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The key idea is that the equity portion in an acquisition is a fixed percentage of the enterprise value (EV). The EV is determined by the EBITDA multiple, since EV = multiple × EBITDA. Here the multiple is 10x, so EV = 10 × EBITDA. The private equity firm’s equity investment is 40% of that EV (with 60% financed by debt).

Putting it together: equity = 0.4 × EV = 0.4 × (10 × EBITDA) = 4 × EBITDA. If EBITDA is 100 million, then equity equals 4 × 100 million = 400 million. This aligns with the given choice of 400 million. In general, the numeric equity depends on the actual EBITDA value, but with the implied EBITDA of 100 million, the initial equity investment is 400 million.