Apply Knowledge of Upper and Lower Bounds

In this worksheet, students will use upper and lower bounds in order to calculate areas and speeds from the given information.

Key stage:  KS 3

Curriculum topic:   Number

Curriculum subtopic:   Round/Approximate Numbers Appropriately

Difficulty level:

QUESTION 1 of 10

This activity is about calculating upper and lower bounds.

Suppose we've rounded a number to the nearest whole number and get 14.

Numbers that round up to 14 include 13.9, 13.75, 13.694, 13.5, etc.

The smallest of these is 13.5.

This is then called the lower bound.

Anything smaller than 13.5 will round down to 13.

Larger numbers that round down to 14 include 14.1, 14.34968, 14.49, etc.

The largest of these is 14.4999999...

This is the upper bound, which we write as 14.5.

Example 1:

A room measures 433 cm by 372 cm, measured to the nearest centimetre.

Find the greatest and smallest possible area of the room.

Upper bounds of measurements: 433.5 cm and 372.5 cm.

Greatest area is 433.5 × 372.5 = 161478.75 cm2.

Lower bounds of measurements: 432.5 cm and 371.5 cm.

Least area is 432.5 × 371.5 = 160 673.75 cm2.

Example 2:

A sprinter runs 100 m in 12.8s.

The distance is measured to the nearest centimetre and the time to the nearest tenth of a second.

Find the greatest and least possible speeds of the sprinter.

Upper bounds of measurements: 100.005 m and 12.85s

Lower bounds of measurements: 99.995 m and 12.75s.

Be careful here!

For the greatest speed, we want the maximum distance in the minimum time:

Greatest speed =
 100.005 12.75
= 7.84 ms-1 (3 s.f.)

For the least speed, we want the minimum distance in the maximum time:

Least speed =
 99.995 12.85
= 7.78 ms-1 (3 s.f.)

Now it's over to you.

Find the greatest and least areas of a room with the following measurements 358 cm by 332 cm.

 119201.25 134321.25 118511.25 133573.25 Greatest area Least area

Find the greatest and least areas of a room with the following measurements: 297 cm by 451 cm.

 119201.25 134321.25 118511.25 133573.25 Greatest area Smallest area

Find the greatest and least speeds from the following information: 100.03 m (to the nearest cm) covered in 11.4 s (to the nearest tenth second).

 8.89 8.81 8.74 8.47 Greatest speed Least speed

Find the greatest and least speeds from the following information: 99.70 m (to the nearest cm) covered in 11.3 s (to the nearest tenth second).

 8.89 8.86 8.78 8.47 Greatest speed Least speed

Find the greatest and least areas of a room with the following measurements: 447 cm by 314 cm.

 119201.25 140738.75 118511.25 139977.75 Greatest area Least area

Find the greatest and least areas of a room with the following measurements: 3 m by 4 m.

 15.75 11.25 12.25 8.75 Greatest area Least area

Find the greatest and least speeds from the following information: 100 m (to the nearest m) covered in 12 s (to the nearest second).

 8.04 8.74 8.65 7.96 Greatest speed Least speed

Find the greatest and least speeds from the following information: 99 miles (to the nearest mile) covered in 3 hours (to the nearest hour).

 28.1 33 39.8 39.4 Greatest speed Least speed

Find the greatest and least speeds from the following information: 100.01 m (to the nearest cm) covered in 12.6 s (to the nearest tenth second).

 7.99 7.97 7.91 7.78 Greatest speed Least speed

Find the greatest and least areas of a room with the following measurements: 268 cm by 320 cm.

 86054.25 140738.75 85466.25 139977.75 Greatest area Least area
• Question 1

Find the greatest and least areas of a room with the following measurements 358 cm by 332 cm.

 119201.25 134321.25 118511.25 133573.25 Greatest area Least area
EDDIE SAYS
How did you get on with your first question? We need to find the two possible greatest values for the sides of the room, then multiply these together. Then we need to repeat this process using the two least values for the sides of the room. Upper bounds of measurements: 358.5 cm and 332.5 cm. Greatest area: 358.5 × 332.5 = 119201.25 cm² Lower bounds of measurements: 357.5 cm and 331.5 cm². Least area: 357.5 cm × 331.5 = 118511.25 cm² Did you calculate those correctly and choose the correct options in the answer grid?
• Question 2

Find the greatest and least areas of a room with the following measurements: 297 cm by 451 cm.

 119201.25 134321.25 118511.25 133573.25 Greatest area Smallest area
EDDIE SAYS
Again, we need to calculate the highest and lowest possible areas of the room. Upper bounds of measurements: 297.5 cm and 451.5 cm. Greatest area: 297.5 × 451.5 = 134321.25 cm² Lower bounds of measurements: 296.5 cm and 450.5 cm². Least area is: 296.5 × 450.5 = 133572.25 cm² A strong start, we have 8 more questions to tackle!
• Question 3

Find the greatest and least speeds from the following information: 100.03 m (to the nearest cm) covered in 11.4 s (to the nearest tenth second).

