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Simplify of Ratios in Different Units – An Ultimate Guide and Examples
Question 1 Simplify the ratio \( 2 \) hours to \( 30 \) minutes. \( \begin{align} 2 \text{ hours} : 30 \text{ minutes} &= 2 \times 60 \text{ minutes} : 30 \text{ minutes} \\ &= 120 : 30 \\ &= 4 : 1 \end{align} \) Question 2 Simplify the ratio \( 50 \) cm to […]
Monotonic Increasing and Decreasing | How to Find the Inverse of Monotonic Functions
Question 1 (a) Sketch \( y = x^2 \). (b) Find the domain that \( y=x^2 \) is restricted to a monotonic increasing curve.. \( x \gt 0 \) (c) Find the inverse of \( y = x^2 \). \( \begin{align} y^2 &= x \\ y &= \pm \sqrt{x} \\ y &= […]
Dividing a Quantity in a Given Ratio – Clear Examples
Question 1 Divide \( 500 \) in the ratio \( 2 : 3 \). \( \displaystyle \begin{align} \frac{2}{2+3} \times 500 &= 200 \\ \frac{3}{2+3} \times 500 &= 300 \\ 500 \text{ is divided by } &200 \text{ and } 300. \end{align} \) Question 2 Divide \( 600 \) in the ratio \( 2 : 3 […]
How to Substitute a Number into an Algebraic Expression
$$ \large \begin{align} a &= 4 \\ b &= 3 \\ x &= 25 \\ y &= -8 \end{align} $$ Question 1 If \( a=4 \) and \( b=3 \), evaluate \( ab \). \( \begin{align} ab &= a \times b \\ &= 4 \times 3 \\ &= 12 \end{align} \) Question 2 If \( […]
Finding Equation of Locus by Locus Definition
A Locus and its Equation A locus is a collection of points with the same relations. The illustration shows a locus of points whose distance from the \( y \)-axis is equal to its distance from the \(x\)-axis. Example 1 A point \( P(x,y) \) moves so that its distance from the \( y \)-axis […]
Applications of Projectile Motion
Example A particle is projected at an angle of \( 45^{\circ} \) and a velocity of \( 60 \text{ ms}^{-1} \), and taking gravity as \( 10 \text{ ms}^{-2} \). (a) Find the equation for the particle for horizontal displacement. \( \begin{align} \displaystyle \require{AMSsymbols} \ddot{x} &= 0 \\ \dot{x} &= 60 \cos 45^{\circ} = […]
Probability by Basic Two-Way Tables | Two-Way Frequency Tables and Probability
\( \begin{array}{|c|c|c|c|} \hline &\text{swimmer} &\text{non-swimmer} &\text{total} \\ \hline \text{Boys} & 11 & 10 & \\ \hline \text{Girls} & 13 & 16 & \\ \hline \text{total} & & \\ \hline \end{array} \) Question 1 (a) How many boys are in the class? \( \require{AMSsymbols} \begin{array}{|c|c|c|c|} \hline &\text{swimmer} &\text{non-swimmer} &\text{total} \\ \hline \text{Boys} & \bbox[yellow,3px]{11} & […]
Conversion from Volume to Capacity
$$ \text{cm}^3 = \text{mL} \xrightarrow{\div 1000} \text{L} \xrightarrow{\div 1000} \text{kL} \xrightarrow{\div 1000} \text{ML}$$ $$\text{cm}^3 = \text{mL} \xleftarrow{\times 1000} \text{L} \xleftarrow{\times 1000} \text{kL} \xleftarrow{\times 1000} \text{ML}$$ Converting cm3 to L Convert \( 2000 \text{ cm}^3 \) to \( \text{L} \). \( \begin{align} 1 \text{ cm}^3 &= 1 \text{ mL} \\ 2000 \text{ cm}^3 &= 2000 \text{ […]
Scale Ratios and Scale Factors – Calculating the Real and Scale Lengths
Simplifying a Map Scale Simplify a map scale \( 2 \text{ cm} : 1 \text{ km} \). \( \begin{align} 2 \text{ cm} : 1 \text{ km} &= 2 \text{ cm} : 1000 \text{ m} \\ &= 2 \text{ cm} : 100 \ 000 \text{ cm} \\ &= 1 \text{ cm} : 50 \ 000 \text{ […]
Mathematical Induction Proof with Factorials | Principles of Mathematical Induction
Prove \( 2 \times 1! + 5 \times 2! + 10 \times 3! + \cdots + (n^2+1)n! = n(n+1)! \). Step 1 Show it is true for \( n=1 \). \( \begin{align} &\text{LHS} = (1^2+1) \times 1! = 2 \\ &\text{RHS} = 1 \times (1+1)! = 2 \\ &\text{LHS} = \text{RHS} \\ &\text{Therefore it is […]