How to Write a Worked Solution

Use this page as a checklist when we solve a new exercise together. Do not try to perform every calculation mentally. Writing down the intermediate steps makes mistakes easier to find and turns one answer into a method that can be reused.

1. Identify the exercise

Record enough information to find the original question again:

book title and author;
edition;
chapter or section;
exercise number.

Then paraphrase the question in one or two sentences. State exactly what must be found.

2. Draw and choose conventions

Draw a small diagram when the problem has geometry, rays, forces, or vectors. Declare the sign convention before substituting numbers. For example, a thin-lens calculation might use positive distances for a real object and a real image.

3. List knowns and unknowns

Write every supplied value with its symbol and unit. A compact list might look like this:

\[f = 100\ \mathrm{mm}, \qquad d_o = 300\ \mathrm{mm}, \qquad d_i = \;?\]

Use one system of units before calculating. Do not silently combine metres, millimetres, degrees, and radians.

4. Select the governing equation

Write the equation in symbols first. Explain briefly why it applies and list any assumptions. For a thin lens in air, for example,

\[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}.\]

This model assumes a paraxial ray and a lens whose thickness can be neglected.

5. Rearrange before substituting

Solve algebraically for the unknown before inserting numbers. This keeps the logic readable and reduces arithmetic errors:

\[\frac{1}{d_i} = \frac{1}{f} - \frac{1}{d_o} = \frac{d_o-f}{f d_o},\]

and therefore

\[d_i = \frac{f d_o}{d_o-f}.\]

6. Substitute values with units

Show the numerical substitution, not only the result:

\[d_i = \frac{(100\ \mathrm{mm})(300\ \mathrm{mm})} {300\ \mathrm{mm}-100\ \mathrm{mm}}.\]

Keep more digits during the calculation than the final answer requires.

7. State and interpret the answer

Put the final result on its own line and explain what its sign and size mean. The statement the image is 150 mm behind the lens is more useful than the number 150 by itself.

8. Check the result

Use at least one independent check:

substitute the answer into the original equation;
check that dimensions and units agree;
estimate the expected size or sign;
solve by a second method or with a limiting case;
compare with a KrakenOS paraxial calculation when appropriate.

Sphinx equation syntax

Use the inline math role for a short expression such as :math:`f=100, which renders as \(f=100\). Use a math directive for a displayed equation:

.. math::

   \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}

The blank line after .. math:: and the indentation of the equation are required. Equation labels can be used when a later paragraph needs a precise reference:

.. math::
   :label: thin-lens-template

   \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}

Equation :eq:`thin-lens-template` relates object and image distance.

Template for the next exercise

Copy these headings into a new .rst page and fill them in:

Exercise title
==============

Source
------

Problem in our own words
------------------------

What is known
-------------

Step 1: Choose the model
------------------------

Step 2: Rearrange the equation
--------------------------------

Step 3: Substitute and calculate
--------------------------------

Step 4: Interpret the result
----------------------------

Check
-----

Final answer
------------

Make each underline at least as long as its heading. Add the new filename, without the .rst suffix, to the toctree in this section’s index.rst.