GREENOUGH, Mont. » At a time when many schools are concerned about overcrowded classrooms, the Sunset school in this ranching community has a different problem — keeping its lone student at her desk so it can remain open.

There are other schools in remote rural areas around the West that have only one teacher and one student, but the situation is even starker here. Amber Leetch, age 11, makes up the entire Sunset School District 30.

And while many one-student schools elsewhere in the West are in far-flung, impoverished areas, the Sunset district — whose entire annual budget is about $83,000 — is in a prosperous, ranching corner of the state. One of the reasons there is only one student is that the cost of the scenic landscape here has risen so high that young, aspiring ranchers, the kind who would be likely to have school-age children, cannot afford to buy the land.

Amber, a cheerful sixth grader, attended the historic Sunset school last year by herself and most of this year as well, though she had a first-grader for company for a few weeks. Once that first-grader left, though, it was just Amber again and her teacher, Toni Hatten.

"The hardest part is getting through the day without feeling too lonely," said Amber, as she drew on a computerized white board.

One way of fighting the loneliness is with Baylee, a 3-month-old husky mix that recently became the new school dog.

The one-on-one situation is challenging for both teacher and student, says Hatten, 45, who began teaching this year. There is little escape for either: No grouping students to work together on their own for a while, and the student cannot disappear for a while, while the teacher works with someone else. It is Amber and Toni together all day long.

"You’re their lunch buddy, their PE buddy, their recess buddy," Hatten said. "You’re their everything."

The school is isolated as well as small. The town of Sunset was once the center of industrial-scale logging for copper mines at Butte, but that is long gone and now the economy is ranching and the region is known as Greenough. The town that once stood here is gone, the general store and post office vanished and only the school remains and an outdoor bank of mailboxes. The nearest town is about 20 miles away. The city of Missoula is 35 miles to the west.

The isolation is offset somewhat by the beauty of the landscape. Sunset sits in a valley surrounded by rolling hills and mountains, with a blanket of towering ponderosa pines. The Blackfoot River, a famous fishing stream, flows near here.

Sunset is the smallest one-room school in Montana (a second room was added, but it is only used for physical education and storage), which has 62 of them, ranging from one to 18 students. It is the only one-room school in the state with a lone student, though there are some with two or three. About 20 small schools have closed in the last decade in the state.

Amber went here in kindergarten, then attended a school eight miles away. But she moved back here in the fifth grade. "She dealt with a lot of bullying and the school wouldn’t address it," said her mother, Wendy Leetch.

Not everyone thinks keeping a school open with one student makes sense. "It ruffles some feathers with other districts who say it’s a lot of money for one student," said Darlene Troutwine, the Sunset district clerk. But while the Sunset district seems to be teetering on the edge of disappearing, perhaps when Amber decamps for high school in two years, it is not the typical rural community in decline. The school sits on land owned by the Resort at Paws Up, a guest ranch that features high-end accommodations. Last summer Daniel Craig and Harrison Ford were at the Paws Up promoting their film "Cowboys and Aliens." An owner of the ranch, Nadine Lipson, is on the Sunset school board.

As recently as 2001 there were 20 students at the school, but escalating land prices here mean young ranch families must get out of the business or ranch elsewhere. Hatten has hopes of more students in the future — at least two, or perhaps three — to make things more interesting and to be sure the sun doesn’t set on the little school. "The Ovando School has 11 and feels like a family," Hatten said. "The older kids help the younger kids and they all play together."

School is four days a week here, from eight until four. When the day ends there is no yellow school bus to take Amber home. She lives off the grid with her parents and brother, near the high-altitude ghost town of Garnet, once a thriving mining town. Her mother is a tour guide there in the summer. The only access is by snowmobile.

On a recent school day, Amber and her mother donned helmets to make the five-mile trip through the mountains to their home, with Amber on the back of Wendy Leetch’s vehicle. Leetch recently got a new snowmobile with heated hand rests, which makes a big difference on the trip, especially when temperatures drop to below zero.

They waved as they departed, then roared off, leaving a cloud of blue exhaust hanging in the air.

© 2012 The New York Times Company