The unemployment rate for black workers rose from 13.6 percent to a seasonally adjusted record 15 percent last month as the figure for black and other minority teen-agers soared to an all-time high of 45.7 percent, the Labor Department reported yesterday.

For all workers, the unemployment rate rose from 7 percent in July to 7.2 percent in August. The rate for white employes fell slightly from 6.2 percent to 6.1 percent, leaving joblessness among blacks 2 1/2 times higher than that for whites. Altogether, 7,657,000 people were looking but unable to find jobs last month, with 414,000 being minority teen-agers.

Government officials cautioned that the August report on minority teen-agers could be a fluke, but offered no explanation for the sharp rise in black unemployment.

Separately, the department also reported that producer prices for finished goods continued their moderate upward climb last month, rising 0.3 percent. In the last year finished goods prices rose 7.9 percent, the smallest increase for a 12-month period since August, 1978.

Unemployment among black teen-agers was 50.7 percent. Even with the somewhat smaller figures for other minorities included, the level was about three times as high as that for teen-aged whites, whose unemployment rate dropped in August from 16.1 to 15.6 percent. The rate for black and other minority teens was 36.4 percent in July.

Asked about the unprecedented unemployment rate for black teen-agers, deputy White House press secretary Larry Speakes said, "Certainly we're conscious of it." But he conceded that it was not specifically addressed during a meeting President Reagan had on the economy yesterday with Murray L. Weidenbaum, chairman of the Council of Economic Advisers.

"Unemployment among black teen-agers has always been uncharacteristically high," Speakes said. He said the administration will "do whatever we can" to bring it down, but he envisioned no new programs to accomplish the task.

"We're concerned about it," Speakes said. "It is a historic problem. We're hopeful that our economic approach will . . . create jobs."

Because of the relatively small number of minority teen-agers included in the national surveys, the figures for them are volatile, and officials at the Bureau of Labor Statistics warned that the 45.7 percent figure might be a fluke.

BLS Commissioner Janet L. Norwood urged "caution in ascribing a great deal of significance to this one-month increase," but said "the data do confirm that unemployment for black teen-agers is a continuing problem."

Norwood noted that the rate had been in the 30-percent-to-40-percent range for months, and added, "data for additional months are needed to understand whether an underlying change in the labor market for black teen-agers is really occurring."

The unemployment rate for adult men, which had dropped in both June and July, rose from 5.6 percent to 5.9 percent in August. The rate for adult women, on the other hand, fell to 6.5 percent from 6.7 percent.

The small increase in producer prices was welcomed at the White House, where Speakes declared, "We continue to be encouraged by progress in bringing inflation under control . . . . What we see is a continuing pattern of steady moderation."

Among finished goods, consumer food prices rose 0.2 percent, much less than the 1.5 percent jump in July. The index for finished energy goods fell 0.7 percent, the third drop in the last four months. Gasoline prices dropped another 1.6 percent, following declines of 2 percent in July and 1.2 percent in June. Heating oil prices fell 0.7 percent after a 2 percent drop in July, but natural gas prices rose 1 percent.

Much of the slower increase in food prices was due to a turnaround in beef and veal prices, which fell 2.1 percent after climbing 3.8 percent in July.

The index for finished consumer goods other than foods and energy rose 0.5 percent, more than the 0.2 percent in July.

The good news on prices extended to intermediate and crude materials as well. The index for intermediate materials rose 0.4 percent, the same as in July. The index for crude materials fell 0.6 percent, largely because of a 0.9 percent decline in prices of foodstuffs and feedstuffs. Other crude materials prices also fell 0.1 percent, however.

Most economists attribute the slower increase in producer prices to tight monetary policies and high interest rates, which have combined to hold down commodity prices and slow economic activity.