At the Conservative party conference two weeks ago, I spoke about a young black woman who had to put a more white-sounding name on her CV before she started getting called for interviews. Such racism in 21st-century Britain, I said, was a disgrace, and I committed our party to ending discrimination and finishing the fight for real equality. The audience rose to their feet. It was a significant moment. And it provoked three reactions.

The first was: “What on earth are the Tories doing speaking about equality? It’s not their issue.” To this I say “nonsense”. We’re not just the party of the first Jewish prime minister and the first female prime minister; we’re the party that introduced the Disability Discrimination Act, legalised gay marriage and reduced stop-and-search, and that today will start forcing companies to publish the gap between men and women’s pay, and making police record Islamophobia as a separate hate crime.

I accept that we haven’t always got this 100% right. No party ever has. And, because of our belief in equality of opportunity, as opposed to equality of outcome, for too long we have thought the only barrier to success was a lack of a decent education. Of course, that’s the most important thing – and it’s why school reform has been at the heart of our plan in government.

But we need to sweep away the other things holding people back, not least discrimination. You can give someone all the opportunities in the world – a brilliant school, first-class training, decent jobs – but it’s no good if they’re prevented from getting on because of their gender, race, religion, sexuality or disability. Getting this right is a natural and logical next step for a modern, compassionate Conservative party that wants to extend social mobility.

The second reaction was: “All right then, the Tories get this, but the only thing you can do is pass laws, and where are your proposals?” Of course, laws are important, like ours on disability discrimination and equal marriage. And we should recognise that the Race Relations Acts of 1965 and 1976, or the Equal Pay Act from 1970 didn’t just outlaw discrimination, they also helped to change attitudes and shift public opinion.

But for all the legislation we have passed, discrimination still persists. It’s no longer signs on doors that say “no blacks allowed”; it’s quieter and more subtle discrimination. It’s the disappointment of not getting your first choice of university place; it’s being passed over for promotion and not knowing why; it’s organisations that recruit in their own image and aren’t confident enough to do something different, like employing a disabled person or a young black man or woman. In my opinion, you won’t change these attitudes simply through more laws, but in smarter, more innovative ways.

The third reaction was: “Right, if you mean this, when and how will you show us how?” We have managed to get some of the biggest graduate employers to pledge to anonymise their job applications – in other words, make them name-blind. That means those assessing applications will not be able to see the person’s name, so the ethnic or religious background it might imply cannot influence their prospects.

The civil service, BBC, NHS, local government, HSBC, Deloitte, KPMG, Virgin Money, learndirect – all these and more will now recruit people solely on merit. The Conservative party HQ will do it too. Taken together, these organisations employ 1.8 million people.

And we’ll go further. Some research has shown that top universities make offers to 55% of white applicants, but only to 23% of black ones. The reasons are complex, but unconscious bias is clearly a risk. So we have agreed with UCAS that it will make its applications name-blind, too, from 2017.

Britain has come so far, but the long march to an equal society isn’t over. Today’s announcement is not the only thing we can do, but it’s a milestone. And it means that a young black woman knows she’ll get a fair shot when she applies for the job of her dreams.