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There is a UK capital gains tax annual exempt amount, currently £11300 https://www.gov.uk/government/publications/rates-and-allowances-capital-gains-tax/capital-gains-tax-rates-and-annual-tax-free-allowances

There is also a UK tax-free personal allowance, currently £11500 https://www.gov.uk/income-tax-rates

Are these two tax allowances independent?

If someone receives £11300 in capital gains and £11500 in income from working do they have to pay any tax?

If someone receives £11300 in capital gains and £30000 in salary do they only pay tax on the £30,000 income?

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Yes, the two allowances are completely separate and independent.

If someone receives £11300 in capital gains and £11500 in income from working do they have to pay any tax?

No.

If someone receives £11300 in capital gains and £30000 in salary do they only pay tax on the £30,000 income?

Yes. More specifically, they would pay 20% tax on £30,000 - £11,500 (i.e. £19,500).

The only place that the two rates / allowances have any interplay is that the rate of CGT you pay (10% or 20%, on gains above the CGT allowance) depends on whether your other income makes you a basic rate tax payer or a higher rate taxpayer.

There is a good guide here: https://www.which.co.uk/money/tax/capital-gains-tax/guides/capital-gains-tax-allowances-and-rates

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