3.1

Rounding Off Numbers to Specified Decimal Places

AThe Idea of Rounding

To approximate is to find a suitable value for a quantity within a specified degree of accuracy. When we approximate values by rounding off whole numbers to the nearest 10, 100 and 1000, the degree of accuracy for each result is different.

We can use a number line to help us round off numbers. Round off 23 645 to the nearest (a) 10, (b) 100, (c) 1000.

(a) Nearest 10. 23 645 is the midpoint of 23 640 and 23 650. By convention, we round up 23 645 to 23 650.

We write 23 645 ≈ 23 650 or 23 645 = 23 650 (correct to the nearest 10).

(b) Nearest 100. 23 645 is nearer to 23 600 than to 23 700. We round 23 645 down to 23 600.

So 23 645 ≈ 23 600 (correct to the nearest 100).

(c) Nearest 1000. 23 645 is nearer to 24 000 than to 23 000. We round up to 24 000.

So 23 645 ≈ 24 000 (correct to the nearest 1000).

We can summarise the process of rounding off whole numbers:

Round off whole numbers to the nearest	Digit to consider	Digit is 5 or more	Digit is less than 5
10	digit in the ones place	round up	round down
100	digit in the tens place	round up	round down
1000	digit in the hundreds place	round up	round down

When we round off numbers to a specified place value,

look for the digit in the specified place value,
then consider the next digit to its right. If this digit is 5 or more, round up the number. If this digit is less than 5, round down the number.

📖 WORKED EXAMPLE 1

The exact population of Singapore at a certain moment in 2017 was 5 610 475. Round off the population figure to the nearest

(a) 1000, (b) million.

Solution

(a) Look for the digit in the thousands place. In 5 610 475, the digit '0' is in the thousands place.

Digit in thousands place

Digit is 4 (less than 5), round down the number.

→

Put zeros as placeholders

∴ 5 610 475 = 5 610 000 (correct to the nearest 1000)

(b) Look for the digit in the millions place. In 5 610 475, the digit '5' is in the millions place.

Digit in millions place

Digit is 6 (more than 5), round up the number.

→

Put zeros as placeholders

∴ 5 610 475 = 6 000 000 (correct to the nearest million)

✏️ TRY IT YOURSELF 1

In 2018, the number of taxis in Singapore at a certain moment was 24 891. Round off the number of taxis to the nearest:

(a) 10 = (b) 100 = (c) 10 000 =

THINK²

A student rounds off 5 610 475 to the nearest 1000 in 3 steps as shown below. Is his working correct? Explain.

5 610 475 = 5 610 480 (to the nearest 10)
5 610 480 = 5 610 500 (to the nearest 100)
5 610 500 = 5 611 000 (to the nearest 1000)

自评：

BRounding Off Numbers to Decimal Places

Extending the idea of approximation for whole numbers to decimals, we can summarise the process of rounding decimals as follows:

Round off to desired number of decimal places	Digit to consider	Digit is 5 or more	Digit is less than 5
1 decimal place	digit in the hundredths place	round up	round down
2 decimal places	digit in the thousandths place	round up	round down
3 decimal places	digit in the ten thousandths place	round up	round down

When we round off numbers to a specified decimal place,

look for the digit in the specified decimal place,
then consider the next digit to its right. If this digit is 5 or more, round up the number. If this digit is less than 5, round down the number.

📖 WORKED EXAMPLE 2

Round off 8.4695 to

(a) the nearest whole number, (b) 1 decimal place,
(c) 2 decimal places, (d) the nearest thousandth.

Solution

(a) Ones place: 8; next digit 4 (less than 5) → round down. 8.4695 = 8 (correct to the nearest whole number).

(b) Tenths place: 4; next digit 6 (more than 5) → round up. 8.4695 = 8.5 (correct to 1 d.p.). There should not be any digit after the first decimal place.

(d) Thousandths place: 9; next digit 5 → round up. 8.4695 = 8.470 (correct to 3 d.p.). We put '0' as a placeholder to indicate that the number is correct to 3 decimal places.

✏️ TRY IT YOURSELF 2

Round off 9.003 50 to

(i) nearest tenth = (ii) nearest hundredth =

(iii) 3 d.p. = (iv) nearest 0.0001 =

(b) Are the answers in (i) and (ii) the same numerical value? If so, what is the difference?

