Steply

Upper and lower bounds of a measurement

Foundations · 20 problems, easy to harder

▶ Watch the first one solved

How we do it

  1. Find the step
  2. Half a step
  3. Find the bounds
  4. Show it on a number line
  5. Check

The steps for A length is 8 cm to the nearest cm. Every solution names its steps, so you always know what comes next.

Worked examples

A length is 8 cm to the nearest cm Easy

The question

Question: the length is 8 cm, to the nearest cm.

Find the upper and lower bounds.

Length: 8 cm, to the nearest cm

The real length was rounded.

We want the smallest and the biggest it could have been.

1 Find the step

It was rounded to the nearest cm.

So the step is 1 cm.

step = 1 cm
upper bound = the biggest it could be
lower bound = the smallest it could be

2 Half a step

Anything less than half a step away rounds to 8.

Half of 1 is 0.5.

half a step = 1 ÷ 2 = 0.5 cm

3 Find the bounds

The lower bound is 8 minus 0.5: 7.5.

lower bound = 8 − 0.5 = 7.5 cm

Careful!

We go only half a step down and half a step up.

7 would round to 7, not to 8.

Careful: not 7 to 9 ✗

The upper bound is 8 plus 0.5: 8.5.

upper bound = 8 + 0.5 = 8.5 cm

4 Show it on a number line

So 7.5 ≤ length < 8.5.

This is called the error interval.

7.5 ≤ length < 8.5

5 Check

Let's check.

7.5, rounded to the nearest cm, is 8.

7.5 rounds to 8

And 8.49, just under the upper bound, also rounds to 8.

So the bounds are right!

8.49 rounds to 8

Let's look back.

First: find the step.

Then: half a step.

Then: find the bounds.

Then: show it on a number line.

Last: check.

Answer
7.5 ≤ length < 8.5

▶ Watch it solved

A mass is 2.4 kg to 1 decimal place Medium

The question

Question: the mass is 2.4 kg, to 1 decimal place.

Find the upper and lower bounds.

Mass: 2.4 kg, to 1 decimal place

The real mass was rounded.

We want the smallest and the biggest it could have been.

1 Find the step

One decimal place: we rounded to the nearest 0.1.

So the step is 0.1.

step = 0.1 kg
upper bound = the biggest it could be
lower bound = the smallest it could be

2 Half a step

Anything less than half a step away rounds to 2.4.

Half of 0.1 is 0.05.

half a step = 0.1 ÷ 2 = 0.05 kg

3 Find the bounds

The lower bound is 2.4 minus 0.05: 2.35.

lower bound = 2.4 − 0.05 = 2.35 kg

Careful!

We go only half a step down and half a step up.

2.3 would round to 2.3, not to 2.4.

Careful: not 2.3 to 2.5 ✗

The upper bound is 2.4 plus 0.05: 2.45.

upper bound = 2.4 + 0.05 = 2.45 kg

4 Show it on a number line

So 2.35 ≤ mass < 2.45.

This is called the error interval.

2.35 ≤ mass < 2.45

5 Check

Let's check.

2.35, rounded to 1 decimal place, is 2.4.

2.35 rounds to 2.4

And 2.44, just under the upper bound, also rounds to 2.4.

So the bounds are right!

2.44 rounds to 2.4

Let's look back.

First: find the step.

Then: half a step.

Then: find the bounds.

Then: show it on a number line.

Last: check.

Answer
2.35 ≤ mass < 2.45

▶ Watch it solved

A mass is 4.25 kg to 2 decimal places Harder

The question

Question: the mass is 4.25 kg, to 2 decimal places.

Find the upper and lower bounds.

Mass: 4.25 kg, to 2 decimal places

The real mass was rounded.

We want the smallest and the biggest it could have been.

1 Find the step

2 decimal places: we rounded to the nearest 0.01.

So the step is 0.01.

step = 0.01 kg
upper bound = the biggest it could be
lower bound = the smallest it could be

2 Half a step

Anything less than half a step away rounds to 4.25.

Half of 0.01 is 0.005.

half a step = 0.01 ÷ 2 = 0.005 kg

3 Find the bounds

The lower bound is 4.25 minus 0.005: 4.245.

lower bound = 4.25 − 0.005 = 4.245 kg

Careful!

We go only half a step down and half a step up.

4.24 would round to 4.24, not to 4.25.

Careful: not 4.24 to 4.26 ✗

The upper bound is 4.25 plus 0.005: 4.255.

upper bound = 4.25 + 0.005 = 4.255 kg

4 Show it on a number line

So 4.245 ≤ mass < 4.255.

This is called the error interval.

4.245 ≤ mass < 4.255

5 Check

Let's check.

4.245, rounded to 2 decimal places, is 4.25.

4.245 rounds to 4.25

And 4.254, just under the upper bound, also rounds to 4.25.

So the bounds are right!

4.254 rounds to 4.25

Let's look back.

First: find the step.

Then: half a step.

Then: find the bounds.

Then: show it on a number line.

Last: check.

Answer
4.245 ≤ mass < 4.255

▶ Watch it solved

Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. a length is 8 cm to the nearest cm
    Answer
    7.5 ≤ length < 8.5
  2. a mass is 6 kg to the nearest kg
    Answer
    5.5 ≤ mass < 6.5
  3. a height is 14 m to the nearest m
    Answer
    13.5 ≤ height < 14.5
  4. a time is 45 s to the nearest s
    Answer
    44.5 ≤ time < 45.5
  5. 350 people to the nearest 10
    Answer
    345 ≤ people < 355
  6. a distance is 70 km to the nearest 10 km
    Answer
    65 ≤ distance < 75
  7. a length is 23 cm to the nearest cm
    Answer
    22.5 ≤ length < 23.5

Medium

  1. a mass is 2.4 kg to 1 decimal place
    Answer
    2.35 ≤ mass < 2.45
  2. a time is 12.5 s to the nearest 0.1 s
    Answer
    12.45 ≤ time < 12.55
  3. a length is 9.6 cm to the nearest mm
    Answer
    9.55 ≤ length < 9.65
  4. a volume is 3.8 l to 1 decimal place
    Answer
    3.75 ≤ volume < 3.85
  5. 1200 people to the nearest 100
    Answer
    1150 ≤ people < 1250
  6. a width is 0.7 m to 1 decimal place
    Answer
    0.65 ≤ width < 0.75
  7. a height is 1.65 m to 2 decimal places
    Answer
    1.645 ≤ height < 1.655

Harder

  1. a mass is 4.25 kg to 2 decimal places
    Answer
    4.245 ≤ mass < 4.255
  2. a length is 5000 m to the nearest 100 m
    Answer
    4950 ≤ length < 5050
  3. a time is 0.36 s to 2 decimal places
    Answer
    0.355 ≤ time < 0.365
  4. a distance is 18.0 km to 1 decimal place
    Answer
    17.95 ≤ distance < 18.05
  5. a mass is 250 g to the nearest 10 g
    Answer
    245 ≤ mass < 255
  6. a length is 12.08 m to 2 decimal places
    Answer
    12.075 ≤ length < 12.085