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1.1.2

Upper and lower bounds

Unit 1 · Number

2  Upper and Lower Bounds

A sign says the bridge is 50 m long, to the nearest 10 m. So how long is it really? Not 50. Somewhere between two numbers — and this lesson is about finding them.

🎯 By the end of this you can…

  • ✓Find the lower and upper bound of any rounded measurement.
  • ✓Write the answer as an inequality, like 45 ≤ x < 55.
  • ✓Use bounds in real problems — smallest and largest possible answers.

💡 The idea

This is Lesson 1 played backwards

Last lesson: you had the real number and found the rounded one. Now you have the rounded one and want to know what the real number could have been.

If someone tells you a box weighs 42 g to the nearest gram, the real weight is not exactly 42. It is anything that would round to 42.

41.5 rounds to 42. ✅   42.4 rounds to 42. ✅   42.6 rounds to 43. ❌

📋 The only method you need

  1. 1
    Find the rounding unit — the thing it was rounded to. Nearest gram → 1. Nearest 10 → 10. Nearest 0.1 → 0.1.
  2. 2
    Halve it. That is your jump. Half of 1 is 0.5. Half of 10 is 5. Half of 0.1 is 0.05.
  3. 3
    Lower bound = the number minus the jump.
  4. 4
    Upper bound = the number plus the jump.
  5. 5
    That is it. Same two moves every time, whatever the unit.

✏️ Worked example

A mass is 26 kg, correct to the nearest kilogram. Find the bounds.

  1. 1
    Rounding unit = 1 kg.
  2. 2
    Half of 1 = 0.5. That is the jump.
  3. 3
    Lower bound = 26 − 0.5 = 25.5 kg
  4. 4
    Upper bound = 26 + 0.5 = 26.5 kg

✅ Answer: 25.5 kg ≤ mass < 26.5 kg

⚠️ The classic trap

Watch the < at the top end. The lower bound really can happen — 25.5 rounds to 26. But 26.5 rounds up to 27, so it is not included.

That is why the correct way to write it is — a ≤ at the bottom and a < at the top. Exams give a mark for getting those two symbols the right way round.

🎮 Nearest whole number — give the LOWER bound

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5 cm
2
26 kg
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13 ml
4
50 mm
5
38 km
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23 (upper bound this time)
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439 (upper bound)
8
67547 (lower bound)
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🎮 Nearest ten, five and hundred

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50 cm, to the nearest ten — lower bound
2
260 kg, nearest ten — upper bound
3
130 ml, nearest ten — lower bound
4
400 km, nearest ten — upper bound
5
50 cm, to the nearest FIVE — lower bound
6
265 kg, nearest five — upper bound
7
200 m, nearest hundred — lower bound
8
2600 kg, nearest hundred — upper bound
9
5900 mm, nearest hundred — lower bound
10
380000 km, nearest hundred — upper bound
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🎮 One decimal place

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5.6 cm — lower bound
2
2.9 kg — upper bound
3
1.3 ml — lower bound
4
5.0 mm — upper bound
5
3.1 km — lower bound
6
77.7 — upper bound
7
951.2 — lower bound
8
1009.6 — upper bound
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🤔 A number x is 30 after rounding to 1 significant figure. Which inequality is right?

💡 The idea

In words, not numbers

Exam questions hide the rounding unit inside a sentence. Your first job is always the same: find the phrase “correct to the nearest …”. That tells you the unit, and the rest is automatic.

🎮 Word problems from the worksheet

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1
Fiona drives 49 miles, to the nearest mile. Minimum distance?
2
A pancake has 155 Calories, to the nearest Calorie. Greatest?
3
Jagdeep’s foot is 21 cm, to the nearest cm. Least length?
4
A tray weighs 9 kg, to the nearest kg. Least weight?
5
Appleby to Bunbury is 35 km, to the nearest km. Greatest?
6
An empty box weighs 42 g, to the nearest gram. Least weight?
7
A bag of apples is 7 kg, to the nearest kg. Least possible weight?
8
A table is 105 cm, to the nearest cm. Smallest length?
9
A car’s speed is 50 mph, to the nearest 10 mph. Lowest speed?
10
Sun to Earth is 150000000 km to 2 sf. Lower bound?
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✏️ Worked example

Find the least and greatest total length of 6 sausages, each 8 cm to the nearest cm.

  1. 1
    One sausage: unit is 1 cm, so the jump is 0.5. Each is between 7.5 and 8.5 cm.
  2. 2
    For the shortest total, make every sausage as short as possible: 6 × 7.5.
  3. 3
    6 × 7.5 = 45 cm
  4. 4
    For the longest total, make every sausage as long as possible: 6 × 8.5 = 51 cm

✅ Answer: Least 45 cm, greatest 51 cm

⚠️ The classic trap

Do not round the bounds themselves. If each sausage is at least 7.5 cm, six of them are at least 45 cm — not “about 45”. Bounds are exact numbers.

✏️ Worked example

A rugby pitch is 74 m wide (nearest metre) and 115 m long (nearest 5 m). Find the upper bound of the perimeter.

  1. 1
    Width: unit 1 m, jump 0.5. Upper bound of width = 74.5 m.
  2. 2
    Length: unit 5 m, so jump is 2.5. Upper bound of length = 115 + 2.5 = 117.5 m.
  3. 3
    For the biggest perimeter, use the biggest of both.
  4. 4
    Perimeter = 2 × (74.5 + 117.5) = 2 × 192

✅ Answer: 384 m

🎮 Inequalities — write the limits

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Length l = 1200 m, nearest 100 m — lower bound
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Mass m = 88 g, nearest gram — upper bound
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n = 90 people, nearest ten — lower bound
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Volume V = 4.1 ml, nearest tenth — upper bound
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Time t = 8.46 s, nearest 0.01 s — lower bound
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x = 1500 to 3 sf — lower bound
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x = 9000 to 1 sf — lower bound
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x = 0.20 to 2 sf — upper bound
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🎮 Rounding to dp and sf are secretly the same thing

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Rounding 7.58 to 1 dp is the same as rounding to how many sf?
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8200 to the nearest thousand is how many sf?
3
1822 to the nearest ten is how many sf?
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867.3 to the nearest whole number is how many sf?
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🏅

You finished this lesson!

One move, every time: halve the rounding unit, then go that far each way. And write it properly — .

Next: the different families numbers belong to.

Keep reading — this lesson marks itself done automatically.