Biology 9700/11 — May/June 2019

Background Concept

The defining relationship in microscopy is

$$\text{magnification} = \frac{\text{image size (with ruler or scale bar)}}{\text{actual size (of the real specimen)}}$$

Image and actual size MUST be in the same units before dividing, or the answer will be wrong by a factor of 1000. The common SI prefixes you will meet in CIE Biology are:

• $1\ \text{mm} = 10^{-3}\ \text{m} = 1000\ \text{µm}$
• $1\ \text{µm} = 10^{-6}\ \text{m} = 1000\ \text{nm}$

So $5\ \text{µm} = 5/1000\ \text{mm} = 1/200\ \text{mm}$.

Understanding the Question

The student is told the actual size of a cell structure is $5\ \text{µm}$. The micrograph gives an image whose length is one of the variables $q$ (in cm) or $r$ (in mm). The question gives the letters $q, r, s, t, u$ where $s = 1000$, $t = 1/5$ and $u = 5$, and asks which algebraic combination of these letters correctly returns the magnification (a dimensionless number).

The key insight is that the image must be expressed in the SAME unit as the actual size before dividing, or equivalently the actual size must be converted to the same unit as the image. The four options test whether the student can:

1. choose the right image variable ($q$ in cm or $r$ in mm), and
2. correctly handle the mm-to-µm conversion (factor of 1000) using $s$, $t$ and $u$.

Approach

Test each option by substituting the numerical values of $s$, $t$ and $u$, then check whether the resulting numerical value matches the correct magnification when applied to a representative image size.

A quicker check is to verify that the units cancel correctly: magnification is dimensionless, so the only unit-bearing variable in the answer ($q$ or $r$) must be cancelled by an $s$ in the denominator, or by a $t$ in the denominator that itself contains a unit conversion.

Step-by-Step Reasoning

Correct formula path:

• Image = $r\ \text{mm}$ (use $r$, not $q$, because the actual size is in µm, much closer to mm than to cm — a cm-sized image of a 5 µm object would imply a magnification of 200 000×, plausible on an EM but harder to set up with $s = 1000$).
• Actual = $5\ \text{µm} = 5/1000\ \text{mm}$.
• Magnification = $r / (5/1000) = r \times 1000/5 = r \times s \times t$ ✓

Option A: $q/s \times u$

Substituting: $q/1000 \times 5 = q \times 5/1000 = q/200$. This has units of cm (since $s$ and $u$ are pure numbers). To get magnification we would need $q$ in mm first, but the formula does the opposite — it scales DOWN by 1000 instead of UP, and multiplies by 5 instead of dividing by 5. This gives a number ~40 000× too small.

Option B: $q \times s \times t$

$q \times 1000 \times (1/5) = q \times 200$. This has units of cm × (pure number) = cm. The cm has not been cancelled, so the answer is off by a factor of 10 (cm vs mm). Also $q$ is image size, not yet divided by actual size, so the structure is wrong.

Option C: $r/s \times u$

$r/1000 \times 5 = r \times 5/1000 = r/200$. Units: mm (correct), but the operation is image/(1000) × 5, which is image × (5/1000). The correct operation is image × (1000/5) = image × 200. Option C inverts both the conversion factor AND the division — wrong by a factor of 40 000.

Option D: $r \times s \times t$

$r \times 1000 \times (1/5) = r \times 200$. Units: mm. To get a dimensionless magnification, the actual size in mm must cancel: $5\ \text{µm} = 5/1000\ \text{mm}$, so $r\ \text{mm} / (5/1000)\ \text{mm} = r \times 200$. This matches the formula exactly.

Key Takeaways

• Magnification is a ratio of like to like: image length / actual length, both in the same unit.
• A mm-to-µm conversion is a factor of 1000; if the image is in mm and the actual size is in µm, you must multiply the image by 1000/actual size — equivalently, divide by the actual size expressed in mm.
• Always check that the units of a candidate equation cancel to give a pure number.
• $s = 1000$ and $t = 1/5$ are designed to look interchangeable with $u = 5$; only the correct combination reproduces the right ratio.

Common Mistakes

• Choosing option A or C because they look like "image divided by something times 5", forgetting the direction of the mm-to-µm conversion.
• Choosing option B because $q \times s \times t$ simplifies to $q \times 200$, not noticing that the $q$ is in cm not mm, so the answer is out by a factor of 10.
• Confusing $t = 1/5$ with $u = 5$ — they are reciprocals, and only one of them is correct in any given position.

Things to Be Careful About

• The actual size is $5\ \text{µm}$, NOT $5\ \text{mm}$. Read the stem carefully — many students misread µm as mm and pick the wrong option.
• Magnification is dimensionless. Any candidate formula that still has length units (cm or mm) at the end cannot be correct, which immediately rules out option B.
• The image variable must be in the same order-of-magnitude unit as the actual size, or the algebra must perform the conversion in the right direction. EM magnifications are typically tens of thousands, so an image in mm (a few mm) of a 5 µm object gives a magnification of a few hundred, which is consistent with option D.