Since 2010 the number of children taken into care has risen by 15 per cent and in 2017/18 the national overspend by councils on children's social care was £872m.
But a National Audit Office (NAO) review of children's social care spending found that ministers do not know why demand has grown.
They also lack an understanding of why there is such a wide variation in local spending, which ranges from £566 to £5,166 per child a year.
The Department for Education has commissioned research to explain the growth in demand and local variation in December 2017 but this will not be ready until this summer, says the report.
The government has estimated that 41 per cent of the increased demand is due to population growth but has so far failed to explain in detail what other factors are influencing demand.
The NAO review suggests that different strategies to tackling demand by councils may be a large factor. Deprivation could be another.
Without such knowledge the DfE is likely to miss its target of ensuring that all vulnerable children have access to high-quality support by 2022, states the report.
It adds that the DfE "still does not fully understand what is driving demand for children's social care or why there is such wide variation between local authorities in their children's social care activity and costs.
"It has not yet done the work to tie together available sources of information and therefore lacks a well-informed pathway to achieve its goal."
The NAO released a similarly critical report of DfE efforts to improve children's social care in October 2016.
"Over two years ago we reported that the Department for Education's progress in improving children's services was not up to scratch," said NAO head Amyas Morse.
"Since then the department has adopted the target of giving all vulnerable children access to high-quality support, no matter where they live, by 2022.
"The department has started to build its understanding of variations in services, but it should know more than it does. Even with this understanding, the department faces a tall order to achieve its goal within three years."
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The NAO's report also details latest funding figures that show that in 2018/19 councils expect to spend £4.2bn on children in care, which is £350m more than they budgeted for the previous year.
Another finding is that there is no link between how much a council spends on each child in need and the quality of support they receive. The NAO found that while some children's services that are rated as good by Ofsted are spending £570 per head, others with the same rating are spending £4,980.
Councils are also struggling to find suitable residential care placements. Only a third (32 per cent) of councils have access to enough children's homes for children aged 14 and 15.
The NAO is asking the DfE to work with councils to ensure there is enough residential places to meet demand.
Meg Hillier, chair of the House of Commons public accounts committee said the government needed to "act swiftly" to ensure "long overdue improvements" are made in children's social care.
"Children's safety and wellbeing must not be subject to a postcode lottery and it is appalling that the department does not fully understand what is driving demand for children's social care or why there are such wide variations between local authorities," she said.
Minister for children and families Nadhim Zahawi said: "We have raised the bar in children's social care and the child protection system, so that children at risk are identified sooner, and we are tackling the reasons why children are in need in the first place.
"The number of local children's services rated outstanding is growing, and the number rated inadequate has dropped by a third since 2017 - from 30 down to 19. By 2022, I want this reduced to fewer than 10 per cent of councils, and we are on track to meet this.
"As the report acknowledges, we are currently working across government to improve our understanding of demand for children's services. And we know there are pressures on councils, which is why we are providing an additional £410m in the Budget for adult and children's social care and an extra £84m to expand innovative practice to support vulnerable families across a further 20 councils."