A Puzzle: Separating a Cost Increase into Price and Quantity variances/pieces

I realize this is probably a simple problem for most of you, but let's say that the price of a product you purchase increases from, say:

--$5 in Year 1 to $10 in Year 2, for a $5 increase in price--

Let's say that the quantity you purchase increases from, say:

--50 widgets in Year 1 to 70 widgets in Year 2, for a 20 widget increase--

So total cost in Year 1 is $5 multiplied by 50 widgets for a total cost of $250.

Total cost in Year 2 is $10 multiplied by 70 widgets for a total cost of $700.

Year 2 cost has increased $450 from $250 in Year 1 to $700 in Year 2.

So I think what would be interesting to see is a breakdown of this $450 increase into a price change factor/contribution amount and a quantity change/contribution amount.

The question I want to answer for the price contribution amount is: what is the contribution of the price increase to the total cost increase? In my mind, this question assumes quantity is held at the Year 1 level because you're wanting to analyze change in price, and changing quantity would muddy the waters. So the answer is price increased by $5.

Multiply this $5 price increase by the Year 1 quantity of 50 and you get $250.

I now want to answer the same question for the quantity contribution portion. Quantity increased by 20 widgets. The Year 1 price is $5. 20 times $5 is $100. So quantity changes alone, without regard to price changes, contributed $100 to the price increase.

So we have accounted for $250 + $100 = $350 of the $450 increase. I want obviously to add up to the total increase. What remains to be added is the increase in price multiplied by the increase in quantity. This is $5 times 20 = $100. $350 + $100 = $450 total increase. But in this case for the $100, I don't know of a way to break this down between quantity and price. It is a combination of both. Is there a standard method to break this out? In a report, it would seem confusing to see a price contribution, a quantity contribution, and then a mixed contribution, so I would like to somehow break this mixed method up. Any advice?

Thanks.