Arithmetic
Fractions, decimals, ratios, percentages, powers and roots, at Level 2 depth.
Arithmetic, summary notes
- 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.
- 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
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.
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.
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.
Arithmetic concept map
Arithmetic
Arithmetic check
Arithmetic
1. Evaluate −5 × (−4) + 3.
23−17172. Evaluate 12 − 4 × 3.
02483. Evaluate (−6) ÷ (−2) − 5.
−2824. Evaluate 3 − (−7) + (−2).
8−6125. Evaluate −2 × 3 × (−4).
24−24146. Evaluate 20 ÷ (2 + 3) × 2.
82107. What is 24 × 15?
3603403808. What is 625 ÷ 25?
2524309. What is 48 × 12?
57656658610. What is 1728 ÷ 12?
14412415411. What is 125 × 8?
1000900102512. What is 3600 ÷ 45?
80907513. Evaluate 2/3 + 3/4.
17/125/76/1214. Evaluate 3/5 × 10/9.
2/33/213/1415. Express 5/8 as a decimal.
0.6250.580.85