Year 9 Number Cheatsheet

• Direct proportion: as one quantity increases, the other increases at the same rate (y = kx)
• Index laws work with negative, zero, and fractional powers too
• Scientific notation writes large or small numbers compactly: a × 10ⁿ
• Simple interest: money earned based on principal, rate, and time
• Rational numbers are fractions; irrational numbers like √2 and π cannot be written as fractions
• Surds are square roots of non-perfect squares; we simplify them

Example 1: Write 0.000456 in scientific notation

1. Move the decimal point to get a number between 1 and 10: 4.56
2. Count how many places the decimal moved: 4 places to the right
3. When moving right, the power of 10 is negative: 0.000456 = 4.56 × 10^-4

Example 2: Calculate simple interest on $5000 at 4% per annum for 3 years

1. Use I = PRT
2. I = 5000 × 0.04 × 3
3. I = $600
4. Total amount = Principal + Interest = $5000 + $600 = $5600

Example 3: Simplify √50

1. Find the perfect square factors of 50: 50 = 25 × 2
2. √50 = √(25 × 2) = √25 × √2 = 5√2

• Mistake: Writing 2^-3 = -8 (thinking the negative sign means multiply by -1)
  Correction: 2^-3 = 1/8. The negative power means reciprocal, not negative.
• Mistake: Writing 5.2 × 10³ = 5200.0 (adding extra zeros)
  Correction: 5.2 × 10³ = 5200 (move decimal 3 places right)
• Mistake: Using simple interest formula as A = P(1 + RT) instead of I = PRT
  Correction: I = PRT gives you just the interest. Add to principal for total amount.
• Mistake: Leaving √12 as √12 instead of simplifying to 2√3
  Correction: Always simplify surds by removing perfect square factors.

1. Calculate 3^-2
   Answer

   3^-2 = 1/9
2. Write 8500 in scientific notation
   Answer

   8.5 × 10³
3. If y is directly proportional to x, and y = 20 when x = 5, find y when x = 8
   Answer

   k = 20/5 = 4, so y = 4x. When x = 8, y = 32
4. Simplify √45
   Answer

   √45 = √(9 × 5) = 3√5
5. Find simple interest on $2000 at 3% per annum for 4 years
   Answer

   I = 2000 × 0.03 × 4 = $240

• Direct Proportion: A relationship where y = kx; as x increases, y increases proportionally.
• Constant of Proportionality: The value k in y = kx.
• Scientific Notation: A way to write numbers as a × 10ⁿ where 1 ≤ a < 10.
• Index (or Exponent): The power to which a base is raised.
• Simple Interest: Interest calculated as a percentage of the principal for a fixed time period.
• Principal: The original amount of money borrowed or invested.
• Surd: The root of a number that cannot be simplified to a whole number (e.g., √5).
• Rational Number: Any number that can be written as a fraction a/b.
• Irrational Number: A number that cannot be written as a simple fraction (e.g., √2, π).