Understanding upper and lower bounds
🎯 In this topic you will
- Work out upper and lower bounds
🧠 Key Words
- lower bound
- upper bound
Show Definitions
- lower bound: The smallest possible value within a given range.
- upper bound: The largest possible value within a given range.
You already know how to round numbers to a given number of decimal places or significant figures. When you are given a number that has already been rounded, have you ever wondered what the original number was, before it was rounded? Read this information about the Super Bowl:
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The Super Bowl is an American football game played to decide the champion of the National Football League (NFL). It is one of the most widely watched sporting events in the world. The Super Bowl in 1979 holds the record for the highest number of fans actually in the stadium at $104\,000$ (to the nearest $1000$). |
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How many fans were actually in the stadium? It is impossible to know the exact number of fans from the information given. However, you can work out the smallest number of fans there could have been. This is called the lower bound. You can also work out the greatest number of fans there could have been. This is called the upper bound.
🔎 Reasoning Tip
Bounds for rounded numbers: For the rounded number of fans \(104{,}000\): Lower bound = \(103{,}500\) fans Upper bound = \(104{,}499\) fans
❓ EXERCISES
1. All of these whole numbers have been rounded to the nearest $10$.
For each part write
i a list of the integer values the number could be
ii the lower bound
iii the upper bound.
a. $30$ b. $90$ c. $270$ d. $850$
👀 Show answer
a. i $25,26,27,28,29,30,31,32,33,34$; ii $25$; iii $34$
b. i $85,86,87,88,89,90,91,92,93,94$; ii $85$; iii $94$
c. i $265,266,267,268,269,270,271,272,273,274$; ii $265$; iii $274$
d. i $845,846,847,848,849,850,851,852,853,854$; ii $845$; iii $854$
2. A number with one decimal place is rounded to the nearest whole number. The answer is $12$.
Copy and complete these sentences.
a. The numbers with one decimal place that round to $12$ are $11.5, 11.6, 11.7, \dots, \dots, \dots, \dots, \dots, \dots, \dots, \dots$
b. The lower bound is …
c. The upper bound is …
👀 Show answer
a. $11.5, 11.6, 11.7, 11.8, 11.9, 12.0, 12.1, 12.2, 12.3, 12.4$
b. $11.5$
c. $12.4$
3. This is part of Aashi’s homework. There are marks on her work, covering some of the numbers.
Question: Increase $\$42$ by $\_\_\_$%. Round your answer to the nearest whole number.
Solution: $42 \times 1\_\_\_ \_\_\_ = \$55$ (to the nearest whole number)
The number she rounds to $55$ has one decimal place.
a. Write:
i. a list of the numbers with one decimal place that round to $55$
ii. the lower bound
iii. the upper bound.
b. The question is "Increase $\$42$ by $30\%$". What are the numbers covered by the marks in Aashi’s solution?
👀 Show answer
a.
i. $54.5, 54.6, 54.7, 54.8, 54.9, 55.0, 55.1, 55.2, 55.3, 55.4$
ii. Lower bound: $54.5$
iii. Upper bound: $55.4$
b. $42 \times 1.3 = 54.6$, so the missing numbers are $1.3$ and $54.6$.
🧠 Think like a Mathematician
Scenario: A decimal number is rounded to the nearest whole number. The rounded answer is 8.
Arun says: “I think the lower bound is 7.5 and the upper bound is 8.4.”
Zara says: “I agree with the lower bound, but disagree with the upper bound.”
- a) Do you agree with the lower bound of 7.5? Explain why.
- b) Zara knows that the upper bound is not 8.4, but has not said what she thinks it is. What do you think the upper bound is? Explain your answer.
- c) Discuss your answers to a and b with others.
Follow-up prompts:
👀 show answer
- a: Yes — when a number is rounded to the nearest whole number and the result is 8, the smallest it could be is halfway between 7 and 8, which is 7.5. Anything smaller would round to 7.
- b: The largest value that would still round to 8 is halfway between 8 and 9, which is 8.5. Since the upper bound is not included, it is $8.5$ (exclusive)$. Arun’s 8.4 is within the range, but not the true bound.
- Correct range:$7.5 \leq \text{number} < 8.5$.
🧠 Think like a Mathematician
Scenario: Sofia and Marcus are discussing how to write the range of values that round to 8 (to the nearest whole number).
Sofia: “I would write the range as $7.5 \leq x \leq 8.499999999...$.”
Marcus: “I would write the range as $7.5 \leq x < 8.5$.”
- a) Who do you think has written the inequality in the better way? Explain why.
- b) Can you think of another way to write the inequality?
- c) Discuss your answers to parts a and b with others.
Follow-up prompts:
👀 show answer
- a: Marcus’s notation $7.5 \leq x < 8.5$ is generally considered clearer and simpler than using 8.499999..., because it precisely expresses that the upper bound is not included, without infinite decimals.
- b: Another way to write it is using set notation: $\{x \in \mathbb{R} \mid 7.5 \leq x < 8.5\}$, meaning “the set of real numbers x such that 7.5 ≤ x < 8.5.”
