Biology 9700/12 — May/June 2016

Background Concept

When viewing a photomicrograph (an image taken through a microscope), the structure shown appears much larger than it really is. The magnification tells you how many times bigger the image is compared to the real object:

$$\text{magnification} = \frac{\text{image size}}{\text{actual size}}$$

To find the true size of an organelle, you rearrange this to:

$$\text{actual size} = \frac{\text{image size}}{\text{magnification}}$$

The other essential skill here is unit conversion. Most measurements made directly off a printed photomicrograph with a ruler will be in millimetres (mm), but cell organelles are so small that biologists usually report their sizes in micrometres (μm). The conversion is:

$1\ \text{mm} = 1000\ \text{μm}$

Understanding the Question

The student has a photomicrograph of a cell organelle and knows the magnification printed on it. They want to calculate the actual (real) length of the organelle, and they want the answer in μm. They have already measured the length of the organelle in the image (in mm). Which option gives the correct formula?

The command word "which calculation... is correct" means we are choosing the formula that, when applied, gives actual size directly in μm.

Approach

1. Write down the magnification formula and rearrange it to make actual size the subject.
2. Identify the units we have (image size in mm) and the units we want (actual size in μm).
3. Insert the correct unit-conversion factor (×1000) at the correct point in the equation.

Step-by-Step Reasoning

Step 1 — Rearrange the formula.
Starting from $\text{magnification} = \dfrac{\text{image}}{\text{actual}}$, invert both sides and multiply:

$$\text{actual} = \frac{\text{image}}{\text{magnification}}$$

This immediately rules out options A and B, which start from "actual size" — they would tell you the magnification given an actual size, not the other way round.

Step 2 — Check the unit conversion.
The image is measured in mm, but we want the answer in μm. Since $1\ \text{mm} = 1000\ \text{μm}$, we must multiply by 1000:

$$\text{actual (μm)} = \frac{\text{image (mm)} \times 1000}{\text{magnification}}$$

This matches option D.

Step 3 — Why not option C?
Option C uses image size in cm. To convert cm to μm you multiply by $10000$ (because $1\ \text{cm} = 10\ \text{mm} = 10000\ \text{μm}$), not by 1000. So $\dfrac{\text{image (cm)} \times 1000}{\text{magnification}}$ would give an answer in a wrong unit (it would convert to $0.1\ \text{μm}$ per cm, not $1\ \text{μm}$ per cm), and is incorrect.

Key Takeaways

• Formula: $\text{actual size} = \dfrac{\text{image size}}{\text{magnification}}$
• Unit conversions to remember:
  • $1\ \text{m} = 10^3\ \text{mm} = 10^6\ \text{μm} = 10^9\ \text{nm}$
  • $1\ \text{mm} = 1000\ \text{μm}$
  • $1\ \text{cm} = 10\ \text{mm} = 10000\ \text{μm}$
• The conversion factor must match the units of the measured image and the units of the desired answer.

Common Mistakes

• Reversing the formula — writing $\text{magnification} \times \text{actual size}$ instead of dividing. This inflates the answer by a factor equal to magnification².
• Using the wrong conversion factor — multiplying by 1000 when converting cm to μm (should be 10000), or omitting the conversion entirely and leaving the answer in mm.
• Dividing by 100 instead of 1000 — confusing the mm-to-μm conversion with cm-to-mm.
• Multiplying the magnification by the conversion factor — the conversion only applies to the image size, not the magnification (magnification is dimensionless).

Things to Be Careful About

• Magnification has no units — it is a pure ratio. Therefore, the only conversion in the equation comes from the image size.
• Always state the units of your final answer explicitly; marks are often lost for a numerically correct answer given without units, especially in calculation questions.
• In practice, on a Paper 3 practical, you may also need to convert the image size when the magnification is given as, e.g., "$\times 400$" — use the scale bar on the micrograph if provided rather than measuring, as the printed image may have been resized.