3.X

Chapter 3 — Let's Sum Up · Review Exercise · Maths Journal

🎯 LET'S SUM UP!

APPROXIMATION

To approximate is to find a suitable value for a quantity within a specified degree of accuracy.

Rules for rounding off numbers to specified place value / decimal places

Look for the digit in the specified place value / decimal place.
Consider the next digit to its right. If it is 5 or more, round up; if less than 5, round down.

Rounding off to place value
34 267 = 34 000 (correct to the nearest 1000)

Rounding off to decimal places
6.0784 = 6.08 (correct to 2 d.p.)

Rules to identify significant digits

All non-zero digits are significant.
All zeros between non-zero digits are significant.
For any integer, the zeros at the end may or may not be significant depending on how the number is approximated.
For any decimal, all zeros before the first non-zero digit are NOT significant.
For any decimal, zeros after the first non-zero digit ARE significant.

Rules for rounding off to specified s.f.

Look for the specified significant figure starting from the left.
Consider the next digit to its right. If 5 or more, round up; if less than 5, round down.

Example. 0.020 47 = 0.0205 (correct to 3 s.f.)

ESTIMATION

Estimation is the process of finding an approximate value of a number or a measurement.

Estimation strategies

(a) In numbers

Round off numbers
Use cluster values

(b) In measurements

Use benchmarks
Use decomposition-recomposition

Follow-through errors

Follow-through errors in numerical calculations might occur when intermediate values are rounded off to different degrees of accuracy.

If the final answer is required to be correct to 3 s.f., then all intermediate values should be given to at least 4 s.f.

Reasons for using approximation and estimation

Difficult to obtain exact values due to limitations of measuring instruments.
Unnecessary to use the exact value.
Too difficult to obtain the exact value.
Useful to check the reasonableness of the answers in computations.

🔁 REVIEW EXERCISE 3

本章 8 道综合复习题——结合 3.1（d.p.）、3.2（s.f.）、3.3（estimation）的所有规则。

Evaluate 3 1/2 × 7 1/3 + (−1 1/8) ÷ 2 1/4, giving your answer:

(a) in the exact value:

(b) correct to 2 decimal places:

(c) correct to 3 significant figures:
Three items cost $16.95, $23.40, $5.15.

(a) Estimate the total by rounding each to 1 s.f.: $

(b) Mr Cai has only $50 in his wallet. Does he have enough?

Explain:

EN1-s.f. estimates: $20 + $20 + $5 = $45. The first two rounded UP, the third was already a 1-s.f. value rounded normally. With the first two rounded UP, the actual total < $20+$20+$5+(adjustments) — but the estimate $45 < $50. We need to be careful: the first two were rounded UP (so actual is less for those two), the third $5.15 → $5 was rounded DOWN by $0.15. Still, |adjustment| is small. Actual = $45.50 < $50 → enough.

⏱ 20 秒后查看中文版

中文1 s.f. 估算：$20 + $20 + $5 = $45 < $50。前两个都向上取了整，第三个 $5.15 → $5 向下取了 $0.15，但调整很小。实际 = $45.50 < $50 → 够用。

自评：
A car park has 6 identical parking lots arranged in 3 columns × 2 rows. A car of 4.5 m × 1.7 m fits in one lot.

(a) Each lot is approximately m by m

(b) The area of the car park = m²
Edmund uses two different weighing scales. Scale A displays his mass as 53.0 kg; Scale B displays his mass as 53.00 kg. Do the two scales measure to the same degree of accuracy? If not, which is more accurate?

ENNo — they measure to different degrees. Scale A's 53.0 kg has 3 s.f. (precision 0.1 kg, range [52.95, 53.05) kg). Scale B's 53.00 kg has 4 s.f. (precision 0.01 kg, range [52.995, 53.005) kg). Scale B is more accurate — it pins the true mass down to a 10× narrower window.

⏱ 20 秒后查看中文版

中文不同。A 的 53.0 kg 是 3 s.f.（精度 0.1 kg）；B 的 53.00 kg 是 4 s.f.（精度 0.01 kg）。B 更精确，把真值锁在比 A 窄 10 倍的范围内。

自评：
The radius of a circle is 18 cm, correct to the nearest cm. The actual radius is a value with one decimal place. Find the largest possible error when we calculate the area using the rounded value of 18 cm. Give your answer correct to 3 s.f.

Largest possible error ≈ cm²
Haoquan and his parents plan to visit Singapore for 4 days, 3 nights from Beijing.
- 3 air tickets, round trip BEIJING ↔ SINGAPORE: SGD $562 each.
- Accommodation (1 room for 3 nights): 4 Apr $198, 5 Apr $189, 6 Apr $209.
(a) Estimate the total cost in SGD (rounded to the nearest $100): $

(b) Convert to Renminbi (RMB). Given 1 SGD = 4.89 RMB, estimate total RMB to the nearest 1000 RMB: RMB
A high-sensitivity balance measures masses to 0.0001 g.

(a) Mass of object A is 40.0005 g. Round to 3 s.f.: g

(b) Mass of object B is 2.01 g, correct to 3 s.f. Find the smallest and greatest possible actual mass.

Smallest = g Greatest = g

💡 思考后展开解释

3 s.f. 的 2.01 g 表示真值在 [2.005, 2.015) g。最小 = 2.005；最大就看测量精度——若用 4 s.f. 表示是 2.0149（因为 2.0150 会进位到 2.015 → 3 s.f. = 2.02，不再是 2.01）。所以 greatest < 2.015 → 实际取 2.0149 g（4 dp）或近似 2.014 g（3 dp）。
The sum of 3 consecutive even numbers is estimated to be 200. If 200 is rounded off to 1 s.f., what are:

(a) the smallest possible values of the 3 numbers:

(b) the largest possible values of the 3 numbers:

💡 思考后展开推导

1 s.f. 的 200 表示真值范围 [150, 250)。设 3 个连续偶数为 n, n+2, n+4，和为 3n+6。
3n + 6 ∈ [150, 250) → n ∈ [48, 81.33)，且 n 是偶数。
最小：n = 48 → 48, 50, 52（和 150）
最大：n = 80 → 80, 82, 84（和 246）

📖 MATHS JOURNAL

用自己的话记录学到的内容——把学习内化为表达。

1 Describe some real-life situations in which approximate values are more relevant than exact values. Explain why you think they are.

自评：

2 Describe two situations where estimation is useful in your daily life.

自评：