Revision Notes: Ratio and Proportion | Mathematics Class 10 ICSE PDF Download

Important Concepts-Ratio

1. The ratio between x and y is the fraction (x/y) and written as x : y, where, y ≠ 0
   Here, the first quantity x is known as Antecedent and second quantity y is known as Consequent.
2. The ratio (x/y) has no unit.
3. Ratio is taken only between positive quantities.
4. Ratio exists only between two quantities of the same kind.
5. The value of a ratio remains unchanged, if both the antecedent (N^r) and the consequent (D^r) are multiplied or divided by the same quantity.
6. The ratio is in its lowest terms if the H.C.F of its both the terms is unity, 1.
7. Ratio a : b is not equal to b:a, unless a = b
8. If a quantity increases or decreases in the ratio a : b, the new resulting quantity is equal to (b/a) times of a the original quantity.
9. Compounded Ratio is the ratio obtained by multiplying two or more ratios together 
   Let a : b and x : y be the two ratios then their compounded ratio be
   
   Compounded ratio of a:b, c:d and e:f is ace : bdf
10. Duplicate ratio is the compound ratio of two equal ratios.
    Duplicate ratio of x : y is x² : y².
11. Triplicate ratio is the compound ratio of three equal ratios.
    Triplicate ratio of x : y is x³ : y³ .
12. Sub-duplicate ratio of x : y is .
    The sub-triplicate ratio of x: y is x^1/3 : y^1/3 .
13. The reciprocal ratio is the ratio between the reciprocals of two quantities.
    The reciprocal ratio of x : y is (1/x) : (1/y) i.e., y : x.

Important Concepts-Proportion

1. a : b : :  c : d
   This is called cross product rule.
   If four quantities a, b, c and d are such that the ratio a:b is equal to the ratio c:d then we say a, b, c and d are in proportion. We express it by writing a:b :: c:d. Here, a and d are called extreme terms while b and c are called the middle terms. Here, d is called the fourth proportional.
2. Three numbers a, b and c are said to be in continued proportion if a:b::b:c that is
3. If a, b and c are in continued proportion then b is called the mean proportional or geometric mean of a and c. Thus,
4. If a, b, c, d, e, f, … are in continued proportion if and only if,
5. If a, b and c are in continued proportion, then ‘a’ is called the first proportional.
6. If a, b and c are in continued proportion, then ‘c’ is called the third proportional.

Some properties of Ratio and Proportion

1. Invertendo: If a:b:c:d then b:a::d:c, that is
2. Alternendo: If a:b:c:d then a:c::b:d, that is
3. Componendo: If a:b:c:d then that is (a + b):b: :(c + d):d, that is
4. Dividendo: If a:b:c:d then (a - b):b: :(c - d):d, that is
5. Componendo and Dividendo: If a:b:c:d then
   (a + b): (a - b):: (c + d): (c - d):, that is
6. Convertendo: If a:b:c:d then a:(a - b)::c:(c - d), that is
7. For any two or more equal ratios, each ratio is equal to the ratio between sum of their antecedents and sum of their consequents.
   Therefore, we have,