Published: 05 Aug 2019

**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 (a^{2} : b^{2}).

**9. Sub-duplicate Ratio** of (a : b) is ( root of a : root of b)

**10. Triplicate Ratio** of (a : b) is (a^{3} : b^{3}).

**11. Sub-triplicate Ratio** of (a : b) is (a^{1/3} : b^{1/3}).

**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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