Deriving and using formulae booklet
Deriving and using formulae booklet
- Unit 1: Expressions, Formulae & Equations
- Unit 2: Sequences & Functions
- Unit 3: Graphs
🎯 In this topic you will
- Write and use formulae
- Change the subject of a formula
🧠 Key Words
- changing the subject
- subject of a formula
Show Definitions
- changing the subject: The algebraic process of rearranging a formula to isolate a different variable as the new subject of the equation.
- subject of a formula: The variable that is expressed alone on one side of an equation, typically representing the quantity being calculated or defined by the formula.
📐 Understanding Formulas
A formula is a mathematical rule that shows the relationship between variables. Let's consider the formula \( v = u at \) as an example.
Formula Example:$v = u at$
In this formula, v is the subject of the formula. We can change the subject of the formula to make a different variable the subject. For example, if we want to make u the subject:
Changing the Subject:$u = v - at$
💡 Quick Math Tip
Formula Subject: Usually, we write the subject of a formula on the left side of the equation.
🔄 Rearranging Formulas
Sometimes we need to rearrange a formula to make a different variable the subject. For example, if we know the values of v, u, and a, but need to find t, we must rearrange the equation to make t the subject.
Step-by-Step Derivation:
Starting with: $v = u at$
Rearrange: $u at = v$
Subtract u from both sides: $at = v - u$
Divide both sides by a: $t = \frac{v - u}{a}$
❓ EXERCISES
1. Time unit conversion (minutes to seconds)
a) Write the formula for converting minutes to seconds.
b) Substitute values to calculate.
c) Rearrange the formula.
👀 Show answer
a) Formula: $seconds = minutes \times 60$
b) Example: 5 minutes = $5 \times 60 = 300$ seconds
c) Rearranged formula: $minutes = \frac{seconds}{60}$
2. Based on the formula $F = ma$
a) Substitute values to calculate force.
b) Rearrange the formula to find acceleration.
c) Rearrange the formula to find mass.
👀 Show answer
a) Example: When mass $m = 5kg$ and acceleration $a = 2m/s^2$, force $F = 5 \times 2 = 10N$
b) Rearranged for acceleration: $a = \frac{F}{m}$
c) Rearranged for mass: $m = \frac{F}{a}$
🧠 Think like a Mathematician
💡 Quick Math Tip
Euler's formula for polyhedra: $F V - E = 2$ (where F = faces, V = vertices, E = edges).
Question: Investigate the relationship between faces (F), vertices (V), and edges (E) in 3D shapes.
Method:
- Complete the table showing faces, vertices, and edges for different 3D shapes.
- Write the formula connecting F, V, and E, then verify with examples.
- Rearrange the formula to make V the subject and calculate V for given E and F.
- Identify the shape from given F, V, and E values.
- Rearrange the formula to make F the subject and calculate F for given V and E.
- Calculate F and discuss the validity of your answer.
Follow-up Questions:
| Shape | Faces (F) | Vertices (V) | Edges (E) |
|---|---|---|---|
| Cube | 6 | 8 | 12 |
| Cuboid | 6 | 8 | 12 |
| Triangular Prism | 5 | 6 | 9 |
| Square Pyramid | 5 | 5 | 8 |
| Tetrahedron | 4 | 4 | 6 |
Show Answers:
- 1: Table completed above.
- 2: Formula: $F V - E = 2$. Verification: Cube (6 8-12=2), Triangular Prism (5 6-9=2).
- 3: Rearranged: $V = 2 - F E$. When E=12, F=6: $V = 2 - 6 12 = 8$.
- 4: Triangular Prism (5 faces, 6 vertices, 9 edges).
- 5: Rearranged: $F = 2 - V E$. When V=6, E=9: $F = 2 - 6 9 = 5$.
- 6: When V=4, E=6: $F = 2 - 4 6 = 4$. This describes a tetrahedron, which is valid as it satisfies Euler's formula.
❓ EXERCISES
4. Sarah is twice as old as her brother Tom. In 5 years, Sarah will be 1.5 times as old as Tom.
(a) Write expressions for their current ages.
(b) Find their current ages.
(c) Write an expression for their total age in 3 years.
👀 Show answer
(a) Let Tom's current age be $x$ years. Then Sarah's current age is $2x$ years.
(b) In 5 years, Tom will be $x 5$ years and Sarah will be $2x 5$ years.
According to the problem: $2x 5 = 1.5(x 5)$
$2x 5 = 1.5x 7.5$
$0.5x = 2.5$
$x = 5$
So Tom is 5 years old and Sarah is 10 years old.
(c) In 3 years, their total age will be $(5 3) (10 3) = 8 13 = 21$ years.
