Thin Lens Equation Height Ratio Problems on the MCAT

Thin lens questions on the MCAT can feel like a physics content question and a math skills question at the same time. You need to know the thin lens equation, but you also need to handle ratios, reciprocals, and fraction subtraction without losing track of what the question actually asked.

That last part matters. In this transcript example, solving for image distance is only the middle step. The question asks for the ratio of image height to object height, so you have to keep going.

Start by Writing the Ratio the Way the Question Asks

The question asks for the ratio of the height of the image to the height of the object.

• height of image / height of object

Always write ratios in the same order as the question or answer choices. If image height comes first, it goes in the numerator. If object height comes second, it goes in the denominator.

This prevents a very common MCAT mistake: calculating the reciprocal of the answer you actually need.

Write the Thin Lens Equation

The thin lens equation is:

• 1/f = 1/do + 1/di

Here:

• f is the focal length.
• do is the object distance from the lens.
• di is the image distance from the lens.

In the transcript example, the object is placed 4 focal lengths from the center of a thin convex lens.

• do = 4f

The focal length is just f. So now we can solve for di in terms of f.

Solve for Image Distance

Plug do = 4f into the thin lens equation:

• 1/f = 1/(4f) + 1/di

Now isolate 1/di:

• 1/f - 1/(4f) = 1/di

To subtract the fractions, use a common denominator. Rewrite 1/f with 4f in the denominator:

• 1/f = 4/(4f)

So:

• 4/(4f) - 1/(4f) = 1/di
• 3/(4f) = 1/di

This is where the reciprocal matters. We solved for 1/di, not di. To find di, flip the fraction:

• di = 4f/3

That image distance is important, but it is not the final answer yet.

Do Not Stop at Image Distance

A tempting wrong answer is 4/3, because that is the coefficient in the image distance.

But the question did not ask for image distance. It asked for:

• height of image / height of object

So now we need the thin lens magnification relationship.

Use the Height Ratio Relationship

For thin lens problems, the ratio of image height to object height is related to the ratio of image distance to object distance.

• hi / ho = di / do

For many MCAT questions, you can use this as a ratio relationship. If the question is asking for magnitude, focus on the size of the ratio. If a sign convention is explicitly tested, pay attention to whether the image is inverted.

In this question, we have:

• di = 4f/3
• do = 4f

Plug those into the ratio:

• hi / ho = (4f/3) / (4f)

The f terms cancel, and the 4 terms cancel:

• hi / ho = 1/3

So the height of the image is one-third the height of the object.

Why the Ratio Is Smaller Than 1

This answer also makes sense conceptually. If the object is placed far outside the focal length, the image formed by a convex lens can be smaller than the object. A height ratio of 1/3 means the image is reduced compared with the object.

That kind of reasonableness check is useful on test day. If your algebra gives a huge magnification for an object placed 4 focal lengths away, pause and check your setup.

Common MCAT Traps

Trap 1: Reversing the Ratio

If the question asks for image height over object height, write hi/ho from the beginning. Do not accidentally solve for ho/hi.

Trap 2: Forgetting to Flip the Reciprocal

After subtracting fractions, you get 3/(4f) = 1/di. That means di = 4f/3, not 3/(4f).

Trap 3: Stopping at di

The image distance is not the height ratio. You still need to compare di to do.

Trap 4: Weak Fraction Skills

Thin lens problems often require common denominators and reciprocal math. If that feels rusty, it is worth practicing before test day.

MCAT Takeaway

For thin lens height ratio problems, move in two stages: first solve for image distance, then use the distance ratio to find the height ratio.

• Write the requested ratio first: hi/ho.
• Use the thin lens equation: 1/f = 1/do + 1/di.
• If do = 4f, then 1/f = 1/(4f) + 1/di.
• Solving gives di = 4f/3.
• Use hi/ho = di/do.
• The final ratio is (4f/3)/(4f) = 1/3.

The content you need is the thin lens equation and the height-distance ratio. The math skills you need are ratio setup, common denominators, and remembering to flip the reciprocal at the end.