The Conservative Party is fractured and Britain’s public finances are battered. That will test the political skills of a leader who has been involved in national politics for only seven years.LONDON — Rishi Sunak took over as Britain’s prime minister on Tuesday, the third in seven weeks, hoping to slow the revolving door at 10 Downing Street and restore stability to a government in turmoil.But as he assembled a cabinet and began to confront a grave economic crisis, Mr. Sunak faced formidable political challenges, for which analysts said his seven-year career in national politics had not fully prepared him. The swift, truncated nature of his election may further complicate his task.Having been elected with the votes of some 200 Conservative Party lawmakers, but not the party’s rank-and-file members, Mr. Sunak could have trouble claiming a mandate to lead a deeply fractured party, let alone the whole country. With his government forced into spending cuts and tax increases, he will have few resources with which to reward either his lawmakers or the public.“He’s inheriting a divided party with a large number of Conservative M.P.s and members who believe he has no legitimate mandate,” said Matthew Goodwin, a professor of politics at the University of Kent. “That’s compounded by the fact that the party is in a free-fall and it’s not clear it has a parachute.”And yet, on a day of now-familiar rituals, as Mr. Sunak, the fifth prime minister in six years, traveled to Buckingham Palace to be anointed by King Charles III, there was also a calm in British politics — something that had been missing since Boris Johnson’s chaotic departure this past summer.Much of that owed to the 42-year-old prime minister himself: His well-received address to the nation on Tuesday showed a degree of political awareness, conceding the mistakes of his predecessor, Liz Truss, and promising improvement, while also reaching out to her and Mr. Johnson.“I will place economic stability and confidence at the heart of this government’s agenda,” a somber and solitary Mr. Sunak said on Downing Street, after returning from the palace. “This will mean difficult decisions to come.”Mr. Sunak and King Charles III in Buckingham Palace on Tuesday.Pool photo by Aaron ChownHis decision to appear there without his wife or daughters, and to dispense with the cheering staff members that greeted Ms. Truss last month, lent his arrival a brisk, businesslike tone. It also underlined the contrast between Mr. Sunak and his predecessor, which he said would extend beyond optics.A former chancellor of the Exchequer, Mr. Sunak is expected to pull Britain back to more mainstream policies after Ms. Truss’s experiment in trickle-down economics, which rattled financial markets and badly damaged Britain’s fiscal reputation.More on the Political Turmoil in BritainMaking History: Rishi Sunak is the first person of color and the first Hindu to become prime minister of Britain — a milestone for a nation that is more and more ethnically diverse but also roiled by occasional anti-immigrant fervor.Economic Challenges: Sunak already has experience steering Britain’s public finances as chancellor of the Exchequer. That won’t make tackling the current crisis any easier.Political Primaries: Are primary elections of British leaders driving Britain’s dysfunction? The rise and fall of Liz Truss offers some lessons.Lifelong Allowance: As a former prime minister, Ms. Truss is eligible for a taxpayer-funded annual payout for the rest of her life. Some say she shouldn’t be allowed to receive it.“Mistakes were made,” Mr. Sunak said. “Not borne of ill will or bad intentions. Quite the opposite, in fact. But mistakes, nonetheless. And I have been elected as leader of my party, and your prime minister, in part, to fix them.”Mr. Sunak quickly set about selecting a cabinet remarkable for its familiar faces. He retained Jeremy Hunt, the chancellor whom Ms. Truss installed after ousting Kwasi Kwarteng, the architect of ill-fated tax cuts. Mr. Hunt, who has soothed the markets, is scheduled to present a more detailed fiscal plan on Oct. 31.Mr. Sunak also kept on Ben Wallace as defense secretary and James Cleverly as foreign secretary, even though both had backed Mr. Johnson over him in the leadership race. And he retained Penny Mordaunt, who mounted a spirited challenge to him in that contest, as leader of the House of Commons.It was a striking contrast to Ms. Truss, whose cabinet consisted almost entirely of people who had backed her for leader, and it seemed to signal a recognition by Mr. Sunak that he could not succeed by drawing dividing lines in the party.Clockwise from top left: Jeremy Hunt, Ben Wallace, Dominic Raab, Michael Gove, Suella Braverman, James Cleverly.AFP — Getty; EPA, via Shutterstock; EPA, via Shutterstock; EPA, via Shutterstock; AFP — Getty; AFP — GettyMost conspicuously, Mr. Sunak reappointed Suella Braverman as home secretary, a job she had been forced out of only a week ago, ostensibly because she breached security rules. Her appointment