2.5

Operations on Real Numbers

AFractions

We have learnt equivalent fractions such as 3/4 = 6/8 = 12/16.

3/4

6/8

12/16

We can see that 6/8 = (3 × 2)/(4 × 2) = 3/4 and 12/16 = (3 × 4)/(4 × 4) = 3/4. The fractions 3/4, 6/8 and 12/16 are equivalent. The factor of 2 (or 4) is a common factor.

In the fraction 3/4, its numerator and denominator have no common factors except 1. This fraction is said to be in its simplest form or in its lowest terms.

We have also learnt the four operations on positive proper fractions, improper fractions and mixed numbers. In this section, we shall extend the four operations to involve all the rational numbers in the form a/b, where a and b are integers and b ≠ 0.

Addition and Subtraction of Fractions

The rules of addition and subtraction on integers hold for fractions.

📖 WORKED EXAMPLE 14

Without using a calculator, find the value of 7/8 − (−2 3/4) + (−1/3).

Solution

LCM of 8, 4 and 3 = 24.

7/8 − (−2 3/4) + (−1/3) = 7/8 + 11/4 − 1/3Convert mixed → improper, simplify signs.
= 21/24 + 66/24 − 8/24LCM of 3, 4 and 8 = 24.
= (21 + 66 − 8) / 24
= 79/24
= 3 7/24

✏️ TRY IT YOURSELF 14

Without using a calculator, find the value of 1 1/4 + (−8/3) − (−5/9).

Multiplication and Division of Fractions

We shall extend what we have learnt about multiplication and division of a fraction by a whole number or a proper fraction to involve mixed numbers, negative fractions and negative mixed numbers. Dividing a fraction by another fraction is equivalent to multiplying the first fraction by the reciprocal of the second fraction.

📖 WORKED EXAMPLE 15

Without using a calculator, find the value of

(a) −2 2/5 × 3/4, (b) 1 7/8 ÷ 1 2/7.

Solution

(a)

−2 2/5 × 3/4 = (−12/5) × (3/4)Change mixed → improper.
= (−³⁄₅) × (3/₁) · (cancel 4)Divide 12 and 4 by HCF = 4.
= −9/5 = −1 4/5

(b)

1 7/8 ÷ 1 2/7 = 15/8 ÷ 9/7Change mixed → improper.
= 15/8 × 7/9Multiply by reciprocal of 9/7.
= ⁵⁄₈ × 7/₃ · (cancel 3)HCF(15, 9) = 3.
= 35/24 = 1 11/24

✏️ TRY IT YOURSELF 15

Without using a calculator, find the value of

(a) −6 2/3 × 2/5 = (b) 1 5/9 ÷ −2 1/3 =

📖 WORKED EXAMPLE 16

Without using a calculator, find the value of (−2/3) ÷ [1/2 − (−1/3)].

Solution

(−2/3) ÷ [1/2 − (−1/3)] = (−2/3) ÷ (1/2 + 1/3)Simplify inside the brackets first.
= (−2/3) ÷ (3/6 + 2/6)LCM of 2 and 3 = 6.
= (−2/3) ÷ (5/6)
= (−2/3) × (6/5)Multiply by the reciprocal of 5/6.
= −4/5

✏️ TRY IT YOURSELF 16

Without using a calculator, find the value of (−3/4) ÷ [(−1/5) + (−1/3)].

📖 WORKED EXAMPLE 17

In a conference, 1/4 of the participants are under the age of 15 and 1/5 of them are above 20 years old. Of those above 20 years old, 4/7 of them are women.

Find the fraction of the participants who are

(a) between 15 and 20 years old, (b) men above 20 years old.

Solution

(a) Required fraction = 1 − 1/4 − 1/5

= 20/20 − 5/20 − 4/20LCM of 4, 5 = 20.
= 11/20

11/20 of the participants are between 15 and 20 years old.

(b) Required fraction = 1/5 × (1 − 4/7)

= 1/5 × 3/7Men = 1 − women's fraction = 3/7.
= 3/35

3/35 of the participants are men above 20 years old.

