Concept: Ratio Proportion, Variation & Partnership Quick Revision

 

Ratio is used to compare 2 or more similar quantities (expressed in same units). It is usually expressed in simplest form, i.e., common factor (if any) is eliminated.

• If $\frac{\mathrm{a}}{\mathrm{b}}$ = $\frac{\mathrm{p}}{\mathrm{q}}$, it means For every p parts of a, there are q parts of b.

• If $\frac{\mathrm{a}}{\mathrm{b}}$ = $\frac{\mathrm{p}}{\mathrm{q}}$, then a = p × x and b = q × x

  where x is the common multiple.

Note: If a question involves multiple ratios, their common factor need not be same.

• If $\frac{\mathrm{a}}{\mathrm{b}}$ = $\frac{\mathrm{p}}{\mathrm{q}}$ and a + b = T, then a = $\frac{\mathrm{p}}{\mathrm{p\; +\; q}}$ × T, and b = $\frac{\mathrm{q}}{\mathrm{p\; +\; q}}$ × T

• If a × x = b × y, then $\frac{\mathrm{a}}{\mathrm{b}}$ = $\frac{\mathrm{y}}{\mathrm{x}}$

Note, here $\frac{\mathrm{a}}{\mathrm{b}}$ ≠ $\frac{\mathrm{y}}{\mathrm{x}}$

• If $\frac{\mathrm{a}}{\mathrm{x}}$ = $\frac{\mathrm{b}}{\mathrm{y}}$ = $\frac{\mathrm{c}}{\mathrm{z}}$, then a : b : c = x : y : z

  Note: Ratio of numerators is same as ratio of denominators

• If a × x = b × y = c × z, then a : b : c = $\frac{\mathrm{LCM}(\mathrm{x}, \mathrm{y}, \mathrm{z})}{\mathrm{x}}$ : $\frac{\mathrm{LCM}(\mathrm{x}, \mathrm{y}, \mathrm{z})}{\mathrm{y}}$ : $\frac{\mathrm{LCM}(\mathrm{x}, \mathrm{y}, \mathrm{z})}{\mathrm{z}}$

• If a : b = $\frac{\mathrm{p}}{\mathrm{x}}$ : $\frac{\mathrm{q}}{\mathrm{y}}$, then to simplify RHS, we multiply all terms in RHS with LCM of denominators i.e., LCM (x, y)

• Inverse or reciprocal ratio of a, b and c = $\frac{1}{\mathrm{a}}$ : $\frac{1}{\mathrm{b}}$ : $\frac{1}{\mathrm{c}}$

 

• If a, b, c and d are in proportion, then
  ⇒ $\frac{\mathrm{a}}{\mathrm{b}}$ = $\frac{\mathrm{c}}{\mathrm{d}}$
  ⇒ a × d = c × b

• If a, b and c are in continuous proportion, then
  ⇒ $\frac{\mathrm{a}}{\mathrm{b}}$ = $\frac{\mathrm{b}}{\mathrm{c}}$
  ⇒ a × c = c²

 

If x is directly proportional to y, then

⇒ x ∝ y,
⇒ x = ky

⇒ $\frac{\mathrm{x}}{\mathrm{y}}$ = k (constant)

⇒ $\frac{{\mathrm{x}}_1}{{\mathrm{y}}_1}$ = $\frac{{\mathrm{x}}_2}{{\mathrm{y}}_2}$

Note: When 2 quantities are directly proportional, their ratio is constant

If x is inversely proportional to y, then

⇒ x ∝ 1/y,
⇒ x = k/y

⇒ x × y = k (constant)

⇒ x₁ × y₁ = x₂ × y₂

Note: When 2 quantities are inversely proportional, their product is constant

If x is proportional to two or more variable, then it is proportional to product of all those variables.

Example: If x directly proportional to p and q while inversely proportional to r, then
x ∝ $\frac{\mathrm{p\; \times \; q}}{\mathrm{r}}$

⇒ $\frac{{\mathrm{x}}_1}{{\mathrm{x}}_2}$ = $\frac{{\mathrm{p}}_1{\mathrm{q}}_1\times {\mathrm{r}}_2}{{\mathrm{r}}_1\times {\mathrm{p}}_2{\mathrm{q}}_2}$

 

Let P_A, I_A and T_A represent the Profit, Investment and Time invested for A [Same goes for B and C]

P_A : P_B : P_C = I_A × T_A : I_B × T_B : I_C × T_C

P_A : P_B = (I_A1 × T_A1 + I_A2 × T_A2 + ...) : (I_B1 × T_B1 + I_B2 × T_B2 + ...)