The family car you bought new for $5,000 five or six years ago has given up the ghost. You liked it and think it would be nice to buy another like it. You head to the nearest showroom and . . . SURPRISE! The corresponding make and model now is over $10,000 and you'll have to part with $1,000-$3,000 up front as a down payment . . .

Few new-car buyers walk into a dealership today, pick a car and pay for it in cash. It's more likely that the buyer makes a down payment and finances the balance--through a credit union, bank or automobile-company finance plan. And now, say car dealers, an increasing number of people are coming in and leasing, as a result of the tight economy, high interest rates and short money supply.

"Five years ago we wouldn't have bothered with an individual customer," says one area leasing official, "but now they're making up a sizeable part of our business."

Sid Rose of the American Automotive Leasing Assn., Milwaukee, Wis., says there are around 1.5 million cars, representing 8 or 9 percent of the domestic new-car market, currently under lease to small businesses and individual consumers. He says new-car leases on the consumer level could grow by 5-7 percent each year over the next five years, assuming there is no major change in the economy.

Another car dealer predicts that leasing may account for as much as 20 percent of new-car deals by 1985, "and maybe even sooner than that."

What makes leasing an attractive option?

* No money down. Leases generally call for one month's payment in advance plus a security deposit of slightly more than one month's payment. Area lending institutions require down payments ranging from 20 to 33 percent for the purchase of new cars; used cars, when they are financed, often take as much as 30 percent down. Although some credit unions may finance the entire cost of a new-car purchase, they are in the minority.

The no-money-down factor represents one of the major advantages of leasing: The consumer can take the money that would have gone for a down payment and invest it in a money-market fund or other high-interest-bearing account.

* Lower monthly payments. Area bank interest rates on new-car purchases run 16 to 18 percent; credit-union rates range from 12.8 to 16 percent; car-company financing rates go from 16 to 24 percent (averaging 19-21 percent). Interest rates on lease vehicles currently run 12.5 to 15.5 percent.

* No car to resell or trade in.

People who lease can turn in their car at the end of the leasing period and lease another or just walk away. No seeking out a buyer for the old car, or negotiating for the best trade-in with a new-car dealer.

Disadvantages to leasing:

The mileage surcharge can add up rapidly if you exceed the stipulated miles-per-year ceiling in the closed-end lease arrangement.

* You don't own the car. If at the end of the lease period you decide to buy the car, the buy-out figure and monthly payments usually will exceed the amount you would have paid to purchase the car in the first place.