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Section 1.1

Arithmetic

Fractions, decimals, ratios, percentages, powers and roots — at Level 2 depth.

Notes

Arithmetic — summary notes

Main ideas
  • Order of operations (BODMAS: Brackets, Orders, Division/Multiplication, Addition/Subtraction) decides the result of any mixed expression — get it wrong and every downstream calculation is wrong.
  • Signs: like signs multiply to a positive, unlike signs to a negative. Subtracting a negative is the same as adding.
  • A fraction, a decimal and a percentage are three ways of writing the same quantity — convert freely, because maintenance data mixes all three.
  • Ratio and proportion govern gear trains, lever arms and mixture ratios. Direct proportion: one rises as the other rises. Inverse: one rises as the other falls.
  • Averages (means) are used to smooth repeated measurements — e.g. taking several micrometer readings and averaging to reduce reading error.
  • Unit conversion is a constant source of error in maintenance. Always carry the units through the calculation, not just the numbers.
Key formulas
Percentage of a quantity
x % of N = N × (x / 100)
Percentage change
% change = (change ÷ original) × 100
Mean (average)
mean = sum of values ÷ number of values
Area of a rectangle
A = length × width
Volume of a cuboid
V = length × width × height
Metric length
1 m = 1000 mm, 1 cm = 10 mm, 1 km = 1000 m
Metric mass / volume
1 kg = 1000 g, 1 litre = 1000 mL
Solved examples

  1. A component is reworked from 250 mm to 244 mm. What is the percentage reduction?

    Change = 250 − 244 = 6 mm. % change = (6 ÷ 250) × 100 = 2.4 % reduction.

  2. A gearbox steps 3000 rpm down in the ratio 5 : 2. Find the output speed.

    Output = 3000 × (2 ÷ 5) = 1200 rpm. The output turns slower, so the smaller number goes on top — a useful sanity check.

  3. Four micrometer readings of a journal are 25.02, 25.04, 25.03 and 25.03 mm. What diameter should be recorded?

    Mean = (25.02 + 25.04 + 25.03 + 25.03) ÷ 4 = 100.12 ÷ 4 = 25.03 mm. Averaging repeated readings reduces the effect of individual reading error.

Mind map

Arithmetic concept map

Arithmetic

Quiz

Arithmetic check

Arithmetic

15 of 50Pass 75%

  1. 1. Evaluate −5 × (−4) + 3.

  2. 2. Evaluate 12 − 4 × 3.

  3. 3. Evaluate (−6) ÷ (−2) − 5.

  4. 4. Evaluate 3 − (−7) + (−2).

  5. 5. Evaluate −2 × 3 × (−4).

  6. 6. Evaluate 20 ÷ (2 + 3) × 2.

  7. 7. What is 24 × 15?

  8. 8. What is 625 ÷ 25?

  9. 9. What is 48 × 12?

  10. 10. What is 1728 ÷ 12?

  11. 11. What is 125 × 8?

  12. 12. What is 3600 ÷ 45?

  13. 13. Evaluate 2/3 + 3/4.

  14. 14. Evaluate 3/5 × 10/9.

  15. 15. Express 5/8 as a decimal.

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