4.1
Introduction to Algebra
ABasic Notations in Algebra
We have learnt to use symbols such as △ and ▢ to represent numbers in arithmetic. In algebra, we may also use letters such as a, b, c, … to represent these numbers.
In arithmetic, we manipulate numbers which have definite values. However, in algebra, the letters may represent any possible numerical values.
Let us consider an example.
Mrs Soh wants to buy a watermelon and some mangoes at the supermarket. The price of a watermelon is $5 and the price of a mango is $2.
- If she buys 1 watermelon and 3 mangoes, the total price = $(5 + 2 × 3) = $11.
- If she buys 1 watermelon and 4 mangoes, the total price = $(5 + 2 × 4) = $13.
- If she buys 1 watermelon and $n$ mangoes, the total price = $(5 + 2 × $n$).
In the expression $5 + 2 \times n$ above, we use the letter $n$ to represent the number of mangoes. We call $n$ a variable and $5 + 2 \times n$ an algebraic expression.
In algebra, we need to follow certain conventions so that everyone understands what we want the notations in the expressions to represent. The operation symbols '+', '−', '×', '÷' and '=' have the same meanings in algebra as in arithmetic.
In the following table, we look at arithmetic expressions and algebraic expressions involving the four operations.
| Operation | Statement | Arithmetic expression | Statement | Algebraic expression |
|---|---|---|---|---|
| Addition |
|
5 + 7 or 7 + 5 |
|
$a + b$ or $b + a$ |
| Subtraction |
|
18 − 11 |
|
$a − b$ |
| Multiplication |
|
2 × 6 5 × 5 or 52 |
|
$x \times y$ or $xy$ $x \times x$ or $x^2$ |
| Division |
|
24 ÷ 3 or $\dfrac{24}{3}$ |
|
$x \div y$ or $\dfrac{x}{y}$ |
When we multiply two variables, $a$ and $b$, we can simply write the product as $ab$. When multiplying a known number and a variable, say 3 and $y$, we write it as $3y$ instead of $y3$. As for $1 \times y$ or $y \times 1$, it is written as $y$ instead of $1y$.
In algebra, the index notation is used in a similar way as in arithmetic. For example, in arithmetic, we write $7 \times 7 \times 7 = 7^3$ and $5 \times 5 \times 5 \times 5 = 5^4$.
In algebra, we write expressions in index notation as follows:
- $a \times a = a^2 \qquad$ read as $a$ squared or square of $a$
- $b \times b \times b = b^3 \qquad$ read as $b$ cubed or cube of $b$
- $c \times c \times c \times c = c^4 \qquad$ read as $c$ to the power of 4
Write down an algebraic expression for each of the following statements.
- (a) Subtract $n$ from the quotient of $m$ divided by $t$.
- (b) Add the product of $a$ and $b$ to the square of $c$.
- (c) Multiply 8 by the product of $a$ and $b$ squared.
- (d) Divide the sum of 2 and the cube of $c$ by $a$.
SOLUTION
(a) Quotient of $m$ divided by $t = m \div t = \dfrac{m}{t}$.
Subtract $n$ from the quotient of $m$ divided by $t$ = $\dfrac{m}{t} - n$.
(b) Product of $a$ and $b = a \times b = ab$. Square of $c = c^2$.
Add the product of $a$ and $b$ to the square of $c$ = $ab + c^2$.
(c) Product of $a$ and $b$ squared $= ab^2$.
Multiply 8 by the product of $a$ and $b$ squared $= 8 \times ab^2 = 8ab^2$.
📌 We can write $8 \times ab^2$ as $8ab^2$, omitting the multiplication sign.
(d) Cube of $c = c^3$. Sum of 2 and the cube of $c = 2 + c^3$.
Divide the sum of 2 and the cube of $c$ by $a = \dfrac{2 + c^3}{a}$.
Write down an algebraic expression for each of the following statements. Type your answer using ^ for powers (e.g., v^3) and / for fractions.
- (a) Multiply $a$ by the cube of $v$.
