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The Cross Product Law as a Linear Equation

The Cross Product Law as a Linear Equation

Proportionality calculations lead to the use of the so-called 'rule of 3', also nammed the 'cross product law'.

Let's study it with an example.

If it lasts 15 minutes to bake 500 g of rastbeef, how much time shalll I bake my 950 g rastbeef?

Let's call x the time in minutes that we are looking for.

Then the time per gram is either 15 min over 500 g, or x min over 950 g:

  x               15
------    =    ------    (1)
950            500

If we multiply both sides of (1) by the product 950 x 500 and simplify the left side by 500 and the right side by 950, we obtain the cross product law:

500 x = 950 x 15 (2)

where the numerator of a side is multiplied by the denominator of the other side.

(2) is a linear equation of the type a x = b, wich solution is x=a/b (a divided by b).

Thus the solution of our problem is the 'rule of 3' formula:

          950 x 15
x =  --------------- =  28.5 min = 28 min 30 s
            500

The rule of 3 can be obtain directly from the proportionality equation (1), multiplying both sides of (1) by 950 and simplifying the left side by 950.

Thus, the rule of 3 solves a proportionality equation, that is in fact a linear equation as well, with coefficients inverses of integers…

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