories about struggling to find customers and even the barest minimum of capital to launch their ­businesses.

Meanwhile, out in California, a company I had never heard of before this month was making success look easy. It had created software to send text messages on smartphones. It sold the software for 99 cents a year — beginning in the second year. WhatsApp didn’t have the first application of its type, and it wasn’t the best. It just got ­popular.

So popular, Facebook claimed when the deal was announced, that it was adding more than 1 million new users each day. One good way to think of how WhatsApp got to a $19 billion valuation was that it got much better at giving away its product than its competitors were at giving away theirs.

In talking with entrepreneurs in Minneapolis last week, it was surprising to hear the WhatsApp deal described as more or less like reading about a schoolteacher winning the lottery.

They are happy for her, but it means nothing to them. The old equation of what it takes to be successful still holds true, and the appeal of entrepreneurship is as strong as ever.

“My first reaction was confusion because I had never even heard of the app,” wrote Mark A. Lazarchic, the founder of the local software firm Otterology, in an e-mail. “Then I researched it and I still have no idea what it does. The amount doesn’t bother me at all. If Facebook wants to pay billions for your company, I say take it. You can buy new friends.”

As for hard work, that by itself has never been a surefire way to succeed. Only long hours on the right product that solves the right problem in the market is meaningful.

Clay Collins of locally based LeadPages explained that technology applications that suddenly blow up into wild popularity are those that the creators give away to consumers for free, like Twitter. Midwestern entrepreneurs are far more likely to have created a product or service that gets sold to other businesses.

He shared a Twitter exchange he had with the founder of a very young start-up, seeking his advice on whether to try to grab users by giving away a software product.

Bad idea, Collins said, as “growth vs. revenue is a false dichotomy. Revenue growth is the only real ­business growth.”

He didn’t say this, but he didn’t need to say it: Getting real paying customers for any new business is hard work.

“Rare is the entrepreneur that is in this solely for the money,” explained Matthew Dornquast, the co-founder and CEO of Code42 of Minneapolis, a software provider. “If you set out to create something with your primary motivation being [a sale], you will most likely fail. More popular motivations include creating something they want in the world, solving a problem that’s driving him nuts, wanting ­others to use it and appreciate it [and] wanting to have more independence in their life.”

And if entrepreneurs have the ­wisdom to think that way, he said, the rewards of creating a product and a business are far more than financial and just as valid as ever.

Dornquast suspects the founders of WhatsApp likely said no to $500 million or $1 billion buyout offers well before they talked about a deal with Facebook. With a venture capital firm as a shareholder, they undoubtedly have known for years that someday they needed to agree to a sale, but he’s all but certain the founders still care more about their products and users than the money and Facebook stock they will receive.

As Dornquast put it, “Hard work on something you are passionate about absolutely equals reward, 100 percent of the time.”