✏️ TRY IT YOURSELF 17

In a playground, 3/8 of the children are Chinese, 2/5 of them are Indians and the rest are Malays. Of the Chinese children, 2/9 of them are boys. Find the fraction of the children who are

(a) Malays = (b) Chinese girls =

BDecimals

Addition and Subtraction of Decimals

Here we shall extend what we have learnt about addition and subtraction of decimals to involve negative decimals.

📖 WORKED EXAMPLE 18

Evaluate each of the following without using a calculator.

(a) 13.65 + 5.938 (b) 47.3 + (−8.415) (c) 15.2 − (−4.33)

Solution

(a) Line up the decimal points. A zero is placed behind '5' so that the two numbers have the same number of decimal places. Add numbers down the columns, working from right to left.

  13.650
+  5.938
  ─────
  19.588

∴ 13.65 + 5.938 = 19.588

(b) 47.3 + (−8.415) = 47.3 − 8.415. Two zeros are placed behind '3' so that the two numbers have the same number of decimal places.

  47.300
−  8.415
  ─────
  38.885

∴ 47.3 + (−8.415) = 38.885

(c) 15.2 − (−4.33) = 15.2 + 4.33. The rules of addition and subtraction on integers hold for all rational numbers.

  15.20
+  4.33
  ─────
  19.53

∴ 15.2 − (−4.33) = 19.53

✏️ TRY IT YOURSELF 18

Evaluate each of the following without using a calculator.

(a) 21.357 + 9.24 = (b) 16.054 − 8.39 =

Multiplication and Division of Decimals

We have learnt the multiplication and division of decimals by whole numbers. Let us look at how we can multiply and divide a decimal by another decimal.

📖 WORKED EXAMPLE 19

Find the value of each of the following without using a calculator.

(a) 3.2 × 4.7 (b) 1.348 × (−0.02)

Solution

(a) Method 1. Convert to fractions:

3.2 × 4.7 = (32/10) × (47/10) = (32 × 47)/100 = 1504/100 = 15.04

Method 2 (vertical). 3.2 has 1 d.p., 4.7 has 1 d.p. → product has 1 + 1 = 2 d.p. Compute 32 × 47 = 1504, then put the decimal point 2 places from the right.

    3.2   ←  1 d.p.
×   4.7   ←  1 d.p.
  ─────
    224   ←  32 × 7 = 224
   1280   ←  32 × 40 = 1280
  ─────
  15.04   ←  2 d.p.

(b) 1.348 × (−0.02): Apply the sign rule (positive × negative = negative).

1.348 (3 d.p.) × 0.02 (2 d.p.) → product has 3 + 2 = 5 d.p.; 1348 × 2 = 2696 → 0.02696. With the sign: −0.02696.

✏️ TRY IT YOURSELF 19

Find the value of each of the following without using a calculator.

(a) 3.47 × 1.2 = (b) 2.93 × (−0.07) =

📖 WORKED EXAMPLE 20

Find the value of each of the following without using a calculator.

(a) (−3.27) ÷ 0.3 (b) (−0.9628) ÷ (−0.004)

Solution

(a) Multiply numerator and denominator by 10 to make the divisor a whole number:

(−3.27)/0.3 = (−3.27 × 10)/(0.3 × 10) = −32.7/3

Long-divide 32.7 by 3 → 10.9. Apply sign: −10.9.

(b) Multiply numerator and denominator by 1000 to make the divisor a whole number:

(−0.9628)/(−0.004) = (−0.9628 × 1000)/(−0.004 × 1000) = −962.8/−4 = 962.8/4

Long-divide 962.8 by 4 → 240.7. So 240.7.

✏️ TRY IT YOURSELF 20

Find the value of each of the following without using a calculator.

(a) (−0.266) ÷ 0.07 = (b) (−0.148) ÷ (−0.005) =

CUse of Calculators

Let us look at how we can perform the four operations on real numbers using a calculator.

📖 WORKED EXAMPLE 21

Use a calculator to evaluate each of the following.