Answer: 检查 - (b) Subtract the quotient of $a$ divided by $b$ from the cube of $u$.
Answer: 检查 - (c) Divide the product of $x$ and $3y$ by the sum of $a$ and $b$ cubed.
Answer: 检查 - (d) Add the cube of $(c+d)$ to the square of $(a-b)$.
Answer: 检查
Let us learn how to translate real-world situations into algebraic expressions.
Kelly is $x$ years old. Her father is twice as old as her, and her mother is $y$ years younger than her father. Kelly's brother is half as old as her mother. Write down an expression, in terms of $x$ and $y$, for the age of Kelly's brother.
SOLUTION
- Age of Kelly: $x$
- Age of Kelly's father: $2x$
- Age of Kelly's mother: $2x - y$
- Age of Kelly's brother: $\dfrac{1}{2} \times (2x - y) = \dfrac{1}{2}(2x - y)$
∴ Kelly's brother is $\dfrac{1}{2}(2x - y)$ years old.
📌 We can also write the age of Kelly's brother as $\dfrac{2x - y}{2}$.
Penny is $m$ years old. Her brother is $n$ years older than her while her father is three times as old as her brother. Write down an expression, in terms of $m$ and $n$, for the age of Penny's father.
Answer (in expanded form): 检查
💡 思路提示
Brother = $m + n$; Father = $3 \times$ brother = $3(m + n) = 3m + 3n$.
A sales promoter gets a basic salary of $1500 a month. She will earn a commission of $100 for every TV set and $50 for every sound system she sells in a month.
- (a) Find her total salary if she sells 6 TV sets and 9 sound systems in a month.
- (b) Write down an expression for her total salary in terms of $x$ and $y$ if she sells $x$ TV sets and $y$ sound systems in a month.
SOLUTION
(a) Her total salary = \$(1500 + 100 × 6 + 50 × 9) = \$2550
(b) Her total salary = \$(1500 + 100 × $x$ + 50 × $y$) = \$$(1500 + 100x + 50y)$
An empty box has a mass of 600 g. It is used to contain some pens and books. Given that each pen has a mass of 3 g and each book has a mass of 150 g, find the total mass if the box contains
- (a) 20 pens and 3 books, mass = g 检查
- (b) $p$ pens and $b$ books, mass = g 检查
💡 思路提示
(a) 600 + 3(20) + 150(3) = 600 + 60 + 450 = 1110 g.
(b) 600 + 3$p$ + 150$b$ g.
BEvaluation of Algebraic Expressions and Formulae
Evaluating Algebraic Expressions
We can evaluate an algebraic expression when we know the value represented by each variable in the expression.
Let us consider the earlier example. The price of a watermelon is $5 and the price of a mango is $2. If Mrs Soh buys 1 watermelon and $n$ mangoes, the total price = \$$(5 + 2n)$.
then the total price = \$$[5 + 2(7)]$
= \$19
∴ Mrs Soh has to pay $19 when she buys 1 watermelon and 7 mangoes.
then the total price = \$$[5 + 2(10)]$
= \$25
∴ Mrs Soh has to pay $25 when she buys 1 watermelon and 10 mangoes.
Let us evaluate and compare different pairs of algebraic expressions by substituting different values into the variables.
Objective: To compare and verify if each pair of algebraic expressions is equivalent.
- Create a spreadsheet. Enter the headings in Row 1 and the values of $n$ in Column A.
