Measure short lengths by taking multiples
Measure a group of similar objects, then divide to estimate the mean size of one.
Section P1.1 · Physical quantities and measurement techniques
Mapped to the named 0654 section using teaching from the corresponding single-science course. A 0654-specific review of every Core and Supplement objective is still in progress. Written practice is self-checked; it does not demonstrate supervised laboratory competence or award an official Cambridge grade.
Before you begin
- Divide a total by a count.
- Read and subtract ruler positions.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
A total length is shared by 20 equal widths. How do you find one width?
What you will learn
- Determine a mean thickness from a stack.
- Explain why a larger measured total reduces fractional reading effects.
Make a small distance measurable
The thickness of one sheet may be too small to read reliably on a ruler. Stack a known number of similar sheets, measure the stack thickness from two clear scale readings, and divide the total by the number of sheets. Fifty sheets measuring 5.0 mm give a mean thickness of 0.10 mm.
The result is a mean for those sheets, not a guarantee that every sheet is identical. Count the objects carefully and align them without gaps. For identical turns of wire, a close-packed row of known turns can similarly give a mean diameter from total width divided by the number of turns.
The method has conditions
If the same absolute reading effect is 0.5 mm, it is large compared with a single thin sheet but much smaller compared with a 20 mm stack. Dividing a longer measured total by the count reduces its contribution to the inferred mean thickness. It does not make the ruler itself more finely graduated.
Avoid compressing soft layers or leaving air gaps, because either changes the quantity being measured. Repeat with another carefully formed stack or winding and compare the means. A wrong count is a systematic calculation error; more repeated ruler readings with that wrong count will not fix it.
Sources for this lesson
Pause and explain
Why measure a large stack instead of one very thin sheet?
Worked example
Two ruler readings for 80 sheets are 12 mm and 20 mm. Find the mean sheet thickness.
Show the worked solution
- The stack is 20 − 12 = 8 mm thick.
- Divide by 80 sheets.
- The mean thickness is 0.10 mm.
Answer 0.10 mm.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A stack of 50 sheets is 5.0 mm thick. Find the mean thickness.
Give me a hint
Divide the measured thickness by the number of sheets.
Compare my reasoning
- There are fifty repeated thicknesses.
- 5.0/50 = 0.10.
- Record the mean as 0.10 mm.
0.10 mm.
Look for these in your work
- Divide by the count.
- Identify a mean thickness.
Thirty close-packed wire turns occupy 18 mm. Find the mean wire diameter.
Give me a hint
Treat the width as thirty repetitions with no gaps.
Compare my reasoning
- The total width is 18 mm.
- 18/30 = 0.60.
- The mean diameter is 0.60 mm.
0.60 mm.
Look for these in your work
- Use all thirty turns.
- Keep the length unit.
A loose stack includes air spaces between sheets. Will dividing its thickness by the count give too high or too low a sheet estimate?
Give me a hint
Separate sheet thickness from unwanted gaps.
Compare my reasoning
- The measured total contains sheets and gaps.
- Dividing assigns some gap width to each sheet.
- The estimate is too high; align the stack to remove gaps without compressing the sheets.
Too high; remove gaps without changing sheet thickness.
Look for these in your work
- Explain the direction of bias.
- Address the stack construction.
Common mistakes
- Subtract the stack's starting reading before dividing.
- Use the actual count of sheets or wire turns.
- Gaps and compression change the measured total.
What can you explain now?
Twenty close-packed wire turns occupy 14 mm. What is the mean diameter?
Compare with the explanation
0.70 mm.
Divide the total width 14 mm by the count twenty.
After trying it yourself, choose your next review. This is your self-assessment.
Your review choice appears on Today. Sign in to sync it across devices.
Make it stick
Explain a multiples method for a small distance and name one error that averaging cannot correct.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.