Errors in measurement · Ages 13–14
Upper and lower limits
Limits = value ± half the unit.
How to do it
Method
Limits = value ± half the unit
12 cm (nearest 1 cm): 11.5 ≤ x < 12.5
Another way
On a number line
Numbers rounding to 12: 11.5 up to 12.5
In full
Every measurement has an error: the value measured is only correct to some unit (the smallest unit of the measurement). Measured as 12 cm to the nearest 1 cm, the true length is anything that rounds to 12: from 11.5 cm up to, but not including, 12.5 cm. Lower limit = the value − half the unit, upper limit = the value + half the unit: 12 − 0.5 = 11.5 and 12 + 0.5 = 12.5. The true value's range is 11.5 ≤ x < 12.5: it may equal the lower limit but not the upper one (12.5 rounds to 13). Read the unit first: to the nearest 0.1 kg the unit is 0.1, half of it 0.05; to the nearest 10 mL, half is 5. Or picture a number line: every number that rounds to 12 lies between 11.5 and 12.5.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
A bucket's capacity is measured as 70 L to the nearest 2 L. The true capacity x lies in ≤ x < .
Fill in each blank
Show the answer and working
Answer: 69, 71
Half the unit is 1 L: 70 − 1 = 69 and 70 + 1 = 71; the upper limit itself would round up, so it is "<".
Example 2
A distance was measured to some unit, and its true value x lies in 11.25 km ≤ x < 11.75 km. What was the distance measured, in km?
Show the answer and working
Answer: 11.5 km
The measured value sits halfway between the limits: (11.25 + 11.75) ÷ 2 = 11.5 km.
Example 3
A watermelon is weighed as 31 kg to the nearest 1 kg. Its true weight could be 31.5 kg.
True or false?
Show the answer and working
Answer: False
The true weight lies in 30.5 ≤ x < 31.5: the lower limit is in, the upper limit is not.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.A bag of flour is weighed as 6 kg to the nearest 0.1 kg. What is the upper limit of its weight, in kg?
Answer:
2.A watermelon is weighed as 0.9 kg to the nearest 0.1 kg. Its true weight could be 0.925 kg.
True or false?
Answer:
3.A distance was measured to some unit, and its true value x lies in 7.55 km ≤ x < 7.65 km. What was the distance measured, in km?
Answer:
4.A carton's width is measured as 13 cm to the nearest 1 cm. Which of these could be its true width?
- A. 13.5 cm
- B. 12.4 cm
- C. 12.5 cm
- D. 13.6 cm
Choose the right answer
Answer:
5.A measurement has a lower limit of 92.5 mL and an upper limit of 97.5 mL. To the nearest how many mL was it measured?
Answer:
All the answers
- 1. 6.05 kg
- 2. True
- 3. 7.6 km
- 4. C. 12.5 cm
- 5. 5 mL
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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