自评：

📖 WORKED EXAMPLE 3

(a) Evaluate 3 4/7 × (−2 1/15). (b) Round off the answer to 4 decimal places.

Solution

(a)

3 4/7 × (−2 1/15) = 25/7 × (−31/15)Convert to improper fractions.
= −155/21Cancel 5: 25/15 = 5/3.
= −7 8/21

(b) −7 8/21 = −7 − 8/21 = −7 − 0.380 952… = −7.3810 (correct to 4 d.p.).

SPOTLIGHT — Calculator check: press (-) 3 □ 4 □ 7 × (-) 2 □ 1 □ 15 = → result displays as −7 8/21.

✏️ TRY IT YOURSELF 3

(a) Evaluate 3 7/11 × (−1 9/20). =

(b) Round to 4 d.p. →

📖 WORKED EXAMPLE 4

Ella bought 2.07 kg of broccoli at $3.85 per kg and 0.735 kg of salmon at $29.90 per kg. Find the total amount Ella spent, correct to the (a) nearest dollar, (b) nearest cent.

Solution

Total = $3.85 × 2.07 + $29.90 × 0.735
= $7.9695 + $21.9765
= $29.9460Exact value.

(a) Nearest dollar: $30.

(b) Nearest cent: $29.95. 'Correct to the nearest cent' is equivalent to 'correct to 2 d.p.' when the unit is dollars.

✏️ TRY IT YOURSELF 4

Lily bought 800 g of chicken breast at $10.80/kg and 820 g of bananas at $4.82/kg.

(a) Nearest dollar = $ (b) Nearest cent = $

📖 WORKED EXAMPLE 5

A Vernier caliper is a measuring instrument with a precision of 0.1 mm. Johan measures the diameter of a pencil and records the value as 6 mm (correct to the nearest integer). What is the (a) largest, (b) smallest possible reading on the caliper?

Solution

(a) Largest = 6.4 mm. The next number with 1 d.p. greater than 6.4 is 6.5; however, 6.5 would round off to 7, not 6.

(b) Smallest = 5.5 mm. The next number with 1 d.p. smaller than 5.5 is 5.4; however, 5.4 would round off to 5, not 6.

✏️ TRY IT YOURSELF 5

A micrometer has precision 0.01 mm. John records a coin thickness as 1.6 mm (correct to 1 d.p.).

(a) Largest possible reading = mm

(b) Smallest possible reading = mm

📝 PRACTICE EXERCISE 3.1

BASIC MASTERY

Round off the following numbers to the nearest whole number.

(a) 13.4 → (b) 321.8 →
Round off the following numbers to the nearest 100.

(a) 7289 → (b) 13 562 →
Round off the following numbers to 1 decimal place.

(a) 23.69 → (b) 0.72 →
Round off the following numbers to 2 decimal places.

(a) 10.7543 → (b) 2.9968 →
Round off the following numbers to 3 decimal places.

(a) 0.040 25 → (b) 17.926 53 →
Round off the following numbers to 4 decimal places.

(a) 3.004 056 → (b) 18.471 984 →

INTERMEDIATE

(a) Evaluate (−13) × [(−17) + (−12)] = (b) Round to nearest 10 =
Evaluate the following and give the answers correct to 3 decimal places.

(a) 2/3 × (−8 + 12) ≈ (b) 4 1/7 − 5 1/8 ≈

(c) −4 2/9 ÷ 1 7/12 ≈ (d) √(17² − 8²) / 13 ≈
The Upper Peirce Reservoir has a water storage capacity of 27 800 000 m³. Round off the capacity to the nearest 1 000 000 m³.

= m³
The area of an apartment is 84.5 m².

(a)(i) Nearest 1 m² = m² (ii) Nearest 10 m² = m²

(b) Price of apartment = $725 000. Price per m² to nearest dollar: $
A metalworker measures the height of an object as 2.1036 cm.

(a)(i) 2 d.p. = cm (ii) 3 d.p. = cm

(b) Which approximation is more accurate? Explain.

EN(ii) is more accurate. 2.104 cm preserves three decimal places (precision ≈ 0.001 cm), while 2.10 cm only preserves two (precision ≈ 0.01 cm). More d.p. → tighter possible range → higher accuracy.