- Key in
= 2 + A2in B2. Similarly, key in the rest as shown in the table for each cell from C2 to K2. - Copy B2 to K2 to the cells in row 3 to row 12.
| A | B | C | D | E | F | G | H | I | J | K | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | $n$ | 2 + $n$ | $n + n$ | $2n$ | $n^2$ | $2n^2$ | $(2n)^2$ | $(\tfrac{1}{2})(2n+1)$ | $n+1$ | $(\tfrac{1}{3})(n+1)$ | $\tfrac{n+1}{3}$ |
| 2 | −5 | = 2 + A2 | = A2+A2 | = 2*A2 | = (A2)^2 | = 2*A2^2 | = (2*A2)^2 | = (1/2)*(2A2+1) | = A2+1 | = (1/3)*(A2+1) | = (A2+1)/3 |
| 3 | −4 | ||||||||||
| 4 | −3 | ||||||||||
| 5 | −2 | ||||||||||
| 6 | −1 | ||||||||||
| 7 | 0 | ||||||||||
| 8 | 1 | ||||||||||
| 9 | 2 | ||||||||||
| 10 | 3 | ||||||||||
| 11 | 4 | ||||||||||
| 12 | 5 |
1. Compare the values of each pair of algebraic expressions. (Fill the column “Are they equivalent?”)
| Expression 1 | Expression 2 | Are they equivalent? |
|---|---|---|
| $2n$ | $2 + n$ | 检查 |
| $n + n$ | $2n$ | 检查 |
| $n^2$ | $2n$ | 检查 |
| $2n^2$ | $(2n)^2$ | 检查 |
| $\tfrac{1}{2}(2n+1)$ | $n + 1$ | 检查 |
| $\tfrac{1}{3}(n+1)$ | $\tfrac{n+1}{3}$ | 检查 |
2. For $2n$ and $n^2$, what do you observe when the value of $n$ changes from negative numbers to positive numbers?
3. (a) What does $(2n)^2$ mean?
(b) What is the relationship between $2n^2$ and $(2n)^2$?
💡 EQUIVALENCE — 等价是什么意思？
Although $2n$ and $n^2$ have the same value when we substitute $n = 0$ and $n = 2$ into each of the expressions, the two expressions do not have the same value when we substitute other values of $n$ such as $n = -1, 3, \tfrac{1}{2}$, etc. Hence we say that these expressions are not equivalent. 两个表达式只有对所有 $n$ 取同样的值，才叫等价。
📌 In a spreadsheet, ^ stands for power and * for multiplication. For example, n^2 stands for $n^2$ and (1/3)*(n+1) stands for $\tfrac{1}{3}(n+1)$.
When $x = 4$ and $y = -1$, find the value of
- (a) $5x + 7y$,
- (b) $x^2 y - 3xy + 5$.
SOLUTION
(a) When $x = 4$ and $y = -1$,
$5x + 7y = 5(4) + 7(-1) = 20 - 7 = \mathbf{13}$.
(b) When $x = 4$ and $y = -1$,
$x^2 y - 3xy + 5 = (4)^2(-1) - 3(4)(-1) + 5 = -16 + 12 + 5 = \mathbf{1}$.
When $x = 5$ and $y = 3$, find the value of
- (a) $2x - 4y = $ 检查
- (b) $xy^2 + 2xy + 7 = $ 检查
💡 思路提示
(a) $2(5) - 4(3) = 10 - 12 = -2$.
(b) $5(3)^2 + 2(5)(3) + 7 = 45 + 30 + 7 = 82$.
When $p = \tfrac{1}{2}$, $q = 5$ and $r = -2$, evaluate the following expressions.
- (a) $4p(q^2 - 3r)$
- (b) $\dfrac{q - pr}{pqr}$
SOLUTION
(a) $4p(q^2 - 3r) = 4(\tfrac{1}{2})[5^2 - 3(-2)] = 4(\tfrac{1}{2})(31) = \mathbf{62}$.
(b) $\dfrac{q - pr}{pqr} = \dfrac{5 - (\tfrac{1}{2})(-2)}{(\tfrac{1}{2})(5)(-2)} = \dfrac{5 + 1}{-5} = -\dfrac{6}{5} = \mathbf{-1\tfrac{1}{5}}$.
When $x = \tfrac{3}{4}$, $y = 6$ and $z = -5$, evaluate the following expressions.
- (a) $z(8x + yz)$.
- (b) $\dfrac{10xy - z + 2}{xy^2 - 3}$.
Formulae
We have learnt that the area of a rectangle is given by area = length × breadth.
