📚 Maths Summary Notes
Significant Figures Table
Type of Zero
Counts as significant?
Example
Number of significant figures
Leading zeros (before first non-zero digit)
No
0.0045
2 ( 4 and 5 )
Captive zeros (between two non-zero digits)
Yes
105
3 (all of them)
Trailing zeros after a decimal point
Yes
2.300
4 (all of them)
Trailing zeros before a decimal point (no decimal shown)
No
2300
2
Trailing zeros with a decimal point
Yes
2300.
4
Writing Fractions as Decimals
Eg. 2 ⅛ ➡️ 8 ⟌1
- Like converting fraction to recurring decimal
- = 2.125
Writing Decimals as Fractions
Eg. 2.55
= 2 55/100
= 2 11/20
(SIMPLIFY)
Recurring Decimals
- Fraction to a recurring decimal
- Eg. ⅙
- Divide as two normal integers, using long division
- Numerator: in the bracket, Denominator: outside
- Recurring decimal to fraction
- Eg. 0.16 (recurring)
- 𝑥 = 0.16
- 10 𝑥 = 1.66666…
- 100𝑥 = 16.6666…
- 90𝑥 = 15
- 𝑥 = 15/90
- = ⅙
Multiplying Decimals
Steps:
- Ignore the decimals and multiply.
- Count the total number of decimal places in both numbers.
- Move the decimal left in your answer that many places.
NOTE: To square decimals, just lay it out as a normal multiplication, eg. to do (0.8)², lay it out as 0.8 x 0.8 and follow the steps above, and regard it as a normal multiplication :)
Dividing Decimals
Steps:
- Ignore the decimal while dividing.
- Divide as usual like it’s a whole number.
- Count the decimal places in the original number.
- Put the decimal in the result by moving it the same number of places you counted.
Writing Fractions as Percentages
- Figure out the decimal (do long division, numerator of the fraction goes inside the long division bracket, denominator outside, results on top (obvi))
- Multiply decimal by 100 to get the percentage.
- Add % sign
- DONE!!!!!
Evaluating 23² - 21² and other such problems
- This is basically the difference of two squares in a normal integer format, like in algebra x² - y².
- Use the difference of two squares formula - a² - b² = (a - b)(a + b)
- So, to solve 23² - 21², it would be
- 23² - 21² = (23-21) (23+21) = 2 x 44 = 88
- This is the fastest way to evaluate such problems
Finding the number halfway between two
- Use the formula: (a + b) / 2
For fractions:
- You still add the two fractions.
- Then you divide the result by 2.
- You might need to use a common denominator, but the rule doesn’t change
Perfect Squares
(a + b)² = a² + 2ab + b²
(a - b)² = a² - 2ab + b²
Difference of 2 Squares
a² − b² = (a − b)(a + b)
Difference of two squares - Factorising it
- Identify the two perfect squares in the expression.
- Take the square root of both terms.
- Apply the formula by writing the result as a product of two binomials: (a−b) (a+b)
Solving Questions like: The ratio of Zoe's savings to Lucy's savings was 9: 4. After Zoe spent $81, the ratio became 3:2. What was the value of Lucy's savings?
- Original ratio: Zoe : Lucy = 9 : 4 → Zoe = 9x, Lucy = 4x
- Zoe spends $81 → New amount = 9x - 81
- New ratio: 9x−814x=32\frac{9x - 81}{4x} = \frac{3}{2}4x9x−81=23
Solve:
* Cross-multiply:
2(9x−81)=3(4x)2(9x - 81) = 3(4x)2(9x−81)=3(4x)
18x−162=12x18x - 162 = 12x18x−162=12x
6x=1626x = 1626x=162 → x=27x = 27x=27
Lucy’s savings = 4x = 4 × 27 = $108
Holiday Pay Loading
* Person: Sarah
* Weekly Salary: $1,200
* Holiday Pay Loading: 17.5%
* Annual Leave Duration: 4 weeks
Calculation:
* Normal Pay for 4 Weeks: $1,200 × 4 = $4,800
* Holiday Pay Loading (17.5%): $1,200 × 17.5% = $210 per week
* Total Holiday Pay Loading for 4 Weeks: $210 × 4 = $840
* Total Holiday Pay (Normal Pay + Loading): $4,800 + $840 = $5,640
Non Monic Quadratics
Ax² + bx + c = 0
1. Multiply a⋅c
2. Find two numbers that multiply to ac and add to b.
3. Split the middle term using these two numbers.
4. Group terms and factor each pair.
5. Factor out the common binomial.
Example:
Factorising Algebraic Fractions
ALWAYS FACTORISE BEFORE CANCELLING STUFF OUT. DO NOT NOT FACTORISE. ALWAYS FACTORISE.
Expanding 3 brackets
To expand three brackets:
1. Multiply the first two brackets together.
2. Take the result and multiply it by the third bracket.
3. Distribute each term carefully, and combine like terms to get the final result.
Assessment 1
Score:
Part A + B = 68/70 🥳
Mentals = 15/20 🥴
Assessment 2