- Reasoning: This range represents all numbers that round to 8 to the nearest whole number. The lower bound 7.5 is included, and the upper bound 8.5 is excluded.
❓ EXERCISES
6. A decimal number is rounded to the nearest whole number.
Write an inequality to show the range of values the number can be when
🔎 Reasoning Tip
Inequality notation: Use Marcus’ method of writing the inequality from Question 5. For part a, the answer is \( 3.5 \leq x < \dots \).
a. the answer is $4$
b. the answer is $12$
c. the answer is $356$
d. the answer is $670$
👀 Show answer
a. $3.5 \le x \lt 4.5$ b. $11.5 \le x \lt 12.5$
c. $355.5 \le x \lt 356.5$ d. $669.5 \le x \lt 670.5$
7. A decimal number is rounded to the nearest ten.
Copy and complete each inequality to show the range of values the number can be when
a. the answer is $20$ $15 \le x \lt \ldots$
b. the answer is $340$ $\ldots \le x \lt 345$
c. the answer is $4750$ $\ldots \le x \lt \ldots$
d. the answer is $6300$ $\ldots \le x \lt \ldots$
👀 Show answer
a. $15 \le x \lt 25$
b. $335 \le x \lt 345$
c. $4745 \le x \lt 4755$
d. $6295 \le x \lt 6305$
8. A decimal number is rounded to the nearest one hundred.
Copy and complete each inequality to show the range of values the number can be when
a. the answer is $300$ $250 \le x \lt \ldots$
b. the answer is $1900$ $\ldots \le x \lt 1950$
c. the answer is $4700$ $\ldots \le x \lt \ldots$
d. the answer is $8000$ $\ldots \le x \lt \ldots$
👀 Show answer
a. $250 \le x \lt 350$
b. $1850 \le x \lt 1950$
c. $4650 \le x \lt 4750$
d. $7950 \le x \lt 8050$
🧠 Think like a Mathematician
Question: What do you notice about the methods you use to work out the lower and upper bounds of a rounded number?
- a) Copy and complete these generalising sentences:
- When you round to the nearest whole number, the lower and upper bounds will be … below and above the rounded number.
- When you round to the nearest ten, the lower and upper bounds will be … below and above the rounded number.
- When you round to the nearest one hundred, the lower and upper bounds will be … below and above the rounded number.
- b) Can you describe a general rule to explain how you work out the lower and upper bounds of a rounded number?
Follow-up prompts:
👀 show answer
- a.i: 0.5 below and 0.5 above.
- a.ii: 5 below and 5 above.
- a.iii: 50 below and 50 above.
- b:General rule: When rounding to the nearest unit of a given place value, the lower bound is found by subtracting half of that unit, and the upper bound is found by adding half of that unit (upper bound is exclusive).
- Example: Rounding to the nearest 0.1 → subtract/add 0.05.
❓ EXERCISES
10. Vihaan works out the circumference of a pond to be $1560 \ \text{cm}$, correct to the nearest $10 \ \text{cm}$.
a) Write:
- the lower bound of the circumference
- the upper bound of the circumference
b) Write an inequality to show the range of values the circumference could have.
👀 Show answer
a)
- Lower bound: $1555 \ \text{cm}$
- Upper bound: $1565 \ \text{cm}$
b) Inequality: $1555 \le C \lt 1565$
11. Saarya works out the mean height of the members in their netball team to be $172 \ \text{cm}$, correct to the nearest centimetre.
a) Write:
- the lower bound of the mean height
- the upper bound of the mean height
b) Write an inequality to show the range of values the mean height could have.
👀 Show answer
a)
- Lower bound: $171.5 \ \text{cm}$
- Upper bound: $172.5 \ \text{cm}$
b) Inequality: $171.5 \le h \lt 172.5$
12. The rectangular cards show a range of values that a rounded number can be.
The oval cards show the degree of accuracy of the rounding.
The hexagonal cards show the rounded numbers.
Match each rectangular card with the correct oval and hexagonal card.
Rectangles: A: $1550 \le x \lt 1650$ B: $550 \le x \lt 650$ C: $55 \le x \lt 65$
D: $15.5 \le x \lt 16.5$ E: $155 \le x \lt 165$ F: $164.5 \le x \lt 165.5$
Ovals: i: nearest $100$ ii: nearest $10$ iii: nearest $1$
Hexagons: a: $16$ b: $60$ c: $160$ d: $165$ e: $1600$ f: $600$
👀 Show answer
A → i (nearest $100$) and e ($1600$)
B → i (nearest $100$) and f ($600$)
C → ii (nearest $10$) and b ($60$)
D → iii (nearest $1$) and a ($16$)
E → ii (nearest $10$) and c ($160$)
F → iii (nearest $1$) and d ($165$)
⚠️ Be careful!
When finding the upper bound, never include the next rounded value. For example, if a number is 104 000 to the nearest 1 000, the upper bound is $104\,500$, not$105\,000$. Always stop just before the next rounding point.