Expert: Bobby Soltani - 5/21/2005

Question
Hi Bobby,

I'm a photographer working with lenses. I'm working out the camera to lens and lens to object distance for my macrophotography. I've got two equations that I can't solve... took high school maths 35 years ago...very rusty.

1/S1 + 1/S2 = 1/5

(Thin lens equation. f=5cm for lens. S1 is camera to lens distance; S2 is lens to object distance.)

S1 + S2 = 24 (distance from camera to object is 24cm)

I've gotten as far as :-

1/(24 - S2) + 1/S2 = 1/5

then I get stuck.

Any help most appreciated!
Regards,
Usuff

Answer
Hi Usuff,

So far, you are on the right track.  You need to get a common denominator so you can combine the terms on the left hand sides.  Here's how you do that.

1/(24 - S2) + 1/S2 = 1/5
multiply the first term by S2/S2 and multiply the second term by (24 - S2)/(24 - S2).  We can do this because anything over itself is equal to 1 and something times 1 is itself.

(S2)/(S2(24 - S2)) + (24 - S2)/(S2(24 - S2)) = 1/5

now we can combine the numerators

(S2 + 24 - S2)/(S2(24 - S2)) = 1/5
the S2's on the top cancel

24/(S2(24 - S2)) = 1/5
multiply both sides by (S2(24 - S2)) to get the S2 terms out of the denominator (bottom).

24 = (1/5)*(S2(24 - S2))
multiply it out
24 = (1/5)(24*S2 - S2^2)
multiply both sides by 5 to get rid of the 1/5 on the right

120 = 24*S2 - S2^2
(note that S2^2 means S2 squared)
subtract (24*S2 - S2^2) from both sides to get the correct form to solve the equation.

S2^2 - 24*S2 + 120 = 0
Using the quadratic formula (you can find that anywhere online), I get the following for S2.
S2 = 16.898979
or
S2 = 7.10102

Pluging this back into equation 2 above, we get
S1 = 16.898979
or
S1 = 7.10102

So, using your two equations, we still don't know if S1 is 7.10102 or if S2 does.  Hopefully, you have some additional information available to make this determination.

Please let me know if you have any other questions.  Good luck with your project.

Bobby