Daniel bakes $420$ cakes.  He bakes only vanilla cakes, banana cakes, lemon cakes and chocolate cakes.  $\dfrac{2}{7}$~of the cakes are vanilla cakes.  $35 \%$ of the cakes are banana cakes.  The ratio of the number of lemon cakes to the number of chocolate cakes is $4:5$  Work out the number of lemon cakes Daniel bakes. 
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