The minimum score required for an A from the top tier of an Edexcel maths exam, taken by 80,000 pupils, was set at 45 per cent for two of the three papers. More than half of those sitting the papers got an A or A*.

Across the three papers, candidates would have needed only an average of 51 per cent to achieve an A, while the mark required to gain a C equated to only 21 per cent. Only 0.7 per cent of those entered failed to achieve a C.

The figures were described as “ludicrous” by one of the country’s leading maths experts, and even Edexcel admitted that it was possible for some students to achieve As without having mastered a central topic such as algebra.

Roger Porkess, who developed the first modular maths A-level syllabus, said: “These grade boundary figures are incredibly low. The idea that to get a grade A on a paper you only need 45 per cent means you can get a grade A without knowing any algebra.”

Doug French, of the Maths Association, said: “To give pupils a paper in which they can get a high-grade pass on less than half marks seems crazy.

It suggests they can achieve A grades by missing out on whole swathes of maths.”

The revelation is likely to provoke more claims of dumbing down. Only last month, when the GCSE and A-level results were announced, the Confederation of British Industry renewed its criticism that many youngsters were leaving school with an inadequate grasp of basic arithmetic.

However, Edexcel firmly rejected any suggestion that standards were falling. It explained that only 14 per cent of students who took GCSE maths through it achieved an A or A*. The figure for those taking the higher tier was 54 per cent, but only the top quarter of pupils take these papers. It said the numbers achieving top grades reflected candidates’ abilities.