If we let $l$ and $b$ represent the length and the breadth of the rectangle respectively in cm, and let $A$ represent the area in cm², then the relation above can be expressed as
$A = lb$.
A formula is an equality relating two or more variables. When the values of $l$ and $b$ are known, we can find the value of $A$ in the formula by substitution.
📌 The plural form of formula is formulae or formulas.
The area, $A$ cm², of a rectangle with length $l$ cm and breadth $b$ cm is given by the formula $A = lb$.
- (a) Find the area of a rectangle with length 9 cm and breadth 6 cm.
- (b)(i) Four quarters, each of radius $r$ cm, are shaded at the four corners of a rectangle. Express the area of the unshaded figure in terms of $l$, $b$, $r$ and $\pi$.
- (b)(ii) Given $l = 9$, $b = 6$ and $r = 2$, find the area of the unshaded figure in terms of $\pi$.
SOLUTION
(a) Substituting $l = 9$ and $b = 6$ into the formula, $A = lb = 9 \times 6 = \mathbf{54}$. The area of the rectangle is $\mathbf{54}$ cm².
(b)(i) The area of the four quarters is the area of the circle of radius $r$ cm, i.e., $\pi r^2$ cm². The area of the unshaded figure $= lb - \pi r^2$.
(b)(ii) Substituting $l = 9$, $b = 6$ and $r = 2$, the area of the unshaded figure $= 9 \times 6 - \pi \times 2^2 = (54 - 4\pi)$ cm².
📌 SPOTLIGHT: When finding the answer in terms of π, we do not evaluate π using a calculator or substitute any approximated values like 3.142 or $\tfrac{22}{7}$. The final answer is given with the notation π.
📌 SPOTLIGHT: $A = lb$ is a formula. $lb - \pi r^2$ is an algebraic expression.
- (a) The volume, $V$ cm³, of a cuboid with length $l$ cm, breadth $b$ cm and height $h$ cm is given by the formula $V = lbh$. Find the volume of a cuboid with length 12 cm, breadth 7 cm and height 10 cm.
Volume = cm³ 检查 - (b)(i) The perimeter, $P$ m, of a square with length $l$ m is given by the formula $P = 4l$. Find the perimeter of a square with sides 9 m each.
Perimeter = m 检查 - (b)(ii) Four quarters, each of radius $r$ m, are shaded at the four corners of a square (side $l$ m). Express the perimeter of the unshaded figure in terms of $l$, $r$ and $\pi$.
Perimeter = m 检查💡 思路提示
每条边的可见长度 = $l - 2r$；四条边的总长 = $4(l - 2r) = 4l - 8r$。再加上四个四分之一圆弧 = 一整个完整圆周 = $2\pi r$。合计：$4l - 8r + 2\pi r$。
- (b)(iii) Given $l = 9$ and $r = 3$, find the perimeter of the unshaded figure in terms of π.
Perimeter = m 检查💡 计算
$4(9) - 8(3) + 2\pi(3) = 36 - 24 + 6\pi = 12 + 6\pi$ m.
The sum, $S$, of the first $n$ positive integers is given by $S = \tfrac{1}{2}n(n+1)$. Find
- (a) $1 + 2 + 3 + \ldots + 100$,
- (b) $51 + 52 + \ldots + 100$.
SOLUTION
(a) Substituting $n = 100$ into the formula, $S = \tfrac{1}{2} \times 100 \times (100+1) = \mathbf{5050}$.
∴ $1 + 2 + 3 + \ldots + 100 = 5050$.
(b) We can find the sum of the first 50 integers. Then the difference between the sum of the first 100 integers and the sum of the first 50 integers is what we want to find.
Substituting $n = 50$: $S = \tfrac{1}{2} \times 50 \times (50+1) = 1275$. ∴ $1 + 2 + 3 + \ldots + 50 = 1275$.
Hence $51 + 52 + \ldots + 100 = (1 + 2 + \ldots + 100) - (1 + 2 + \ldots + 50) = 5050 - 1275 = \mathbf{3775}$.
