Y-KNOT

Arithmetic Cheatsheet

1. Ratios, Proportions, and Variations

1.1 Core Concepts

  • Ratio: A comparison of two quantities, written as \(a : b\) or \(a/b\).
  • Proportion: An equality of two ratios. If \(a:b = c:d\), it is written as \(a:b::c:d\).
  • Fundamental Property: Product of extremes = Product of means (\(ad = bc\)).

1.2 Properties of Proportions

Given \(\frac{a}{b} = \frac{c}{d}\):

  • Invertendo: \(\frac{b}{a} = \frac{d}{c}\)
  • Alternendo: \(\frac{a}{c} = \frac{b}{d}\)
  • Componendo: \(\frac{a+b}{b} = \frac{c+d}{d}\)
  • Dividendo: \(\frac{a-b}{b} = \frac{c-d}{d}\)
  • Componendo & Dividendo: \(\frac{a+b}{a-b} = \frac{c+d}{c-d}\)

1.3 Variation

  • Direct Variation: \(y \propto x \implies y = kx\)
  • Inverse Variation: \(y \propto \frac{1}{x} \implies xy = k\)

2. Mixtures and Alligation

2.1 Rule of Alligation

Helps find the ratio in which two ingredients must be mixed to achieve a desired mean value.

$$ \frac{\text{Quantity of Cheaper}}{\text{Quantity of Dearer}} = \frac{d-m}{m-c} $$

Where \(c\) = value of cheaper, \(d\) = value of dearer, and \(m\) = mean value.

2.2 Weighted Average

$$ M_A = \frac{Q_1M_1 + Q_2M_2}{Q_1 + Q_2} $$

2.3 Repeated Dilution / Replacement

If a container holds \(X\) units of a liquid, and \(Y\) units are withdrawn and replaced, repeated \(n\) times:

$$ \text{Final Quantity of Original Liquid} = X \left(1 - \frac{Y}{X}\right)^n $$

3. Profit, Loss, and Discount

3.1 Core Formulas

Profit (\(P\)) = SP - CP
Loss (\(L\)) = CP - SP
$$ \text{Profit \%} = \left(\frac{\text{Profit}}{\text{CP}}\right) \times 100 $$

3.2 Marked Price (MP) and Discount (D)

Discount = MP - SP
$$ \frac{\text{CP}}{\text{MP}} = \frac{100 - D\%}{100 + P\%} $$

3.3 Special Cases

  • Successive Discounts of \(d_1\%\) and \(d_2\%\):
$$ \text{Equivalent Discount} = \left(d_1 + d_2 - \frac{d_1 d_2}{100}\right)\% $$
  • "Buy X, Get Y Free" Offers:
$$ \text{Effective Discount \%} = \left(\frac{Y}{X+Y}\right) \times 100 $$

4. Simple and Compound Interest

4.1 Simple Interest (SI)

$$ SI = \frac{P \times R \times T}{100} $$
Amount (\(A\)) = \(P + SI\)

4.2 Compound Interest (CI)

$$ A = P \left(1 + \frac{R}{100}\right)^T $$
CI = A - P

4.3 Difference between CI and SI

\(\text{Diff}_2\) (For 2 Years) = \(P \left(\frac{R}{100}\right)^2\)
\(\text{Diff}_3\) (For 3 Years) = \(P \left(\frac{R}{100}\right)^2 \left(3 + \frac{R}{100}\right)\)

5. Time, Speed, and Distance (TSD)

5.1 Core Formulas

Distance = Speed \(\times\) Time

5.2 Average and Relative Speed

  • Average Speed (Constant Distance, 2 speeds): \(\frac{2s_1s_2}{s_1+s_2}\) (Harmonic Mean)
  • Average Speed (Constant Time, 2 speeds): \(\frac{s_1+s_2}{2}\) (Arithmetic Mean)
  • Relative Speed (Opposite Directions): \(S_{rel} = s_1 + s_2\)
  • Relative Speed (Same Direction): \(S_{rel} = |s_1 - s_2|\)

5.3 Application: Trains

Crossing a Pole: Time = \(\frac{\text{Length of Train}}{\text{Speed of Train}}\)
Crossing a Platform: Time = \(\frac{\text{Length of Train} + \text{Length of Platform}}{\text{Speed of Train}}\)

5.4 Application: Boats and Streams

  • Downstream Speed: \(S_D = S_B + S_S\)
  • Upstream Speed: \(S_U = S_B - S_S\)
  • Speed of Boat: \(S_B = \frac{S_D + S_U}{2}\)
  • Speed of Stream: \(S_S = \frac{S_D - S_U}{2}\)

5.5 Application: Circular Motion

  • First Meeting (Same Direction): Time = \(\frac{L}{|s_A - s_B|}\)
  • First Meeting (Opposite Direction): Time = \(\frac{L}{s_A + s_B}\)
  • First Meeting at Starting Point: Time = LCM of \(\left(\frac{L}{s_A}, \frac{L}{s_B}\right)\)

6. Time and Work

6.1 Core Concepts

Work Done = Rate of Work \(\times\) Time
Rate of Work (Efficiency) = \(\frac{1}{T}\)

6.2 Combined Work

Two People (x and y days): Time together = \(\frac{xy}{x+y}\)

6.3 MDH Formula (Chain Rule)

$$ \frac{M_1 \times D_1 \times H_1}{W_1} = \frac{M_2 \times D_2 \times H_2}{W_2} $$

6.4 Pipes and Cisterns

  • Inlet Pipe (Fills in x hours): Rate = \(+\frac{1}{x}\)
  • Outlet Pipe (Empties in y hours): Rate = \(-\frac{1}{y}\)