Approximations and Errors
Measurements · Approximations and Errors
Syllabus tag: Kenya CBC | Grade 9 Mathematics | Strand 3.0 Measurements | Sub-Strand 3.6 Approximations and Errors (4 Lessons)
Lesson objectives
By the end of this sub-strand, you should be able to:
- Approximate numbers by rounding off and by truncating.
- Express numbers to a stated number of decimal places and significant figures.
- Determine absolute, relative and percentage error in measurements.
- Judge how errors accumulate when approximated values are combined.
Approximations and Errors
No measurement is ever exact. A tape measure, a weighing scale and a measuring cylinder each have a limit. None can read beyond it.
This topic is about giving sensible rounded answers, and knowing how far off they might be.
a) Rounding
Rounding replaces a number with a simpler one that is close to it.
The rule is the digit that follows. If it is 5 or more, round up. If it is less than 5, leave the digit as it is.
Round from the original number every time. Rounding 4.46 to one decimal place gives 4.5. Rounding that 4.5 again to give 5 is wrong.
b) Estimating
An estimate is a quick rough answer used to check a calculation.
Round each number to one significant figure, then work with those. If 38 × 61 is estimated as 40 × 60, you expect about 2400. An answer of 232 would clearly be wrong.
Estimating before calculating catches misplaced decimal points, which are the most costly errors in practice.
c) Absolute error
A length is given as 8 cm to the nearest centimetre. The true length is not exactly 8 cm. It is somewhere close.
Anything from 7.5 cm up to 8.5 cm rounds to 8 cm. So the true value lies in that range.
The two ends behave differently. A length of exactly 7.5 cm rounds up to 8, so it is included. A length of exactly 8.5 cm rounds up to 9, so it is not. That is why one circle is filled and the other is open.
The absolute error is half of the smallest unit used. Measuring to the nearest centimetre gives an absolute error of 0.5 cm.
The lower and upper ends of the range are called the bounds.
d) Relative and percentage error
Absolute error alone does not tell you whether a measurement is good. An error of 0.5 cm is serious on an 8 cm pencil. On an 8 metre wall it is nothing.
Relative error compares the error to the size of the thing measured. Multiply it by 100 and you have the percentage error.
The smaller the percentage error, the more reliable the measurement.
To reduce it, measure with a finer instrument, or measure something larger.
e) Where this is used
A pharmacist measuring a dose works to a tight percentage error. A carpenter cutting timber rounds to the nearest millimetre. A shopkeeper weighing sugar accepts a larger error than a laboratory would.
Words to know
- Rounding off -- shortening a number by adjusting the last retained digit according to the digit that follows.
- Truncating -- shortening a number by discarding digits without any adjustment.
- Significant figures -- the meaningful digits of a number, counted from the first non-zero digit.
- Absolute error -- half the smallest unit of measurement, expressed in the same units as the measurement.
- Percentage error -- the absolute error expressed as a percentage of the measured value.
:::checkpoint Check yourself
- Round 3 847 to the nearest hundred.
- Estimate 49 × 21 by rounding each number to one significant figure.
- A mass is given as 25 g to the nearest gram. What are the lower and upper bounds?
- Find the percentage error when a 20 cm length is measured to the nearest centimetre. :::
Bridge to practice
The exercises begin with rounding, truncating and significant figures, move to absolute and percentage error, and finish with the way errors accumulate when measurements are combined. Read each instruction with particular care, since decimal places and significant figures give quite different answers for the same number and several questions below exploit exactly that difference.