The proportion was significantly higher than those who received the special measure in state schools where the figure is fewer than one in eight.
The difference has come to light following an analysis of official exam data by BBC Radio 4's Today programme.
Exams watchdog Ofqual said this may be because of the readiness of schools to find students entitled to more time.
HMC, a group representing independent schools, said the higher rate was down to "proper resourcing" which it said "can be be lacking in state maintained schools".
But the head teacher of a comprehensive school in Suffolk said the different rates raised "serious questions".
Students receive extra time, typically more than three hours to complete a two-and-a-half hour exam, because of special needs or disabilities like dyslexia.
The system has previously faced claims that better resourced schools may be more likely to receive the measure, which is available as part of rules to ensure a level playing field for exam candidates.
Last year in GCSE and A-level exams, extra time was awarded to more than 27,000 independent school students -  which represented nearly 20% of all candidates in the sector.
In state institutions, around 200,000 students received extra time, which represented less than 12% of all state sector students taking the exams.
The total number of students awarded extra time in England, Wales and Northern Ireland has increased every year since 2011.
Ofqual has previously introduced measures to try to reduce requests for so-called Access Arrangements after claims the system was being abused.
The Joint Council for Qualifications (JCQ) which represents the UK's biggest exam boards, said it was important the arrangements were approved only for those who needed them.
In a statement, it said: "There's a rigorous process to ensure this, including annual inspections, and we review this process every year.
"Fairness is ensured by the process being the same, irrespective of location or type of school or college."
Geoff Barton, head teacher of King Edward VI comprehensive school in Bury St Edmunds, blamed a "high stakes" accountability system including results and league tables for influencing schools' use of the measure.
He said: "Whether they are getting special consideration does seem to be something that isn't only done in the interests of the youngster, but also can have an effect for the school.
"So there's no surprise that schools will reflect on that as they are planning the exam season."
He added: "What the data does is to raise some serious questions for schools."
Peter Hamilton, HMC's academic policy spokesman and headmaster of Harberdashers' Aske's Boys' School in Borehamwood, said the independent sector was able to apply for help for pupils "because of ever-improving monitoring and awareness of special needs".
He said: "We are also fortunate to have proper resourcing and specialist departments, which can be lacking in state maintained schools.
"All heads want to see learning support staff given the time and money necessary to ensure all pupils are able to claim their rights."
The watchdog said the data did indicate some variation between types of examination centres.
It said the difference could be partly explained "by the ability and readiness of centres to identify students who are entitled to a reasonable adjustment".
Both state and private sectors involved in the data comparison include special schools, whose students are more likely to require access arrangements.
Ofqual said the private sector figures included independent special schools which may impact upon the percentages when making comparisons.
An independent special schools source said the numbers of students involved in such exams was relatively low. It's thought unlikely to have a significant impact on the comparison between sectors.
A charity that supports people with dyslexia has previously said schools needed better training to get pupils assessed, adding that students did not need formal or costly diagnoses to qualify for access arrangements.
The Today programme first used freedom of information laws in 2014 to ask JCQ for figures on access arrangements in various types of schools.
Following a ruling by the Information Commissioner, it emerged that the data would be supplied by exam boards to Ofqual.
One in five students in independent schools received extra time to complete GCSE and A-level exams last year.