Ratio
The ratio of two numbers x and y, is the fraction xy and we write it as x:y.
In the ratio x:y, x is the 1st term or antecedent and y, the 2nd or consequent.
Eg. The ratio 2:3 represents 23 with antecedent = 2, consequent = 3.
Rule: If each term of a ratio is multiplied or divided by the same non-zero number then it does not affect the ratio.
Eg. 2:4 =4:8 =8:16. Also, 4:6 =2:3.
Proportions:
If two ratios (or fractions are equal) then we say that they are in proportion.
If P:Q=R:S, we write P:Q::R:S and we say that P, Q, R, S are in propotion.
Here P and S are called extrems, Q and R called mean terms.
Product of means = Product of extremes.
Thus, P:Q::R:S ⇔(Q×R)=(P×S)
Fourth Proportional:
If P:Q=R:S, then S is called the 4th proportional to P, Q, R.
Third Proportional:
P:Q=R:S, then R is called the 3rd proportional to P and Q.
Mean Proportional:
Mean proportional between P and Q is √PQ.
Comparison of Ratios:
(P:Q)>(R:S) ⇔PQ>RS.
Duplicate Ratios:
Duplicate ratio of (P:Q) is (P2:Q2).
Sub-duplicate ratio of (P:Q) is (√P:√Q).
Triplicate ratio of (P:Q) is (P13:Q13).
If PQ=RS, then
P+QP−Q=R+SR−S
Variations:
x is directly proportional to y means, if x=ky, where k is some constant and we write, x∝y
x is inversely proportional to y means, if xy=k, where k is some constant and we write, x∝1y.