1. Upper and Lower Bounds

2. Upper and Lower Bounds of Measurement. If a length is measured as 25cm to the nearest cm this does not mean that the length is exactly 25cm. It could be 24·9cm or 25·1 or many other lengths. The bounds of measurement are the largest and smallest numbers that will round to number and degree of accuracy given in the question.

3. Finding the Upper and Lower Bounds of Measurement. Find the upper and lower bounds of a shelf that is 100cm to the nearest 10 cm. In other words: what is the smallest measurement that will round up to 100cm and what is the largest measurement that will round down to 100cm. To the nearest 10cm, 95cm is the smallest measurement that will round up to 100cm (Lower Bound) and 105cm is the largest measurement that will round down to 100cm (Upper Bound).

4. The width of a calculator is 80mm to the nearest mm. The smallest value that will round up to 80mm is 79·5mm. This is the lower bound The largest value that will round down to 80mm is 80·5mm (notice we do not say 80·499999….., which is really the largest). This is called the upper bound. We can also think of a number line to answer the questions.

5. How to answer the questions. Find the upper and lower bounds for a measurement of 25cm measured to the nearest cm. 1.Draw a line and write 25cm in the middle of it. 2.Since the degree of accuracy is to the nearest cm, on the left side the line subtract 1cm and on the right add 1cm. 3.Find the two mid-points which are the upper and lower limits. 24cm26cm 25cm 24·5cm25·5cm

6. What does this mean? For 25cm to the nearest cm any value more than or equal to 24·5cm will round up to 25cm Also any value below 25·5cm will round down to 25cm. 24cm26cm25cm24·5cm25·5cm

7. How to answer the questions. Find the upper and lower bounds for a measurement of 140cm measured to the nearest 10cm. 1.Draw a line and write 140cm in the middle of it. 140cm 2.Since the degree of accuracy is to the nearest 10cm, on the left side the line subtract 10cm and on the right add 10cm. 130cm150cm 3.Find the two mid-points which are the upper and lower limits. 135cm145cm

8. What does this mean? For 140cm to the nearest 10cm any value more than or equal to 135cm, will round up to 140cm Also any value below 145cm will round down to 140cm. 130cm150cm140cm135cm145cm

9. How to answer the questions. Find the upper and lower bounds for a measurement of 5·47s measured to the nearest one-hundredths of a second. 1.Draw a line and write 5·47s in the middle of it. 5·47s 2.Since the degree of accuracy is to the nearest one-hundredths of a second, on the left side the line subtract one-hundredths of a second and on the right add one-hundredths of a second. 5·46s5·48s 3.Find the two mid-points which are the upper and lower limits. 5·465s5·475s

10. Practice Questions Find the upper and lower bounds for the following questions.

11. 1.Given the the nearest centimetre. a)25cm b) 50cm c) 200cm d) 2·13m Lower Limit is 24·5cm Upper Limit is 25·5cm 25·5cm 24·5cm 24cm 26cm 25cm

12. 1.Given the the nearest centimetre. a)25cm b) 50cm c) 200cm d) 2·13m Lower Limit is 49·5cm Upper Limit is 50·5cm 49·5cm 50·5cm 49cm 51cm 50cm

13. 1.Given the the nearest centimetre. a)25cm b) 50cm c) 200cm d) 2·13m Lower Limit is 199·5cm Upper Limit is 200·5cm 199·5cm 200.5cm 199cm 201cm 200cm

14. 1.Given the the nearest centimetre. a)25cm b) 50cm c) 200cm d) 2·13m Lower Limit is 2·125m Upper Limit is 2·135m 2·125m 2·135m 2·12m 2·14m 2·13m

15. 2.Given the the nearest millimetre. a)3·2cm b) 4·9cm c) 6·0cm d) 5·162m Lower Limit is 3·15mm Upper Limit is 3·25mm 3·15mm 3·25mm 3·1mm 3·3mm 3·2mm

16. 2.Given the the nearest millimetre. a)3·2cm b) 4·9cm c) 6·0cm d) 5·162m Lower Limit is 4·85cm Upper Limit is 4·95cm 4·85cm 4·95cm 4·8cm 5·0cm 4·9cm

17. 2.Given the the nearest millimetre. a)3·2cm b) 4·9cm c) 6·0cm d) 5·162m Lower Limit is 5·95cm Upper Limit is 6·05cm 5·95cm 6·05cm 5·9cm 6·1cm 6·0cm

18. 2.Given the the nearest millimetre. a)3·2cm b) 4·9cm c) 6·0cm d) 5·162m Lower Limit is 5·1615m Upper Limit is 5·1625m 5·1615m 5·1625m 5·161m 5·163m 5·162m

19. 3.Given the the nearest 10cm. a)60cm b) 500cm c) 2·3m d) 200mm Lower Limit is 55cm Upper Limit is 65cm 55cm 65cm 50cm 70cm 60cm

20. 3.Given the the nearest 10cm. a)60cm b) 500cm c) 2·3m d) 200mm Lower Limit is 495cm Upper Limit is 505cm 495cm 505cm 490cm 510cm 500cm

21. 3.Given the the nearest 10cm. a)60cm b) 500cm c) 2·3m d) 200mm Lower Limit is 2·25m Upper Limit is 2·35m 2·25m 2·35m 2·2m 2·4m 2·3m

22. 3.Given the the nearest 10cm. a)60cm b) 500cm c) 2·3m d) 200mm Lower Limit is 150mm Upper Limit is 250mm 150mm 250mm 100mm 300mm 200mm

23. 4.Given the the nearest hundredth of a second. a) 2·83s b) 72·08s c) 63·80s Lower Limit is 2·825s Upper Limit is 2·835s 2·825s 2·835s 2·82s 2·84s 2·83s

24. 4.Given the the nearest hundredth of a second. a) 2·83s b) 72·08s c) 63·80s Lower Limit is 72·075s Upper Limit is 72·085s 72·075s 72·085s 72·07s 72·09s 72·08s

25. 4.Given the the nearest hundredth of a second. a) 2·83s b) 72·08s c) 63·80s Lower Limit is 63·795s Upper Limit is 63·805s 63·795s 63·805s 63·79s 63·81s 63·80s