(a) (−8) × (−3) ÷ (−6)²
(b) (−5 1/4) × (2 1/2) ÷ (2/3)³
(c) √(3.85 + 1.5 × 3) / ∛(4 1/3 − 5/8)

Solution

Expression	Key sequence (Casio-style)	Result
(−8) × (−3) ÷ (−6)²	(−) 8 × (−) 3 ÷ ( (−) 6 ) x² =	2/3 (or 0.666 666 666 7)
(−5 1/4) × (2 1/2) ÷ (2/3)³	(−) 5 □ 1 □ 4 × 2 □ 1 □ 2 ÷ ( 2 □ 3 ) x³ =	−2835/64 = −44 19/64 ≈ −44.296 875
√(3.85 + 1.5 × 3) / ∛(4 1/3 − 5/8)	√ 3.85 + 1.5 × 3 ▶ ÷ ∛ 4 □ 1 □ 3 − 5 □ 8 =	≈ 1.866 882 492

注：不同型号计算器按键略有差异；按键流程仅作示意。考试用 Casio fx-991ES 或同类型即可。

✏️ TRY IT YOURSELF 21

Use a calculator to evaluate each of the following.

(a) 2.33 × [5.75 − (−1.68)²] =

(b) (5/8) × (−1 1/2)³ ÷ (−2 2/3) =

📝 PRACTICE EXERCISE 2.5

BASIC MASTERY

Evaluate the following, leaving your answer in the simplest form.

(a) 1/5 + 2/7 = (b) (−3/4) + (−1/2) =

(c) 2 3/4 − 3 11/12 = (d) −1 − (−5/6) + (−6/5) =

(e) (−24/25) × (45/16) = (f) (−7/11)² − 7/11 =

(g) 9/32 × 1 3/5 × 3 1/3 = (h) (−4 3/8) ÷ (5 1/4) =

(i) (−1 3/5) ÷ (−1 1/3) = (j) 3 + 2 1/2 × (−1 1/4) =

Use a calculator to check your answer.
Find the value of each of the following.

(a) 32.7 + 9.84 = (b) 25.02 − 9.765 =

(c) 11.47 × 2.8 = (d) (−3.45) × 1.06 =

(e) 10.68 ÷ 0.6 = (f) (−0.1935) ÷ 0.009 =

Use a calculator to check your answer.

INTERMEDIATE

Find the value of each of the following, leaving your answers in the simplest form.

(a) (−1/2 + 1/3 + 1/6) × (−2/5)² =

(b) 1 5/7 × (1/4 − 1/9) ÷ (−1 1/4) =

(c) (−2 2/3) × (−9/8) + (−7/4) × 2/21 =

(d) [−1 1/4 − (−1/6)] ÷ [1/3 + (−1/2)]³ =
Use a calculator to evaluate each of the following.

(a) √(37² − 35²) =

(b) ∛3375 − (−4 1/3)² =

(c) [−5 3/4 − (−6 1/2)]³ =

(d) √(3.21 + 0.4) / (4.7 − 2.2) =
Of the Secondary 1 students in a school, 1/5 of them live in Clementi and 4/9 of them live in Queenstown. Find the fraction of the students who live in neither Clementi nor Queenstown.

Fraction =
Jim spent 5/8 of his savings to buy a pair of sports shoes. He donated 1/3 of the remaining savings to the Red Cross Society. Find the fraction of his savings that was donated.

Donated fraction =
Marcus has 0.5 kg of flour. He wants to bake a cake. According to a recipe, he needs 4 1/4 cups of flour with 1/8 kg of flour in each cup. Does Marcus have enough flour to bake the cake? If not, how much more flour does he need?

(a) Does he have enough?

(b) How much more flour does he need? kg

ADVANCED

[OPEN] Can the difference between any two decimals be

(a) a terminating decimal?

Example (if YES):

ENYES. e.g., 0.7 − 0.2 = 0.5 (terminating). Two terminating decimals always subtract to a terminating decimal.

⏱ 15 秒后查看中文版

中文是。例：0.7 − 0.2 = 0.5（终止小数）。两个终止小数之差仍是终止小数。

自评：

(b) a recurring decimal?

