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Reading Scales & Tolerance0%

Measurement & Geometry · Topic 2 of 9

Reading Scales & Tolerance

Video coming soon3 worked examples

Theory

The Golden Rule: When reading a scale, you must always count the gaps between the numbers, not the physical lines. When dealing with tolerance, always calculate and write down your explicit maximum and minimum limits before attempting to make any decisions.

1. Reading Minor Unnumbered Divisions

You will frequently be asked to read from dials, thermometers, or measuring jugs that have unnumbered markings.

  • Step 1: Find the difference between two printed numbers on the scale.
  • Step 2: Count the number of gaps (minor divisions) between those two numbers.
  • Step 3: Divide the difference by the number of gaps to find what exactly one minor division is worth.

Note: If a dial has two different scales (e.g., mph on the outside and km/h on the inside), ensure you are reading the correct one.

2. Calculating Tolerance Limits

Tolerance is the amount by which a measurement is allowed to vary while still being acceptable. It is written using the plus-minus symbol (±), and gives two limits:

Maximum=Target+Tolerance\text{Maximum} = \text{Target} + \text{Tolerance}
Minimum=Target−Tolerance\text{Minimum} = \text{Target} - \text{Tolerance}

So a measurement written as 400±5400 \pm 5 g is acceptable anywhere from 395395 g to 405405 g.

  • Maximum Limit: Add the tolerance to the target value.
  • Minimum Limit: Subtract the tolerance from the target value.

Percentage Tolerance: If the tolerance is given as a percentage (e.g., 400g ± 3%), you must calculate that percentage of the target weight first, and then add/subtract it to find your limits.

3. Decision Making & Compatibility

Once you have your upper and lower limits, you will often need to evaluate a list of items to see if they are accepted or rejected.

  • An item is only accepted if its measurement falls strictly between or exactly on your calculated limits.
  • Implications for compatibility: You may be asked to consider if a manufactured part will fit safely into another component based on their respective tolerance limits.

⚠️ Common Examiner Traps

  • The "Counting the Lines" Trap: When finding what a scale goes up in, candidates frequently count the physical dashes on the page instead of the "jumps" (gaps) between them, resulting in the wrong scale.
  • The "Percentage of the Sample" Trap: In percentage tolerance questions, candidates occasionally calculate the percentage based on the measured sample weight rather than the target weight. Always calculate the percentage of the main target number.
  • The "Fraction Rejected vs Accepted" Trap: A question may give you a list of 10 items and ask you for the fraction of items that were rejected. Candidates often work out the limits perfectly but accidentally write down the fraction of items that were accepted. Always read the final sentence twice.

Worked examples

Example 1

A mechanic is inflating the tyres on a van. The manufacturer states the tyres must be inflated to a pressure of 45 psi ± 8%. The mechanic checks the pressure using a dial gauge. The dial has major markings every 10 psi, with 5 gaps between each major marking. The needle is pointing exactly 2 marks below the 50 psi line. Determine if the tyre pressure is within the safe tolerance limits. Use your working to justify your answer. (4 marks)

Step 1: Work out the scale on the dial.

Difference =10,Gaps =5\text{Difference } = 10, \quad \text{Gaps } = 5
10÷5=2 psi per gap10 \div 5 = 2 \text{ psi per gap}

Step 2: Read the gauge. The needle is 2 marks (4 psi) below 50.

50−4=46 psi (This is the current tyre pressure).50 - 4 = 46 \text{ psi (This is the current tyre pressure).}

Step 3: Calculate the exact tolerance amount.

8% of 45→45÷100×8=3.6 psi8\% \text{ of } 45 \rightarrow 45 \div 100 \times 8 = 3.6 \text{ psi}

Step 4: Calculate the maximum and minimum safe limits.

Maximum: 45+3.6=48.6 psi\text{Maximum: } 45 + 3.6 = 48.6 \text{ psi}
Minimum: 45−3.6=41.4 psi\text{Minimum: } 45 - 3.6 = 41.4 \text{ psi}

Step 5: Make a decision comparing the reading to the limits.

Yes, the pressure is safe because 46 psi is between 41.4 psi and 48.6 psi.

Example 2

A bakery produces premium loaves of bread. The loaves are accepted for sale if they weigh 800g ± 2.5%. The weights, in grams, of a sample of 12 loaves are shown below:

778, 815, 821, 785, 790, 805, 770, 819, 801, 782, 825, 795

Calculate the fraction of loaves that will be rejected. (3 marks)

Step 1: Calculate the tolerance amount.

2.5% of 800→800÷100×2.5=20g2.5\% \text{ of } 800 \rightarrow 800 \div 100 \times 2.5 = 20\text{g}

Step 2: Calculate the upper and lower acceptable limits.

Maximum: 800+20=820g\text{Maximum: } 800 + 20 = 820\text{g}
Minimum: 800−20=780g\text{Minimum: } 800 - 20 = 780\text{g}

Step 3: Identify which loaves fall outside these limits (less than 780g or more than 820g).

Rejected loaves: 778, 821, 770, 825 (4 loaves).

Step 4: Write as a fraction of the total sample.

412=13\frac{4}{12} = \frac{1}{3}

Example 3

An engineering company manufactures a steel pin and a matching circular slot.

  • The steel pin is manufactured to a diameter of 12.5mm ± 0.15mm.
  • The circular slot is manufactured to a diameter of 12.6mm ± 0.1mm.

For the components to be compatible, the maximum diameter of the pin must be strictly less than the minimum diameter of the circular slot. Determine if these components are compatible. (3 marks)

Step 1: Calculate the maximum diameter of the pin.

12.5+0.15=12.65mm12.5 + 0.15 = 12.65\text{mm}

Step 2: Calculate the minimum diameter of the circular slot.

12.6−0.1=12.50mm12.6 - 0.1 = 12.50\text{mm}

Step 3: Compare the two values against the compatibility rule.

No, they are not compatible because the maximum pin diameter (12.65mm) is greater than the minimum slot diameter (12.50mm), meaning the pin might not fit.

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10 Reading Scales and Tolerance questions with answers and video solutions