**:**

*Important Facts***1.Ratio**: The ratio of two qualities a and b in the same units, is the fraction a/b and we write it as a:b.

In the ratio, a:b, we call as the first term of antecedent and b, the second term consequent.

Ex: The ratio 5:9 represents 5/9 with antecedent=5 ,consequent=9

**Rule**: The multiplication or division of each term of 9 ratio by the same non-zero number does not affect the ratio.

**Proportion**: The equality of two ratios is called proportion. If a:b=c:d, we write a:b::c:d and we say that a,b,c and d are in proportion. Here a and b are called extremes, while b and c are called mean terms.

Product of means=product of extremes

Thus, a:b::c:d => (b*c)=(a*d)

**Fourth proportional**: If a:b::c:d, then d is called the fourth proportional to a,b and c.

**Third proportional**: If a:b::b:c, then c is called third proportional to a and b.

**Mean proportional:**Mean proportional between a and b is SQRT(a*b).

**COMPARISION OF RATIOS**:

We say that (a:b)>(c:d) => (a/b)>(c/d)

**Compounded ratio**: The compounded ratio of the ratios (a:b), (c:d),(e:f) is (ace:bdf).

**Duplicate Ratio**: If (a:b) is (a2: b2 )

Sub-duplicate ratio of (a:b) is (SQRT(a):SQRT(b))

Triplicate ratio of (a:b) is (a3: b3 )

Sub-triplicate ratio of (a:b) is (a1/3: b1/3 ).

If a/b=c/d, then (a+b)/(a-b)=(c+d)/(c-d) (componend o and dividend o)

**VARIATION:**

we say that x is directly proportional to y, if x=ky for some constant k and we write.

We say that x is inversely proportional to y, if xy=k for some constant and we write.

X is inversely proportional to y.

If a/b=c/d=e/f=g/h=k then

k=(a+c+e+g)/(b+d+f+h)

If a1/b1,a2/b2, a3/b3..............an/bn are unequal

fractions then the ratio.

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