 8.89 8.81 8.74 8.47 Greatest speed Least speed
EDDIE SAYS
Upper bounds of measurements: 100.035 m in 11.45 s. Lower bounds of measurements: 100.025 m in 11.35 s. Be careful here, as we need to mix our measurements! For the greatest speed, we want to use the maximum distance with the minimum time: Greatest speed = 100.025 ÷ 11.35 = 8.81 ms-1 (We have rounded our answers to two decimal places to match with the possible options given in the answer grid.) For the least speed, we need to use the minimum distance and the maximum time: Least speed = 100.035 ÷ 11.45 = 8.74 ms-1 Don't worry if you found this one more challenging; it required more knowledge!
• Question 4

Find the greatest and least speeds from the following information: 99.70 m (to the nearest cm) covered in 11.3 s (to the nearest tenth second).

 8.89 8.86 8.78 8.47 Greatest speed Least speed
EDDIE SAYS
Upper bounds of measurements: 99.705 m in 11.35 s. Lower bounds of measurements: 99.695 m in 11.25 s. Let's mix our measurements again! For the greatest speed, we want to use the maximum distance with the minimum time: Greatest speed = 99.705 ÷ 11.25 = 8.86 ms-1 For the least speed, we need to use the minimum distance and the maximum time: Least speed = 99.695 ÷ 11.35 = 8.78 ms-1 Take a deep breath, you've got this.
• Question 5

Find the greatest and least areas of a room with the following measurements: 447 cm by 314 cm.

 119201.25 140738.75 118511.25 139977.75 Greatest area Least area
EDDIE SAYS
Upper bounds of measurements: 447.5 cm and 314.5 cm. Greatest area: 447.5 × 314.5 = 140738.75 cm² Lower bounds of measurements: 446.5 cm and 313.5 cm. Least area: 446.5 × 313.5 = 139977.75 cm² Great focus, is this beginning to get easier?
• Question 6

Find the greatest and least areas of a room with the following measurements: 3 m by 4 m.

 15.75 11.25 12.25 8.75 Greatest area Least area
EDDIE SAYS
Don't let the single units confuse you here - we just need to consider what could be the highest and lowest possible values which have been rounded up/down to reach these measurements. Upper bounds of measurements: 3.5 m and 4.5 m. Greatest area: 3.5 × 4.5 = 15.75 m² Lower bounds of measurements: 2.5 m and 3.5 m. Least area: 2.5 × 3.5 = 8.75 m² Did you get that?
• Question 7

Find the greatest and least speeds from the following information: 100 m (to the nearest m) covered in 12 s (to the nearest second).

 8.04 8.74 8.65 7.96 Greatest speed Least speed
EDDIE SAYS
Upper bounds of measurements: 100.5 m in 12.5 s. Lower bounds of measurements: 99.5 m in 11.5 s. Let's mix our measurements again! For the greatest speed, we want to use the maximum distance with the minimum time: Greatest speed = 100.5 ÷ 11.5 = 8.74 ms-1 For the least speed, we need to use the minimum distance and the maximum time: Least speed = 99.5 ÷ 12.5 = 7.96 ms-1 Are you getting up to speed with these ones yet?!
• Question 8

Find the greatest and least speeds from the following information: 99 miles (to the nearest mile) covered in 3 hours (to the nearest hour).

 28.1 33 39.8 39.4 Greatest speed Least speed
EDDIE SAYS
Upper bounds of measurements: 99.5 m in 3.5 h. Lower bounds of measurements: 98.5 m in 2.5 h. For the greatest speed, we want to use the maximum distance with the minimum time: Greatest speed = 99.5 ÷ 2.5 = 39.8 mph-1 For the least speed, we need to use the minimum distance and the maximum time: Least speed = 98.5 ÷ 3.5 = 28.1 mph-1 Did you notice that we were using miles per hours (mph) as our units of measurement this time?
• Question 9

Find the greatest and least speeds from the following information: 100.01 m (to the nearest cm) covered in 12.6 s (to the nearest tenth second).

 7.99 7.97 7.91 7.78 Greatest speed Least speed
EDDIE SAYS
Upper bounds of measurements: 100.015 m in 12.65 s. Lower bounds of measurements: 100.005 m in 12.55 s. For the greatest speed, we want to use the maximum distance with the minimum time: Greatest speed = 100.015 ÷ 12.55 = 7.97 ms-1 For the least speed, we need to use the minimum distance and the maximum time: Least speed = 100.005 ÷ 12.65 = 7.91 ms-1 Just one challenge remains now!
• Question 10

Find the greatest and least areas of a room with the following measurements: 268 cm by 320 cm.

 86054.25 140738.75 85466.25 139977.75 Greatest area Least area
EDDIE SAYS
Upper bounds of measurements: 268.5 cm and 320.5 cm. Greatest area: 268.5 × 320.5 = 86054.25 cm² Lower bounds of measurements: 267.5 cm and 319.5 cm. Least area: 267.5 × 319.5 = 85466.25 cm² Fantastic work! You can now identify the upper and lower bounds of data sets.
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