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MIAMI — Nearly one-fourth of recent high school graduates who try to join the Army fail its entrance exam, painting a grim picture of an education system that produces graduates who can’t answer basic math, science and reading questions, according to a study released Tuesday.

The report by The Education Trust bolsters a growing worry among military and education leaders that the pool of young people qualified for military service will become too small.

“Too many of our high school students are not graduating ready to begin college or a career, and many are not eligible to serve in our armed forces,” U.S. Education Secretary Arne Duncan told The Associated Press. “I am deeply troubled by the national-security burden created by America’s underperforming education system.”

The effect of the low eligibility rate might not be noticeable now — the Defense Department says it is meeting its recruitment goals — but that could change as the economy improves, said retired Navy Rear Adm. Jamie Barnett.

“If you can’t get the people that you need, there’s a potential for a decline in your readiness,” said Barnett, who is part of the group Mission: Readiness, a coalition of retired military leaders working to bring awareness to the high ineligibility rates.

The report found that 23 percent of recent high school graduates applying for the Army don’t get the minimum score needed on the enlistment test. Questions are often basic, such as: “If 2 plus x equals 4, what is the value of x?”

The results are worrisome also because the test is given to a limited pool of people: Pentagon data show 75 percent of applicants ages 17 to 24 don’t qualify to take the test because they are physically unfit, have a criminal record or didn’t graduate from high school.

Educators expressed dismay that so many high school graduates are unable to pass a test of basic skills.

“It’s surprising and shocking that we are still having students who are walking across the stage who really don’t deserve to be and haven’t earned that right,” said Tim Callahan with the Professional Association of Georgia Educators, a group that represents more than 80,000 educators.

This is the first time the Army has released this test data publicly, said Amy Wilkins of The Education Trust, a Washington-based children’s advocacy group. The study examined the scores of nearly 350,000 high school graduates, ages 17 to 20, who took the Armed Services Vocational Aptitude Battery (ASVAB) between 2004 and 2009. About half of the applicants went on to join the Army.

Recruits must score at least 31 out of 99 on the first stage of the three-hour test to get into the Army. The Marines, Air Force, Navy and Coast Guard require higher scores.