The sum, $T$, of the first $n$ positive even integers is given by the formula $T = n(n+1)$.
- (a)(i) Find the sum of the first 3 positive even integers.
$T = $ 检查💡 验证
$n=3$: $3(3+1) = 12$. Check: $2 + 4 + 6 = 12$ ✓
- (a)(ii) Find the sum of the first 50 positive even integers.
$T = $ 检查💡 验证
$n=50$: $50(51) = 2550$.
- (b) Using the results in (a)(ii) and Worked Example 7(a), find the sum of $1 + 3 + 5 + 7 + \ldots + 99$.
Sum = 检查💡 思路
$1+2+3+\ldots+100 = 5050$（即 WE7(a)）。其中偶数 $2+4+\ldots+100 = 2550$（即 (a)(ii)）。所以奇数和 $= 5050 - 2550 = 2500$。事实上，前 $n$ 个奇数之和 $= n^2 = 50^2 = 2500$ ✓
PPractice Exercise 4.1
- Q1. Mr Lin is 5 cm taller than his wife. Find Mr Lin's height if the height of his wife is
(a) 160 cm. → cm 检查
(b) $h$ cm. → cm 检查 - Q2. Find the number of days in
(a) 5 weeks → days 检查
(b) $n$ weeks → days 检查 - Q3. John's score is three quarters of Mary's score. Find John's score if Mary's score is
(a) 76 → 检查
(b) $s$ → 检查 - Q4. Simplify the following.
(a) $6h \times 3k = $ 检查
(b) $9m \div 9n = $ 检查
(c) $b \times b \times 4 = $ 检查
(d) $e \div f \times g = $ 检查
(e) $3p \times p \times 5p = $ 检查
(f) $4q \times 5r \times q = $ 检查
(g) $s \div 6 + 1 \times t = $ 检查
(h) $u + 6v \div 9w = $ 检查 - Q5. Write down an algebraic expression for each of the following statements.
(a) Subtract $3m$ from the quotient of $n$ divided by $p$. → 检查
(b) Divide the sum of $2t$ and $3u$ by $v$. → 检查
(c) The sum of twice $x$ and the cube of $y$. → 检查
(d) Divide the negative of $y$ by the sum of the square of $x$ and the square of $a$. → 检查 - Q6. The time taken by an aeroplane to travel from City A to City B is 20 minutes more than $\tfrac{1}{8}$ of the time taken by a train. Find the time taken by the aeroplane if the time taken by the train is
(a) 12 hours → minutes 检查
(b) $t$ hours → hours 检查💡 注意单位
(a) 12 hours = 720 min; plane = $720/8 + 20 = 110$ min.
(b) Answer in hours: 20 minutes $= \tfrac{1}{3}$ hour. Plane time $= \tfrac{t}{8} + \tfrac{1}{3}$ hours. - Q7. Find the value of $25 - 4y$ when
(a) $y = 0$ → 检查
(b) $y = \tfrac{5}{2}$ → 检查 - Q8. Find the value of $(2m + n)(m - n + 1)$ when
(a) $m = 5$ and $n = 3$ → 检查
(b) $m = 11$ and $n = -6$ → 检查💡 检查 (b)
$(2 \cdot 11 + (-6))(11 - (-6) + 1) = (22-6)(11+6+1) = 16 \cdot 18 = 288$.
- Q9. Find the value of $y$ in each of the following formulae.