⏱ 15 秒后查看中文版

中文(ii) 更精确。3 位小数（精度 0.001）比 2 位小数（精度 0.01）能锁定更窄的真值范围。

自评：

ADVANCED

An integer n, when rounded off to the nearest 10, is 920.

(a)(i) From the number line, find a (midpoint of 910 and 920) and b (midpoint of 920 and 930).

a = b =

(a)(ii) Smallest possible n = Largest possible n =

(b) If n is a decimal with one decimal place, find the smallest and largest:

Smallest = Largest =
The population of Jurong West in 2018 was 267 000, correct to the nearest hundred.

(a) Largest possible value =

(b) Smallest possible value =
Michael's 100-m race time is recorded as 9.96 seconds, correct to 2 d.p. Find the greatest possible value if the chip timing system has degree of accuracy:

(a) 0.001 s → s

(b) 0.0001 s → s
Determine if each of the following is correct or incorrect. Explain your answer.

(a) 0.959 = 1 (correct to the nearest 0.1)

(b) 508.66 = 51 (correct to the nearest 10)

(c) 14.235 = 14.035 (correct to the nearest 0.1)

(d) 90.92 = 100 (correct to the nearest 100)

💡 思考后展开详细解释

(a) 对。0.959 → 1.0（看百分位 5，进位）
(b) 错。508.66 → 510，不是 51（看个位 8，进位）
(c) 错。14.235 → 14.2，不是 14.035（看百分位 3 < 5，舍去）
(d) 对。90.92 → 100（看十位 9，进位）
[OPEN] A number is rounded off to 38 000.

(a) What are the possible degrees of accuracy of this approximation?

ENPossible degrees: nearest 1, 10, 100, 1000. (Nearest 10 000 would give multiples of 10 000 like 30 000 or 40 000, NOT 38 000, so this is excluded.)

⏱ 15 秒后查看中文版

中文可能的精度：精确到 1、10、100、1000。（精确到 10 000 只能得到 30 000 或 40 000，不能是 38 000。）

自评：

(b) For each degree, give an example of the actual value:
ENExamples:
- nearest 1 → 38 000 exactly
- nearest 10 → 37 998 (in [37 995, 38 004])
- nearest 100 → 38 045 (in [37 950, 38 049])
- nearest 1000 → 37 612 (in [37 500, 38 499])
⏱ 20 秒后查看中文版

中文示例：精确到 1 → 38 000；精确到 10 → 37 998；精确到 100 → 38 045；精确到 1000 → 37 612。
自评：
[OPEN] Sulin wants to determine the thickness of a $1 coin by measuring the height of 25 stacked coins, which is 58 mm.

(a) Using Sulin's method, find the thickness of a $1 coin to the nearest 0.1 mm.

= mm

(b) Is Sulin's method more accurate than measuring the thickness of one coin directly? Explain.

ENYes, Sulin's method is more accurate. Measuring one coin directly is limited by the caliper's precision (~0.1 mm) — any error of ±0.05 mm affects that one coin entirely. By measuring 25 coins together and dividing, the measurement error is also divided by 25, so per-coin error becomes ~0.002 mm instead of 0.05 mm. Averaging reduces random error.

⏱ 30 秒后查看中文版

中文更精确。直接测一枚硬币受卡尺精度（约 0.1 mm）限制，每次误差 ±0.05 mm。一次测 25 枚再除以 25，误差也被除以 25——每枚误差降到 0.002 mm。这是"多次测量取平均"的标准技巧。

自评：

(c) Sulin also wants to determine the mass of a $1 coin accurately using a spring kitchen scale. How can she do it?

ENSame idea: weigh a large number of coins (e.g., 100) on the kitchen scale, then divide by the count. A kitchen scale typically has precision 1 g — too coarse for one ~7 g coin, but weighing 100 coins (~700 g) lets you read to 1 g out of 700 (≈0.15% error per coin).

⏱ 20 秒后查看中文版

中文同样思路：一次称 100 枚硬币，再除以 100。厨房秤精度通常 1 g，一枚约 7 g 直接称误差大；100 枚约 700 g，1 g/700 ≈ 0.15% 误差大幅降低。

自评：

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