5. Use the formula $v = u at$ to solve the following:
(a) Find $v$ when $u = 5$, $a = 2$, and $t = 4$
(b) Find $u$ when $v = 25$, $a = 3$, and $t = 6$
(c) Find $a$ when $v = 40$, $u = 10$, and $t = 5$
👀 Show answer
(a) $v = u at = 5 2 \times 4 = 5 8 = 13$
(b) $v = u at$
$25 = u 3 \times 6$
$25 = u 18$
$u = 25 - 18 = 7$
(c) $v = u at$
$40 = 10 a \times 5$
$40 - 10 = 5a$
$30 = 5a$
$a = 6$
💡 Profit Formula
Percentage Profit: Profit% = $\frac{\text{Selling Price} - \text{Cost Price}}{\text{Cost Price}} \times 100\%$
6. A shopkeeper buys a television for $400 and sells it for $480.
(a) Calculate the profit made.
(b) Calculate the profit percentage.
(c) If the shopkeeper wants to make a 25% profit, at what price should he sell the television?
👀 Show answer
(a) Profit = Selling Price - Cost Price = $480 - $400 = $80
(b) Profit% = $\frac{\text{Profit}}{\text{Cost Price}} \times 100\% = \frac{80}{400} \times 100\% = 20\%$
(c) For a 25% profit:
Profit = 25% of Cost Price = $0.25 \times 400 = $100
Selling Price = Cost Price Profit = $400 $100 = $500
❓ EXERCISES
💡 Quick Math Tip
Note: 9 stones means 9 stones and 0 pounds.
7. In some countries, weight is measured in stones (S) and pounds (P). The formula to convert stones and pounds to kilograms (K) is:
$K = \frac{5(14S P)}{11}$
Calculate the weight in kilograms for each of the following:
(a) 9 stones
(b) 10 stones and 5 pounds
(c) 12 stones and 8 pounds
(d) 15 stones and 3 pounds
👀 Show answer
(a) For 9 stones: S=9, P=0
$K = \frac{5(14 \times 9 0)}{11} = \frac{5 \times 126}{11} = \frac{630}{11} \approx 57.27$ kg
(b) For 10 stones and 5 pounds: S=10, P=5
$K = \frac{5(14 \times 10 5)}{11} = \frac{5 \times 145}{11} = \frac{725}{11} \approx 65.91$ kg
(c) For 12 stones and 8 pounds: S=12, P=8
$K = \frac{5(14 \times 12 8)}{11} = \frac{5 \times 176}{11} = \frac{880}{11} = 80$ kg
(d) For 15 stones and 3 pounds: S=15, P=3
$K = \frac{5(14 \times 15 3)}{11} = \frac{5 \times 213}{11} = \frac{1065}{11} \approx 96.82$ kg
🧠 Think like a Mathematician
Question: Make x the subject of each formula and determine the correct answer.
Method:
- Rearrange each equation to make x the subject.
- Compare your result with the given options (A, B, C).
- Select the correct answer for each equation.
- Analyze and reflect on potential errors in the rearrangement process.
Part a: Make x the subject and choose the correct option (A, B, or C) for each formula.
i.$y = 2x 3$
A. $x = \frac{y-3}{2}$ B. $x = \frac{y 3}{2}$ C. $x = 2y - 3$
ii.$3y = 6x - 9$
A. $x = \frac{y 3}{2}$ B. $x = \frac{y-3}{2}$ C. $x = 2y 3$
iii.$y = \frac{x 5}{3}$
A. $x = 3y - 5$ B. $x = 3y 5$ C. $x = \frac{y-5}{3}$
iv.$4x - 2y = 8$
A. $x = \frac{8 2y}{4}$ B. $x = \frac{8-2y}{4}$ C. $x = 2y - 4$
v.$y = \frac{5}{x-1}$
A. $x = \frac{5}{y} 1$ B. $x = \frac{5}{y} - 1$ C. $x = \frac{5 y}{y}$
Part b: Analyze and think about the reasons for errors that might occur when making x the subject.
Follow-up Questions:
Show Answers
- 1: Common mistakes include: not performing the same operation on both sides, incorrect handling of fractions, sign errors when moving terms across the equals sign, and forgetting to apply operations to every term.
- 2: You can verify by substituting a value for x into the original equation and the rearranged equation to see if they produce the same result for the other variable.
- 3: Making a variable the subject allows us to express that variable in terms of others, which is useful for solving problems, graphing, and understanding the relationship between variables.
❓ EXERCISES
9. Make t the subject of each formula:
a)$v = u at$
👀 Show answer
b)$s = \frac{1}{2}at^2$
👀 Show answer
c)$A = \pi r^2 \pi rt$
👀 Show answer
d)$P = 2(l b t)$
👀 Show answer
10. The figure shows a composite shape made of a square and a rectangle. The sides are labeled a, b, c, a as shown:
a) Write a formula for the area of the shape.
👀 Show answer
b) If a = 5 cm, b = 8 cm, and c = 3 cm, calculate the area.
👀 Show answer
c) Rearrange the formula to make b the subject.
👀 Show answer
d) If Area = 60 cm², a = 4 cm, and c = 5 cm, find b.