Capital asset pricing model

Dear sir,

In past exam question 4a of june 2012. I got stuck in one part and I need your help.

The question has mentioned about proxy company and given details as follows:

lfu Co’s equity beta is 1.40, and it is estimated that the equivalent equity beta for its other activities, excluding the component production, is 1.25. Elfu Co has 400 million 25c shares in issue trading at 120c each. Its debt finance consists of variable rate loans redeemable in seven years. The loans paying interest at base rate plus 120 basis points have a current value of $96 million. It can be assumed that 80% of Elfu Co’s debt finance and 75% of Elfu Co’s equity finance can be attributed to other activities excluding the component production.

In the solution:

They have firstly find the Elfu Co portfolio asset beta which is like in this way:
1.40 × $480m/($480m + $96m × (1 – 0.25)) = 1.217

Then they have find other activities assets beta:
1.25 × $360m/($360m + $76.8m × (1 – 0.25)) = 1.078

In the next step they are finding the component asset beta as:
1.217 = component asset beta × 0.25 + 1.078 × 0.75

My confusion:

Why they have applied 25% and 75% proportions percentage in the last step as already above they have calculated assets betas on the basis of actual proportions. I think they should have directly calculated as 1.217-1.078 = component assets beta.
I don’t understand this last step proportion base as already above in step 2 they have proportionately calculated asset beta as 1.078.