Quick change Kate: Duchess of Cambridge wears £50 frock to meet street children at New Delhi railway station – then steps out in £795 chic mint lace for lunch with Indian prime minister 

Just an hour or two earlier, the Duchess of Cambridge was playing games with India's most vulnerable children in a £50 frock and flat pumps. 

But after a swift costume change that would impress many a West End theatre, she was stepping out to have lunch with the Indian Prime Minister in an £800 lace dress paired with a formal chignon.

It was all part of another busy day on the Royal tour, which started with the Duchess looking radiant in a casual printed maxi dress as she joined her husband at a centre for street children in New Delhi.

The royal couple visited a facility run by the Salaam Baalak Trust, an organisation supporting some of the youngsters who arrive into the city's railway station each day in search of a better life.

Unusually on a public engagement, Kate, 34, opted for flat cream pumps which she paired with a long burgundy ethnic-style dress with a belt tied around the middle. She wore her hair in loose curls half tied up off her face to finish her relaxed look. 

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Art class: The Duchess of Cambridge sat cross-legged as she drew pictures with some of the street children helped by the charity

Costume change: Kate was whisked away to have lunch with the Indian Prime Minister in an £800 lace dress paired with a formal chignon

William and Kate, who are on the third day of their week-long tour of India and Bhutan, today met some of the 6,000 children who travel to New Delhi each year, often fleeing from abuse or appalling poverty.

Once they arrive in the Indian capital, the youngsters become a target for prostitution, people trafficking or sexual and physical abuse. 

The royal couple heard how the Trust's outreach workers try to identify the most vulnerable young children as soon as they arrive, encouraging them to seek support. 

The Duke and Duchess heard how some of the children have their eyes gouged out and hands hacked off to make them more 'attractive ' as professional beggars.

But the horrors of life on the street seemed far removed from the children's minds as they invited the visiting royals to join them for a round of carrom, a traditional table game. 

William and Kate appeared in good spirits as they crouched down on the floor with the boys, who live in the nearby station. 

Relaxed: Kate wore flat cream pumps underneath a long burgundy ethnic style dress with a belt tied around the middle

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Low-key style: The Duchess wore her hair in soft curls that were half tied back from her face for the visit to the train station

Playtime: The Duke and Duchess of Cambridge enjoyed a game of the traditional table game Carrom with street children in New Delhi

Greeting: The couple couldn't keep the smiles off their faces as they met with some of the thousands of children helped by the charity

Namaste: Kate gives the children a traditional greeting (left) while one of the boys looks on as the Duchess draws a picture (right)

A picture of a house drawn by the Duchess of Cambridge during her visit to a children's centre run by the charity Salaam Baalak in Delhi

Light-hearted moment: The Duke and Duchess of Cambridge play a game of carrom with street children at a centre in New Delhi

Joining in: The Duke and Duchess then also visited a temporary shelter for boys, located close to the train station

The Duke and Duchess of Cambridge visited a contact centre run by the charity Salaam Baalak, which provides emergency help and long term support to homeless children at New Delhi railway station

Prince William, 33, asked one of the boys: 'What's the game you're playing? Ah, carrom board. Can you show us how to play?' Flicking the draught, he laughed as it went too far and invited his wife to have a go. 

The charity's director Sanjoy Roy told the couple about the charity's work. 'The boys come here for four hours of lessons and some food every day,' he said. 'When they're not here, they're at the railway station.'

When Prince William asked whether it was dangerous, Mr Roy replied: 'Yes, so they try to stick together. We look after around 7,000 kids a year but every day around 40 to 50 new children arrive at the station.

'They often have to deal with trauma, learning difficulties, ADHD and we have special programmes to help them with that. These children that we look after are the most vulnerable. Some may have their eyes gauged out or hands hacked off.

'The primary reasons they run away from home are misunderstanding with step-parents, physical and mental abuse, incredible poverty or a life event such as forced marriage.' 

The charity has six homes, 21 contact centres and three Childline centres near stations, bus stands and railway stations across Delhi.

Prince William asked what he could do to help, and Mr Roy replied: 'Spread the word. People think of them as street kids, beggars, thieves but they are just children. They deserve an education, future and a life. They have a right to a childhood.' 

The Duke and Duchess of Cambridge arrive for a visit to a children's centre run by the charity Salaam Baalak, which provides emergency help and long term support to homeless children at New Delhi railway station on day three of the Royal visit to India and Bhutan

Crowds gather before the arrival of Kate and William at a contact centre run by the charity Salaam Baalak at New Delhi railway station

The Royal couple receive a drawing on their visit to a children's centre that supports homeless children at New Delhi railway station

The couple also heard how Salaam Baalak Trust always tries to reunite children with their families but where that's not possible they provide permanent care in one of their shelters, where they are given medical care as well as educational, creative and social opportunities. 

Dr Amit Sen, who started the charity's mental health programme, explained to the couple why this support is vital to help children coping with the trauma of life on the streets adapt to life in a nurturing environment.

Aides said the Duke and Duchess were particularly interested to see children's mental health being treated as a key priority in helping children to seek physical healthcare, shelter, and eventually education. Young people's mental health is a major focus of their charitable work. 

Prince William and Kate also met two Salam Baalak City Walk Guides. The Trust is famous for its city walks, guided tours of the Delhi streets by the children who used to live there before being rescued.

The tours provide revenue for the Trust, but more importantly allow the young people to tell their stories of hardship and survival as well as helping them with their English and communication skills.

