**RATIO:**

A ratio is simply a fraction. The following notations all express the ratio of x to y

x:y , x÷ y , or x/y.

In the ratio x:y, we call x as the first term or *antecedent* and y, the second term or *consequent*.

Writing two numbers as a ratio provides a convenient way to compare their sizes.

A ratio compares two numbers. Just as you cannot compare apples and oranges, so the numbers you are comparing must have the same units.

For example, you cannot form the ratio of 2 feet to 4 meters because the two numbers are expressed in different units - feet vs. meters.

**DEMONSTRATION:**

What is the ratio of 2 feet to 4 yards?

(A) 1 : 2 (B) 1 : 8 (C) 1 : 7 (D) 1 : 6 (E) 1 : 5

**Solution:**

The ratio cannot be formed until the numbers are expressed in the same units. Let’s turn the yards into feet.

Since there are 3 feet in a yard, 4 yards = 4 * 3 feet = 12 feet .

Forming the ratio yields 2 feet:12 feet= 1/6=1:6

The answer is (D).

**Duplicate Ratios:**

*Duplicate ratio* of (a:b) is (a^{2}:b^{2})

**Sub-duplicate:**

*Sub-duplicate* ratio of (a:b) is (a^{1}^{/}^{2}:b^{1}^{/}^{2})

**Triplicate Ratio:**

*Triplicate ratio *of (a:b) is (a^{3}:b^{3})

**Sub-triplicate Ratio:**

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

**Theorem:**

Componendo and Dividendo is a theorem on proportions that allows for a quick way to perform calculations and reduce the amount of expansions needed

If a/b = c/d , then (a+b) / (a-b) = (c+d) / (c-d)

Proof of Compenendo Dividendo theorem:

If a/b=c/d

then we can say that a:b=c:d=k(universally recognized as constant).

If a/b=c/d then a+b/a-b=c+d/c-d.

By transferring b on the other side we can say that

a=kb,c=kd.

So we can re write equation in terms of k as:

kb+b/kb-b=kd+d/kd-d

Lets take LHS first:

Lets take b common now we can say that

b(k+1)/b(k-1)

cancel out b from numerator and denominator

we obtain (k+1)/(k-1)

RHS:

kd+d/kd-d

take d as common we can say that

d(k+1)/d(k-1) .

Cancel out d from both numerator and denominator we obtain

(k+1)/(k-1).

Hence Proved

LHS = (k+1)/(k-1)= RHS =(k+1)/(k-1)

**PROPORTION**

The equality of two ratios (fractions) 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,

In a:b::c:d

(b∗c) = (a∗d)

**DEMONSTRATION**

What is the ratio of the number of boys to the number of girls in a group of 8 boys and 12 girls?

The required ratio = number of boys/number of girls

= 8/12

= (2 × 4)/(3 × 4)

= 2/3

**Terms in Proportions**

**Fourth Proportional**

If a:b=c:d then d is called the *fourth proportional *to a,b,c

**Third Proportional**

a:b=c:d, then c is called the *third proportional* to a and b.

**Mean Proportional**

*Mean proportional* between a and b is √a x b

Consider Y is the mean Proportional

I.e. a/y=y/b , y^{2}=a x b ,y = √a x b

**Comparison of Ratios:**

To compare two ratios, follow these steps:

**Step 1:** Make the second term of both the ratios equal.

For this, determine the LCM of the second terms of the ratios. Divide the LCM by the second term of each ratio. Multiply the numerator and the denominator of each ratio by the quotient.

**Step 2:** Compare the First Terms (Numerators) of the new ratios

**Demonstration:**

Compare 3:4 and 1:2

LCM of 4 and 2 is 4

4/4=1 and 4/2 =2

Thus 3/4 = (3 x1 )/( 4 x1) = 3 /4 and 1/2 = (1x2)/(2x2) = 2/4

Thus now we can tell

3/4> 2/4

**Compounded Ratio**

The *compounded ratio* of the ratios:

(a:c) and (b:d) is ac:bd

(a:b),(c:d),(e:f) is (ace:bdf)