ULTIMATE MATH CHEAT SHEET: FORMULAS, TRICKS, AND TIPS TO ACE ANY EXAM!

When it comes to preparing for competitive exams or refreshing our knowledge, a concise and well-organized cheat sheet can make all the difference. This cheat sheet is a quick revision guide for several important topics in mathematics and problem-solving, covering everything from percentages, areas, and volumes to algebra, statistics, and time-related concepts. Whether you’re a student gearing up for exams or someone who needs to brush up on essential formulas and tricks, this guide provides all the necessary information in one place for a streamlined review. Let’s dive in!

Unit 1: Percentages

• Basic Percentage Formula:
  Percentage = (Part / Whole) × 100

• Common Fractions as Percentages:

  1. 10% = 1/10
  2. 15% = 3/20
  3. 20% = 1/5
  4. 25% = ¼
  5. 33.33% = 1/3
  6. 50% = ½
  7. 66.67% = 2/3
  8. 75% = ¾
  9. 100% = 1

• Percentage Increase/Decrease:

  • New Value = Old Value (1 + Inc% / 100)
  • New Value = Old Value (1 – Dec% / 100)

• Successive Percentage Changes:
  a + b + (ab)/100

• Finding the Original Value After Percentage Increase/Decrease:

  • Original Value = Increased Value / (1 + Percentage Increase / 100)
  • Original Value = Decreased Value / (1 – Percentage Decrease / 100)

• Percentage Change Formula:
  Percentage Change = [(New Value – Old Value) / Old Value] × 100

Unit 2: Problem on Ages

• Age Difference Trick:
  The age difference between two people does not change over time.

• Sum of Ages Trick:
  If you know the sum of their ages now, you can also calculate future or past sums using similar techniques.

Unit 3: Area, Shapes, and Perimeter

• Area Formulas:

  • Square: s2s^2s2
  • Rectangle: $L\times B$
  • Triangle: 12×Base×Height\frac{1}{2} \times \text{Base} \times \text{Height}21​×Base×Height
  • Equilateral Triangle: 34×s2\frac{\sqrt{3}}{4} \times s^243​​×s2
  • Parallelogram: $B\times H$
  • Rhombus: 12×d1×d2\frac{1}{2} \times d1 \times d221​×d1×d2
  • Trapezium: 12×(p1+p2)×H\frac{1}{2} \times (p1 + p2) \times H21​×(p1+p2)×H
  • Circle: πr2\pi r^2πr2
  • Ellipse: $\pi \times a\times b$ where a and b are the semi-major and semi-minor axes.

• Volume Formulas:

  • Cube: s3s^3s3
  • Cuboid: $L\times W\times H$
  • Sphere: 43πr3\frac{4}{3} \pi r^334​πr3
  • Cylinder: πr2h\pi r^2 hπr2h
  • Cone: 13πr2h\frac{1}{3} \pi r^2 h31​πr2h

Unit 4: Equations

• Linear Equation:
  $ax+b=0$

• Quadratic Equation:
  ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0

• Discriminant (D) Formula:
  D=b2−4acD = b^2 – 4acD=b2−4ac

• Roots based on the discriminant:

  • If $D>0$: Two distinct real roots.
  • If $D=0$: Two equal real roots.
  • If $D<0$: No real roots (complex roots).

Unit 5: Simplifications

• BODMAS Rule:
  Use the BODMAS rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction) for simplifying expressions.

Unit 6: Mean, Median, Mode, Variance, and Standard Deviation

• Mean: Average of a data set.
• Median: Middle term when data is arranged in ascending order.
• Mode: Most frequent term.
• Variance and Standard Deviation: Measures of spread in the data.

Unit 7: Progressions

• Geometric Progression (GP):
  an=a×rn−1a_n = a \times r^{n-1}an​=a×rn−1
  Sum of first $n$ terms:

  • If r≠1r \neq 1r=1:
    Sn=a(1−rn)1−rS_n = \frac{a(1 – r^n)}{1 – r}Sn​=1−ra(1−rn)​
  • If $r=1$:
    Sn=n×aS_n = n \times aSn​=n×a

• Arithmetic Progression (AP):
  an=a+(n−1)×da_n = a + (n – 1) \times dan​=a+(n−1)×d
  Sum of the first $n$ terms:
  Sn=n2×(a+l)S_n = \frac{n}{2} \times (a + l)Sn​=2n​×(a+l) where $l$ is the last term.

Unit 8: Profit and Loss

• Relation Between C.P., S.P., and Profit/Loss:
  $\text{Profit}=\text{S.P.}-\text{C.P.}$

  • C.P. = S.P. / (1 + Profit% / 100)
  • C.P. = S.P. – Loss
  • S.P. = C.P. / (1 – Loss% / 100)

• Mark-up and Discounts:
  Discount is always calculated on the marked price.
  Successive Discounts:
  S.P.=MP×(1−D1/100)×(1−D2/100)\text{S.P.} = \text{MP} \times (1 – D_1 / 100) \times (1 – D_2 / 100)S.P.=MP×(1−D1​/100)×(1−D2​/100)

Unit 9: HCF and LCM

• Relation Between HCF and LCM:
  $\text{HCF}(a,b)\times \text{LCM}(a,b)=a\times b$

Unit 10: Speed, Time, and Distance

• Conversion:

  • $\text{Km/h}$ to $\text{m/s}$
  • $\text{m/s}$ to $\text{Km/h}$

• Relative Speed:

  • If two objects move towards each other: Relative Speed=S1+S2\text{Relative Speed} = S_1 + S_2Relative Speed=S1​+S2​
  • If two objects move in the same direction: Relative Speed=∣S1−S2∣\text{Relative Speed} = |S_1 – S_2|Relative Speed=∣S1​−S2​∣

Unit 11: Boat and Streams

• Speed of the Boat in Still Water (B):
  Downstream Speed: $D=B+S$
  Upstream Speed: $U=B-S$

Unit 12: Clocks and Calendars

• Clock Problems:

  • Angle between hour and minute hands:
    $|(30H-5.5M)|$
  • Time when hands overlap:
    Time=H×60+15+30H11\text{Time} = H \times 60 + \frac{15 + 30H}{11}Time=H×60+1115+30H​

• Calendar:
  Leap year calculation: Divisible by 4, but not 100 unless divisible by 400.

Conclusion:

This cheat sheet offers a comprehensive overview of essential formulas, tricks, and concepts to help you quickly revise and solve problems in mathematics. By keeping these key formulas at your fingertips, you’ll be able to tackle a wide range of questions efficiently. Whether you’re prepping for an exam or looking to strengthen your problem-solving skills, revisiting these core concepts will ensure you’re well-prepared for whatever comes your way

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