ENYES. e.g., 0.5 − 0.16• = 0.33• (i.e., 0.5 − 1/6 = 1/3).

⏱ 15 秒后查看中文版

中文是。例：0.5 − 0.16̇ = 0.333… = 0.3̇，即 1/2 − 1/6 = 1/3。

自评：

(c) an integer?

ENYES. e.g., 1.5 − 0.5 = 1; or 3.7 − 3.7 = 0.

⏱ 15 秒后查看中文版

中文是。例：1.5 − 0.5 = 1，或 3.7 − 3.7 = 0。

自评：

(d) an irrational number?

💡 思考后再展开（为什么？）

不能（在"decimal"指有限或循环小数这种有理数意义下）。两个有理数之差永远是有理数，所以不可能得到无理数。
注：如果允许"decimal"包含 π = 3.141 592… 这种无理小数表示，那答案变成"可以"——例如 π − 0.1 = 无理数。这取决于教材如何定义"小数"。
Alex, Boris and Carol invest in a business together. Based on the amount each person has invested, Alex owns 1/3 of the business and Boris owns 2/7 of it.

(a) What fraction of the business does Carol own?

(b) Find the total investment in the business if Alex has invested $6000 more than Boris. $
There are two identical jugs, A and B. Jug A is 3/7 full of water and Jug B is 8/11 full. What fraction of the capacity of a jug of water should be poured from B to A so that they both have the same volume of water?

Fraction =
[OPEN] Consider the rational numbers 3/4 and 9/11.

(a) Which number is larger?

(b) Find two rational numbers between them.

EN3/4 = 33/44 = 0.75; 9/11 = 36/44 ≈ 0.818. Any rationals in (0.75, 0.818) work. Examples: 7/9 ≈ 0.778, 4/5 = 0.8, 69/88 (the midpoint, also rational).

⏱ 20 秒后查看中文版

中文3/4 = 0.75；9/11 ≈ 0.818。在 (0.75, 0.818) 内的任何有理数都行。例：7/9 ≈ 0.778、4/5 = 0.8、中点 (3/4 + 9/11)/2 = 69/88。

自评：

(c) Compare your answers in (b) with those of your classmates. (No need to submit.)

(d) How many rational numbers lie between 3/4 and 9/11? Give a reason.

ENInfinitely many. Between any two distinct rationals a < b you can always take the midpoint (a + b)/2 — that's a new rational strictly between them. Then take the midpoint of (a, (a+b)/2), and so on. This process never ends, so there are infinitely many rationals between 3/4 and 9/11.

⏱ 30 秒后查看中文版

中文无穷多个。两个不同的有理数 a 和 b 之间，永远可以取中点 (a+b)/2，这个中点仍是有理数，且严格介于 a 和 b 之间。然后再取 a 和这个中点的中点……这样无限地"插入"下去，每次都得到一个新的有理数，所以介于 3/4 和 9/11 之间有无穷多个有理数。

自评：

🎯 Chapter 2 SUM UP — Classification of Numbers

本章我们建立了完整的"数的世界"：

实数（Real Numbers） 包含一切——可在数轴上表示的数
├── 有理数（Rational Numbers）：可写成 a/b，等价地说是"终止小数 OR 循环小数"
│ ├── 整数（Integers）：…, −3, −2, −1, 0, 1, 2, 3, …
│ │ ├── 自然数（Whole Numbers）：0, 1, 2, 3, …
│ │ └── 负整数：−1, −2, −3, …
│ └── 分数：½, −3⁄8, 0.4, …
└── 无理数（Irrational Numbers）：不终止不循环——例如 π, √2, 0.101 001 000 1…

运算规则：

同号得正、异号得负（乘除）；
减一个数 = 加它的相反数；
运算顺序：括号 → 幂&根 → 乘&除 → 加&减；
整数运算规则同样适用于所有实数（分数、小数、负数）。

章末概念检查 · Concept Checkpoints

5 道封闭题，自动判分。做对这 5 道就算完成 Chapter 2 全章。