How To Cross-Multiply

Here is a typical problem involving an inclined plane:

A car drives up along a 25 degree incline, increasing its altitude by 85 meters. How far has the car driven along the inclined surface ?

The equation of interest is:

sin(25o) = (y/d)
sin(25o) = (85/d)

The equation now has to be rearranged to solve for d. There are a couple of different ways to do this; one way would be to first multiply both sides by d:

d sin(25o) = d(85/d)
d sin(25o) = 85

Then you have to divide both sides by sin(25o) to isolate d. Another way to proceed would be to write both sides as fractions, and then invert each fraction:

[sin(25o)]/1 = (85/d)
1/[sin(25o)] = (d/85)

Then you have to multiply both sides by 85 to isolate d. It is probably a source of confusion that there are many ways to rearrange the same equation; students get into trouble when they have mastered only one way and then have to deal with something new, causing them to guess.

Try this: eliminate fractions from both sides of the equation by cross multiplying the denominator of one fraction with the numerator of the other, then apply the standard rules of dividing out factors to finish the isolation process.

[sin(25o)]/1 = (85/d)
d[sin(25o)] = (85 x 1) = 85
d[sin(25o)]/[sin(25o)] = 85/[sin(25o)]
d = 85/[sin(25o)]

This doesn't look any different from what I showed you at the top of this page, but as a method I have found it to be much more reliable in a variety of problems where fractions are present.

Here is another example: in Elastic Properties of Solids there is an equation involving the change in length of a solid rod under stress:

E = stress/strain = (F/A)/(DL/Lo)

Rearranging this to solve for DL is a real nightmare for some students. Applying the cross-multiplication technique a couple of times makes the equation "look" more manageable by getting rid of all the fractions:

(E/1) = (F/A)/(DL/Lo)
E x (DL/Lo) = (F/A) x 1 = (F/A)
E x DL x A = F x Lo

Then finish off by dividing both sides by EA.

Practice with these problems:

(24/C)(A) = [645(B + 16)]    solve for B.
(75xD)/E = F/3    solve for E.
(G + 7H)/13 = 47    solve for H.

Answers: B = 0.0384(A/C) - 16    E = 225(D/F)     H = (611-G)/7

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