(a) $y = m(3 - 2n)$; given $m = 4$ and $n = 6$ → 检查
(b) $y = 4d(d + 5)$; given $d = -8$ → 检查
(c) $y = \dfrac{p^2 - q^2}{2}$; given $p = -7$ and $q = 3$ → 检查
(d) $y = \dfrac{r}{s} + \dfrac{s}{r}$; given $r = 6$ and $s = -3$ → 检查
(e) $y = \dfrac{1}{e^3 - 1}$; given $e = 2$ → 检查
(f) $y = \dfrac{3f + 4}{5f - 2}$; given $f = -1$ → 检查
(g) $y = uv^2$; given $u = -2$ and $v = -9$ → 检查
(h) $y = \dfrac{(tx)^2}{3t - 5x}$; given $t = 4$ and $x = 2$ → 检查
- Q10. The mass of Book A is 759 g and the mass of Book B is 400 g. Find the total mass of
(a) 4 copies of Book A and 3 copies of Book B → g 检查
(b) $p$ copies of Book A and $q$ copies of Book B → g 检查 - Q11. Mr Tan wants to go for a movie with his family. He has 2 free tickets. Find the amount he has to pay for the tickets if
(a) there are 5 family members and each ticket costs $11 → $ 检查
(b) there are $n$ family members and each ticket costs $p$ → $ 检查 - Q12. Find the value of $b^2 - 4ac$ when
(a) $a = 1$, $b = -5$ and $c = 3$ → 检查
(b) $a = -2$, $b = 7$ and $c = \tfrac{3}{4}$ → 检查 - Q13. Find the value of $p^2 q - (2q)^2$ when
(a) $p = 8$ and $q = -3$ → 检查
(b) $p = -\tfrac{9}{2}$ and $q = 4$ → 检查 - Q14. If $z = k t^{n-1}$, find the value of $z$ when $k = -2$, $t = 5$ and $n = 4$. → $z = $ 检查
- Q15. The daily wage of a worker is given by the expression $15t$, where $t$ is the number of working hours.
(a) Find the daily wage of the worker when he works for 8 hours. → $ 检查
(b) What do you think the number 15 in the expression stands for?
💡 提示
15 是每小时的工资（hourly wage / pay rate per hour），单位是 $/hour。
- Q16. The three sides of a triangle are $x$ cm, $y$ cm and $z$ cm respectively. The perimeter of the triangle is $P$ cm.
(a) Write a formula connecting $x$, $y$, $z$ and $P$. → 检查
(b) Given that $x = 15$, $y = 13$ and $z = 8$, find $P$. → $P = $ cm 检查 - Q17. (a) The perimeter of a square is $L$ cm. Express the length of a side of the square in terms of $L$. → cm 检查
(b) The length of a rectangle is $x$ cm and its perimeter is 20 cm. Express the breadth of the rectangle in terms of $x$. → cm 检查💡 思路
(a) 4 边相等：side = $L \div 4$。
(b) $P = 2(l + b) = 20 \Rightarrow l + b = 10 \Rightarrow b = 10 - x$. - Q18. A car salesman gets a basic salary of $2000 a month. For every car sold, he gets a commission of $800. Let $n$ be the number of cars that he sells in a month.
(a) Find his total monthly salary when $n = 18$. → $ 检查
(b) Express his total monthly salary in terms of $n$. → $ 检查
- Q19. Given the formula $S = \tfrac{1}{2} n(n + 1)$, where $S$ is the sum of the first $n$ positive numbers, find the value of
(a) $2 \times (1 + 2 + \ldots + 100)$ → 检查
(b) $1 + 3 + 5 + 7 + \ldots + 199$ → 检查💡 思路
(a) 直接代入公式 $S(100) = \tfrac{1}{2}(100)(101) = 5050$，所以 $2 \times 5050 = 10100$。
(b) 两种解法都行：
方法 1（奇数和 = 前 199 整数之和 − 偶数和）：
$1 + 3 + \ldots + 199 = (1 + 2 + 3 + \ldots + 199) - (2 + 4 + \ldots + 198)$
$= \tfrac{199 \cdot 200}{2} - 2 \cdot (1 + 2 + \ldots + 99)$
$= 19900 - 2 \cdot \tfrac{99 \cdot 100}{2} = 19900 - 9900 = \mathbf{10000}$。
方法 2（前 $n$ 个奇数之和 $= n^2$）：奇数 $1, 3, \ldots, 199$ 共 100 个，所以和 $= 100^2 = \mathbf{10000}$。 - Q20. A grocer has $p$ cartons of oranges. Each carton contains $q$ oranges and $r$ of them are rotten. Express, in terms of $p$, $q$ and $r$,
(a) the total number of oranges in all the cartons. → 检查
(b) the total number of good oranges. → 检查 - Q21. Mariani is $x$ years old. Suhaini, her brother, is 19 years older than her. Their mother is 3 times as old as Mariani. Their father is twice as old as Suhaini. Write down the expressions, in terms of $x$, for
(a) Suhaini's age. → 检查
(b) their mother's age. → 检查
(c) their father's age. → 检查 - Q22. (OPEN) Describe a real-life situation that could be represented by the expression $2n + 4m$.