Salaam Baalak Trust was set up in 1998 with the proceeds from the film Salaam Bombay, a film depicting the lives of vulnerable street children. 

The name Salaam Bombay means 'salutes the child'. Salaam Baalak Trust runs five children's shelters,which house over 500 children,and 13 contact points in and around Delhi, supporting children living and working on the streets. 

Formal: Prince William (left) shakes hands with Indian Prime Minister Narendra Modi as he and wife Kate arrive for lunch in New Delhi

Official engagement: The Duke of Duchess of Cambridge changed into more formal wear for lunch with Prime Minister Narendra Modi

Demure: The Duchess of Cambridge opted for a mid-length teal dress with cut-out detailing for her lunch with the prime minister

The Duke and Duchess of Cambridge are on a week-long tour of India and Bhutan taking in Mumbai, Delhi, Assam, Bhutan and Agra

Good spirits: The royal couple shared a joke with the prime minister before joining the leader for lunch at Hyderabad House in New Delhi

Kate's ethereal sea green lace dress was high necked and the severity of its cut was echoed by the Duchess's formal chignon

Kate and William then visited a boys' home near Delhi station where around 50 boys live in the four-storey building.

They were greeted in traditional Indian style with red tika spots and flower garlands.

After climbing four flights of stairs to the roof terrace, they chatted to some of the boys and joined in their art work.

Having been presented with some of their drawings, Kate said: 'Did you do this? It's beautiful, well done. Shall I do a drawing for you?'

Kate sat down next to Shansad Abdul, 12, who asked her to draw a picture of her house.

It's unknown exactly which of her two houses she drew, but the resulting picture with large chimney, slated roof and big front door did bear an uncanny resemblance to their Norfolk mansion, Anmer Hall.

Shansad helped her colour it in-doing blue clouds and a large yellow sun. He said afterwards (through a translator): 'I liked doing it with her very much and I learnt how to draw trees and greenery.

'She was a very good lady and very happy to sit and draw with me. I ran away from home because my family are very poor and couldn't look after me.

'I came to Delhi from Purniya in the state of Bihar because I knew my older brother was married and living here. I came on my own and all I knew was that I wanted to find him.

'But when I went to his address he had moved and I had no knowledge of where he was and nowhere to go. Childline found me at the station and brought me here-I've been here for two months. My brother doesn't know I'm here.

'I'm happy but I do miss my parents. They don't know I'm here as I have no way of contacting them.' 

Close links: A Kensington Palace spokesman said the couple are 'grateful to have this chance' to meet the Prime Minister, adding that the tour was first announced following Mr Modi's lunch with The Queen in January

William and Kate appear to share a joke as they sign a book during their meeting with Mr Modi at Hyderabad House in New Delhi

The Duchess smiles while posing for cameras during their private lunch with Mr Modi at a former palace in New Delhi

The royal couple also met the charity's founder Praveen Nair, 85, who used money from her daughter's Oscar-nominated film Salaam Bombay to establish it 28 years ago.

Mrs Nair was very impressed with their interest.

She said: 'It's really very heartening to see well-to-do people are aware of the problems and they come and see it for themselves. It's very fulfilling for me' the staff and the kids to come and see us. It's very good for the children as it makes them feel important and goes a long way to building their self-confidence.

'It was a revelation to see that they were like anyone else, like you or I. They didn't say 'oh it's dirty' or refuse to shake people's hands. That is the pre-conceived opinion for all well-to-do people and Britishers.'

Kate and William then returned to the High Commissioner's residence for private meetings with two other NGOs. They also met embassy staff and children who live in the compound.

The couple later had lunch with Indian Prime Minister Modi, where William discussed 'the pressures facing steel manufacturers in the UK and India', say sources.

There had been heightened speculation that the Duke of Cambridge would raise the issue of Indian firm Tata's decision to pullout of the British steel industry which has put 40,000 jobs in jeopardy.

Happy Birthday, Gran! Kate and William use a sword to cut a special cake for the Queen at a party to celebrate her 90th in New Delhi

William paid tribute to the Queen, describing her as a 'wonderful great-grandmother' for his children and very much his 'boss'

Kate changed into an ethereal sea green lace dress which was high necked and the severity of its cut echoed by the Duchess's formal chignon, courtesy of her personal hairdresser Amanda Cook Tucker.

The 'Desdemona' dress costs £795 and is by Temperley London.

A Kensington Palace spokesman said of the meeting: 'Their Royal Highnesses are grateful to have this chance to meet the Prime Minister - this tour was first announced following Mr Modi's lunch with The Queen in January.' 

They will then fly to Assam where they will spend two days in Kaziranga National Park to highlight rural and conservation issues.

As well as enjoying a safari in the World Heritage Site, which boasts tiger and the Rhinoceros Unicornis - Indian one-horned rhino - as elephant and swamp deer, they will take part in a colourful sunset ceremony.

Their visit comes at the time of the Bohag Bihu festival, the celebration of Assamese New Year. William and Kate will meet local people around a campfire and and watch dance and musical performances. They may even take part themselves. 

Guests of honour: The Duke and Duchess of Cambridge arrive for a reception at the British High Commissioner's residence in New Delhi to celebrate the Queen's 90th birthday at the end of the second day of their Royal visit to India

William was expected to speak to a large gathering of prominent Indians at the reception in tribute his grandmother

The Duke and Duchess will meet hundreds of guests from Indian society at the official garden party celebrating the Queen's 90th Birthday