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The way John Kasich tells it, he inherited a troubled state that was hemorrhaging jobs, saddled with a huge budget shortfall and in desperate need of a rescue.

Five years later, he says he has turned Ohio around. That story is a cornerstone of his bid for president, as he tells voters he can bring his Ohio success story to the rest of the country. “Let’s take some Ohio to Washington to help the entire United States of America,” Mr. Kasich said at a cavernous auto glass plant near Dayton on Friday.

Ohio has indeed gained several hundred thousand jobs since Mr. Kasich took office, and he turned an imposing budget gap into a surplus while also cutting income taxes, all accomplishments that back up his boasts.

But a closer review of his record shows the reality is more complicated. Other states recovered from the recession more quickly than Ohio did. He closed the budget shortfall in part by cutting aid to local governments, forcing some of them to raise their own taxes or cut services. And increasing sales taxes helped make the income tax cuts possible. “Ohio was in intensive care, and Governor Kasich came in and really stabilized the patient,” said Rea S. Hederman Jr., the executive vice president of the Buckeye Institute for Public Policy Solutions, a right-leaning group. “But we’re still a sick state, economically speaking.”