💡 参考答案
开放题。示例：
• 一根铅笔 2 元，一支笔 4 元；买 $n$ 根铅笔和 $m$ 支笔，共花 $(2n + 4m)$ 元。
• 一周中爸爸做 $n$ 天家务（每次 2 小时）+ 妈妈做 $m$ 天家务（每次 4 小时），总时长 $= (2n + 4m)$ 小时。
只要场景合理、单位一致即可。 - Q23. The formula to convert a temperature of $H$ °F (Fahrenheit) to $S$ °C (Celsius) is given by $S = \tfrac{5}{9}(H - 32)$.
(a) The boiling point of water is 212 °F. Express this in °C. → °C 检查
(b) The temperature in New York at a certain time was 23 °F. Express this temperature in °C. → °C 检查 - Q24. The price $P$ for a birthday cake of radius $r$ cm and height $h$ cm is given by the formula $P = \tfrac{1}{25} r^2 h$.
(a) Find the price for a cake of radius 10 cm and height 8 cm. → $ 检查
(b) If the height of the cake in (a) is increased to 10 cm, what is the increase in price? → $ 检查 - Q25. When a box contains $x$ identical plates and $y$ dozens of spoons, its total mass is given by $(2800 + 200x + 300y)$ g. Find the mass of
(a) the empty box. → g 检查
(b) a plate. → g 检查
(c) a spoon. → g 检查💡 思路 (c)
$y$ 是 dozens (打)，1 打 = 12 个。一打 spoon 重 300 g，所以每只 spoon 重 $300 \div 12 = 25$ g。
- Q26. A piece of wire, 12 cm long, is bent to form a rectangle.
(a) By letting $x$ cm be the length of one side, express the area of the rectangle, $A$ cm², in terms of $x$. → 检查
(b)(i) Find the value of $x$ for which $A$ is the greatest, given that $x$ is an integer. → $x = $ 检查
(b)(ii) What type of rectangle is this when $A$ is maximum? → 检查💡 思路
(a) Perimeter $= 12 \Rightarrow 2(x + y) = 12 \Rightarrow y = 6 - x$；面积 $A = xy = x(6-x)$。
(b)(i) Integer $x$ 取值 1..5，对应 $A$ 为 5, 8, 9, 8, 5 → $x = 3$ 时 $A = 9$ 最大。
(b)(ii) $x = 3, y = 3$ → 正方形 / square。 - Q27. (a) When two resistors of resistance $a$ ohms and $b$ ohms are connected to two points $X$ and $Y$ by using different wires in a circuit as shown in Figure 1, the equivalent resistance $R$ ohms is given by the formula $R = \dfrac{ab}{a+b}$. Find the value of $R$ when $a = 20$ and $b = 30$.
$R = $ ohms 检查
(b) When three resistors (20, 30, 15 ohms) are connected as shown in Figure 2 (all parallel), find their equivalent resistance.
$R = $ ohms 检查💡 思路
(a) 并联公式 $R = \dfrac{ab}{a+b} = \dfrac{600}{50} = 12$ ohms.
(b) 三个并联：$\dfrac{1}{R} = \dfrac{1}{20} + \dfrac{1}{30} + \dfrac{1}{15} = \dfrac{3+2+4}{60} = \dfrac{9}{60} \Rightarrow R = \dfrac{60}{9} = \dfrac{20}{3} = 6\tfrac{2}{3}$ ohms.