# Concept: Ratio Proportion, Variation & Partnership Quick Revision

BASIC RATIO CALCULATIONS

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

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

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

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

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

• If a : b = $\frac{p}{x}$ : $\frac{q}{y}$, 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 = $\frac{1}{a}$ : $\frac{1}{b}$ : $\frac{1}{c}$

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

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

VARIATION

Direct Variation

If x is directly proportional to y, then

⇒ x ∝ y,
⇒ x = ky

$\frac{x}{y}$ = 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

Inverse Variation

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

Joint Variation

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 ∝ $\frac{p × q}{r}$

$\frac{{\mathrm{x}}_{1}}{{\mathrm{x}}_{2}}$ = $\frac{{\mathrm{p}}_{1}{\mathrm{q}}_{1}×{\mathrm{r}}_{2}}{{\mathrm{r}}_{1}×{\mathrm{p}}_{2}{\mathrm{q}}_{2}}$

PARTNERSHIP

Let PA, IA and TA represent the Profit, Investment and Time invested for A [Same goes for B and C]

When investment does not change

PA : PB : PC = IA × TA : IB × TB : IC × TC

When investment changes

PA : PB = (IA1 × TA1 + IA2 × TA2 + ...) : (IB1 × TB1 + IB2 × TB2 + ...)

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