How Many Singular Values to Keep

An embedding matrix $A$ used by a retrieval system has exactly five non-zero singular values:

$\sigma_1 = 12, \quad \sigma_2 = 5, \quad \sigma_3 = 4, \quad \sigma_4 = 3, \quad \sigma_5 = 2$

Truncating the SVD after $k$ terms gives the best rank-$k$ approximation $A_k$. Two standard facts you may use:

$\|A\|_F^2 = \sum_i \sigma_i^2, \qquad \|A - A_k\|_F^2 = \sum_{i > k} \sigma_i^2$

The compression rule is: keep the smallest $k$ for which $A_k$ retains at least $90\%$ of $A$'s squared Frobenius norm (its "energy").

What is that $k$, and what relative error $\dfrac{\|A - A_k\|_F}{\|A\|_F}$ does it leave?