# Thread: Equation needed for Analysis...

1. ## Equation needed for Analysis...

Hello all, I'm trying to do an analysis and am having trouble coming up with an equation/calculation that I feel confident addresses what I need...

Assuming in period one that [ A1 * B1 = C1 ].

during the second period, A1 and B1 have both changed (increased/decreased) to A2 and B2, respectively, so that now [ A2 * B2 = C2 ].

My problem is - how can I quantify how much of the change in C was attributable to each the change in A and the change in B? Said another way, what percentage (or amount) of the total change in C was driven solely by the change in A? (and the same for B)?

Does that make sense? I tried posting a similar question in a different forum a couple of weeks ago, but the answer didn't address exactly what i was looking for, probably due to my lack of clarity; hopefully it's more understandable this time...

Thanks in advance for any responses - any help would be greatly appreciated.

Lewis

2. Re-express A2, B2, and C2 as, respectively,
$\displaystyle A_1 + \Delta A,\ \ B_1 + \Delta B,\ \ C_1 + \Delta C$

where $\displaystyle \Delta A,\ \Delta B,\ \Delta C$ simply denote the change in each original quantity.

Your "post-change" expression can then be rendered as
$\displaystyle (A_1 + \Delta A)\ \times \ (B_1 + \Delta B) \ = \ (C_1 + \Delta C)$

Multiplying out the LHS gives you
$\displaystyle A_1B_1 + \Delta AB_1 + A_1\Delta B + \Delta A\Delta B = C_1 + \Delta C$

Finally deduct your first equation from this last one, leaving
$\displaystyle \Delta AB_1 + \Delta BA_1 + \Delta A \Delta B = \Delta C$

In other words, the change in C is the sum of three influences: The change in A operating on the 'original' B; the change in B operating on the 'original' A; and the "cross product" of A's and B's change.

There's a geometric interpretation of your question...

...think of your original quantity C1 as the green shaded area, it having an area equal to the product of A1 and B1 (with A1 and B1 playing the roles of length and width). Then length and width change by some quantities, resulting in the larger rectangle. Your question amounts to asking what gives rise to the change in the area of the rectangle, and this area-change is the yellow-shaded area. You can see that the yellow area is the sum of three components, each one corresponding to those three 'influences' in my final equation.

3. ## Perfect

Thanks a lot, LochWulf - that's exactly what I was looking for!

Lewis