was a gesture to the Conservative Party’s right flank: Ms. Braverman is a hard-liner who wants to cut immigration numbers. She said her “dream” was to see flights deporting asylum seekers from Britain to Rwanda.Mr. Sunak did reward some loyalists, naming Dominic Raab, who campaigned faithfully for him, as deputy prime minister and justice minister, posts he held under Mr. Johnson.Ms. Truss made her own appearance at Downing Street in the morning with her family, after formally submitting her resignation to the king, just seven weeks after she had been anointed by his mother, Queen Elizabeth II, in one of her last official acts, two days before her death.In defiant, unapologetic farewell remarks, Ms. Truss took credit for protecting people from rising energy bills. Reiterating her belief in lower taxes and a fast-growing economy, she said, “I am more convinced than ever we need to be bold and confront the challenges that we face.”Taking a page from Mr. Johnson, who likened himself to the retiring fifth-century Roman politician Cincinnatus, Ms. Truss quoted the Roman philosopher Seneca: “It is not because things are difficult that we do not dare. It is because we do not dare that they are difficult.”Liz Truss after her farewell remarks on Downing Street on Tuesday.Justin Tallis/Agence France-Presse — Getty ImagesMs. Truss’s misfires have made Mr. Sunak’s job even more difficult. Britain’s straitened public finances and its higher borrowing costs — a consequence, in part, of rising interest rates in reaction to her policies — will require painful spending cuts. That will further test Mr. Sunak’s political skills. Last summer, he struggled to sell his tough-love message to party members, who preferred Ms. Truss’s supply-side remedies.“The ideological riddle that Sunak has to try to solve is how the Conservative Party, amid a profound and prolonged economic crisis, can reconnect with the voters it attracted after Brexit,” Professor Goodwin said.Mr. Sunak did reappoint Michael Gove, a seasoned minister, to a post overseeing efforts to “level up” struggling cities in the Midlands and north of England with more prosperous London. That is important to retaining working-class voters who propelled the Conservatives to their landslide general election victory in 2019.As chancellor, Mr. Sunak was lionized when he doled out billions of pounds to people who had lost their jobs because of the coronavirus pandemic. He sponsored another good-news program, “Eat Out to Help Out,” which subsidized meals at restaurants to revive the industry after lockdowns.But when it came to withdrawing those benefits and raising taxes, Mr. Sunak’s reputation unsurprisingly suffered. During his campaign against Ms. Truss, he struggled to stick to his message of fiscal conservatism. Under pressure from her promises of tax cuts, he said he would temporarily suspend the value-added tax, a sales tax, on energy bills — something that he had earlier rejected.“He doesn’t have a lot of what I’d call trench-fighting experience,” said Tim Bale, a professor of politics at Queen Mary University of London. “His progress through the party has been so rapid that he hasn’t spent years forging friendships with colleagues who’ve got his back come what may.”Mr. Sunak and his wife Akshata Murty at the British Asian Trust Reception at the British Museum in London, in February.Vickie Flores/EPA, via ShutterstockProfessor Bale said Mr. Sunak was also thin-skinned about criticism he faced last spring of his wife, Akshata Murty, the daughter of an Indian technology billionaire, for her privileged tax status. Her so-called non-domicile status allowed her to avoid paying taxes in Britain on millions of pounds of her global income (she eventually agreed to pay British taxes).While Mr. Sunak’s sensitive reaction to the attacks against his wife may have been understandable, he is likely to face many more of them in the coming months from an opposition Labour Party that will seize on his extreme wealth to paint him as out of touch with the anxieties of ordinary people.“They don’t care that he and his family are filthy rich,” Professor Bale said. “They do care they didn’t seem to be paying their fair share. That — and his heated outdoor swimming pool and his house in Santa Monica — is going to make it difficult for him to argue, ‘We’re all in this together.’”Political analysts said the sheer magnitude of Ms. Truss’s failure was Mr. Sunak’s biggest asset. The Conservatives are trailing Labour by more than 30 percentage points in some polls. Even those who ardently opposed Mr. Sunak recognize that he is likely their last hope of avoiding a general election rout that would sweep hundreds of Conservative lawmakers out of their seats.“His M.P.s have looked over the edge of the precipice and know that, unless they get behind the guy, who is basically their last chance, they’re heading for a huge fall,” Professor Bale said. “Basically, it’s Rishi or bust.”Mr. Sunak is Britain’s third prime minister in seven weeks. Hannah Mckay/Reuters More