KM TUITION

1 Y KM TUITION r upper and lower Page 1 of 8 upper and lower Bounds In this exercise we a re going to look at upper and lower bounds. We shall explore numbers rounded to the nearest 10, 100 or 1 and work out the lowest and highest values of the original rounded values. World s strongest man claims he can lift 70kg with his little finger rounded to the nearest 10kg. What is the smallest amount he can lift? Here we have to think of all the numbers that can be rounded up to give 70. A number line would be useful here to observe these numbers. The numbers shown in red can be rounded up to give 70. This means the smallest amount the world s strongest man claims to lift is 65kg which is the smallest number shown on the number line.

2 This number is called the lower bound How about the greatest amount that he can lift? Here we need to think about the numbers that can round down to give 70. These numbers have been indicated on the number line below. These numbers are between 70 and and so on. Notice that 75 is not included as this would round to 80. The world s strongest man claims to lift a greatest amount of 74. This is called the upper bound . If you looked carefully about you will have realised that to find the upper and lower bounds of a number rounded to the nearest 10 you could divide 10 by 2 (half the degree of accuracy) and then add or subtract from 70 to find the upper and lower bound as shown below.

3 Find the upper and lower bounds of 70 rounded off to the nearest 10. y KM TUITION r upper and lower Page 2 of 8 To the nearest 100 The same principle described previously applies here. Suppose the distance from London to Chester is 700km rounded to the nearest 100km. What are the lower and upper bounds? The degree of accuracy here is to the nearest 100km and half of that is 50km. This upper and lower bounds are worked out below; And this can also be represented on the number line as shown below; To the nearest whole number Above we have seen that when a value is given to the nearest 10 we find the upper and lower bounds by adding and subtracting 5.

4 We have also seen that when a number is rounded to the nearest 100 we find the upper and lower bounds by adding or subtracting 50. In both cases we added and subtracted half the degree of accuracy. To round to the nearest whole number we go through the same steps above. For example; A pen is 16 cm to the nearest cm, find the upper and lower bounds. As we did above; we a re going to find out half of the degree of accuracy divide 1 by 2 to get This is because we are asked to round 16 to the nearest 1. Now we subtract or add to 16 to find the lower and upper bounds as shown below. y KM TUITION r upper and lower Page 3 of 8 This is also shown on the number line below.

5 The lower bound is and the upper bound is Whenever measurements are expressed in the real world they are often given an upper bound (highest value) and a lower bound ( lower value). Example 1 A length of wood is 1500 mm long, correct to the nearest mm. The upper bound is therefore: mm. The lower bound is therefore: mm. Note The upper and lower bounds are HALF the degree of accuracy. In our example, mm, that is half of 1 mm. Example 2 A poster measures cm x cm correct to the nearest cm. Find the upper and lower bounds for the poster's dimensions, hence find its maximum and minimum area ( to 2 ) upper bounds , and bounds , maximum area = x = = ( ) sq.

6 Minimum area = x = = (2 )sq. y KM TUITION r upper and lower Page 4 of 8 Example 3 A rocket travels a vertical distance of (correct to the nearest ) in seconds (correct to the nearest second). What are the upper and lower limits to the rocket's average speed? upper bounds , and bounds , The highest average speed is with the largest distance divided by the smallest time. y KM TUITION r upper and lower Page 5 of 8 Question Length lower bound Length upper bound Length Width lower bound Width upper bound Width Area lower bound Area upper bound Worked example 6cm to nearest cm 5 5cm 6 5cm 5cm to nearest cm 4 5cm 5 5cm Length LB X Width LB = 5 5 x 4 5 =24 75cm Length UB X Width UB =6 5 x 5 5 =35 75cm 1)** 7cm to nearest cm 3cm to nearest cm 2)** 4cm to nearest cm 2cm to nearest cm 3) 12cm to nearest cm 9cm to nearest cm 4) 23cm to nearest cm 17cm to nearest cm 5) 53cm to nearest cm 39cm to nearest cm 6)

7 ** 40m to nearest 10m 30m to nearest 10m 7)** 80m to nearest 10m 20m to nearest 10m 8) 130m to nearest 10m 110m to nearest 10m y KM TUITION r upper and lower Page 6 of 8 9) 240m to nearest 10m 140m to nearest 10m 10) 300m to nearest 10m 200m to nearest 10m 11)** 2 3cm to 1dp 1 2cm to 1dp 12)** 7 7cm to 1dp 3 8cm to 1dp 13) 12 6cm to 1dp 11 4cm to 1dp 14) 9 5cm to 1dp 7 1cm to 1dp 15) 17 8cm to 1dp 13 2cm to 1dp y KM TUITION r upper and lower Page 7 of 8 Question Length lower bound Length upper bound Length Width lower bound Width upper bound Width Perimeter lower bound Perimeter upper bound Worked example 6cm to nearest cm 5 5cm 6 5cm 5cm to nearest cm 4 5cm 5 5cm (2 x Length LB) + (2 x Width LB) = (2 x 5 5) + (2 x 4 5) = 20cm (2 x Length UB) + (2 x Width UB) = (2 x 6 5) + (2 x 5 5) = 24cm 16)** 7cm to nearest cm 3cm to nearest cm 17)** 4cm to nearest cm 2cm to nearest cm 18) 12cm to nearest cm 9cm to nearest cm 19)

8 23cm to nearest cm 17cm to nearest cm 20) 53cm to nearest cm 39cm to nearest cm 21)** 40m to nearest 10m 30m to nearest 10m y KM TUITION r upper and lower Page 8 of 8 22)** 80m to nearest 10m 20m to nearest 10m 23) 130m to nearest 10m 110m to nearest 10m 24) 240m to nearest 10m 140m to nearest 10m 25) 300m to nearest 10m 200m to nearest 10m 26)** 2 3cm to 1dp 1 2cm to 1dp 27)** 7 7cm to 1dp 3 8cm to 1dp 28) 12 6cm to 1dp 11 4cm to 1dp 29) 9 5cm to 1dp 7 1cm to 1dp 30) 17 8cm to 1dp 13 2cm to 1dp