1. Ratio: We use ratios to make comparisons between two things. When we express ratios in words, we use the word "to"--we say "the ratio of something to something else." Ratios can be written in several different ways: as a fraction, using the word "to", or with a colon.
The ratio of two quantities 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 a as the first term or antecedent and b, the second term or consequent.
Ex. The ratio 5 : 9 represents 5/9 with antecedent = 5, consequent = 9.
Rule: The multiplication or division of each term of a ratio by the same non-zero number does not affect the ratio.
Ex. 4 : 5 = 8 : 10 = 12 : 15. Also, 4 : 6 = 2 : 3.
2. 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, d are in proportion.
Here a and d are called extremes, while b and c are called mean terms.
Product of means = Product of extremes.
Thus, a : b :: c : d ⇔ (b x c) = (a x d).
3. Fourth Proportional: If a : b = c : d, then d is called the fourth proportional to a, b, c.
4. Third Proportional: a : b = c : d, then c is called the third proportional to a and b.
5. Mean Proportional: Mean proportional between a and b is root ab.
6. Comparison of Ratios: we say that (a : b) > (c : d) ⇔ a/b > c/d.
7. Compounded Ratio: The compounded ratio of the ratios: (a : b), (c : d), (e : f) is (ace : bdf).
8. Duplicate Ratios of (a : b) is (a2 : b2).
9. Sub-duplicate Ratio of (a : b) is ( root of a : root of b)
10. Triplicate Ratio of (a : b) is (a3 : b3).
11. Sub-triplicate Ratio of (a : b) is (a1/3 : b1/3).
12. If a/b = c/d, then a+b/a-b = c+d/c-d (componendo and dividendo)
Variations:
We say that x is directly proportional to y, if x = ky for some constant k and we write, x proportional y.
We say that x is inversely proportional to y, if xy = k for some constant k and we write, x proportional 1/y.
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