## Formulas for Ratio and Proportion:-

**Definitions:**

#### Ratio:-

The ratio of two quantities in the same units is the fraction that one quantity is of the other. It is a relationship between two numbers by the division of the same kind. Therefore, the ratio of 2 to 3 is 2/3 or can be written as 2 : 3. The first term of a ratio is known as antecedent and the second term is called consequent. Furthermore, it may be noted that 10: 15 equals to 10/15 or 2/3 is equals to 2 : 3. Therefore, multiplication of each term of a ratio by the same number does not affect the ratio.

#### Proportion

The equality of two ratios is known as proportion. It is used to find out the quantity of one class over the total. In other words, the proportion is a part that describes the comparative relation with the overall part.

Subsequently, 2 : 3 equals to 4 : 6, we will write 2 : 3 :: 4 : 6 and we can say that 2, 3, 4, and 6 are in proportion. Consequently, 2, 3, 4, and 6 are called 1^{st}, 2^{nd}, 3^{rd}, and 4^{th} proportional respectively. The first and the fourth proportional are called the extreme terms while the second and the third proportional are called mean terms.

The product of the means equals the product of the extremes.

#### Fourth proportional

If x : y = z: a, then a is called the fourth proportional to x, y, z.

#### Third proportional

If x: y = y: z, then z is called the third proportional to x and y.

#### Mean proportional

Mean proportional between x and y is √xy

#### Comparison of ratios

We say that (x: y) > (z: a), then (x/y) > (z /a).

#### Compounded ratio

The compounded ratio of the ratios (x: y), (z: a), (b : c) is (xzb: yac)

### Ratio and Proportions Formulas & Properties of Ratio

- The ratio of two people a and b is denoted as a : b.
- a : b = ma : mb, where m is a constant.
- x : y : z = X : Y : Z is equivalent to x/X=y/Y=z/Z
- If x/y = z/a then, x+y/x-y = z+a/z-a

### Ratio and Proportions Formulas Property of Proportion

- x/y=z/a , this means x : y ::z:a

**RATIO**

Ratio is strictly a mathematical term to compare two similar quantities expressed in the same units. The ratio of two terms ‘x’ and ‘y’ is denoted by x:y.

The numerator of the ratio is called the antecedent (x) and the denominator is called consequent (y) of the ratio.

**COMPARISON OF TWO OR MORE RATIOS**

Two or more ratios may be compared by reducing the equivalent fractions to a common denominator and then comparing the magnitudes of their numerator. Thus, suppose 2:5, 4:3 and 4:5 are three ratios to be compared then the fractions and are reduced to equivalent fractions with a common denominator. For this, the denominator of each is changed to 15 equal to the L.C.M. their denominators. Hence the given ratios are expressed and or 2:5, 4:3, 4:5 according to magnitude.

**Example : Which of the ratio 2:3 and 5:9 is greater?**

**Solution: **

In the form of fractions, the given ratios are 2/3 and 5/9, Reducing them to fractions with a common denominator they are written as 6/9 and 5/9

Hence the greater ratio is 6/9 or 2:3.

**Example: Are the ratio 3 to 4 and 6:8 equal?**

**Solution: **

The ratio are equal if 3/4 = 6/8.

These are equal if their cross products are equal; that is, if 3×8 = 4×6. Since both of these products equal 24, the answer is yes, the ratios are equal.

Remember to be care full order matters! A ratio of 1:7 is not the same as a ratio of 7:1.

**REMEMBER**

- The two quantities must be of the same kind and in same unit.
- The ratio is a pure number, i.e. without any unit of measurement.
- The ratio would stay unaltered even if both the antecedent and the consequent are multiplied or divided by the same number.

**Compound ratio:**

Ratios are compounded by multiplying together the antecedents for a new antecedent and the consequent for a new consequent.

The compound of a: b and c: d Is a*c/b*d, i.e., ac: bd.

**Example : Find the compound ratio of the four ratios: 4:5, 15:13, 26:3 and 6:17**

**PROPORTION**

When two ratios are equal, the four quantities composing them are said to be in proportion.

If , then a, b, c, d are in proportion.

This is expressed by saying that ‘a’ is to ‘b’ as ‘c’ is to ‘d’ and the proportion is written as

a:b::c:d or a:b=c:d

The terms a and d are called the extremes while the terms b and c are called means.

**TO FIND THE MEAN PROPORTIONAL**

**Example : Find the mean proportional between 3 and 75.**

**Solution:**

Let x be the required mean proportional. Then, 3:x::x:75 ∴x=√(3×75)=15

**Example: A courier charge to a place is proportional to the square root of the weight of the consignment. It costs ` 54 to courier a consignment weighing 25 kilos. How much more will it cost (in rupees) to courier the same consignment as two parcels weighing 16 kilos and 9 kilos respectively?**

**TO FIND THE VALUES OF AN UNKNOWN WHEN FOUR NUMBERS ARE IN PROPORTION**

**Solution:**

Let the number to be added to each of the four numbers be x.

By the given condition, we get (10+x):(18+x):(22+x):(38+x)

⇒(10+x)(38+x)=(18+x)(22+x)

⇒380+48x+x^2=396+40x+x^2

Cancelling x^2 from both sides, we get

380+48x=396+40x

⇒48x-40x=396-380

⇒8x-16x=16/8=2

Therefore, 2 should be added to each of the four given numbers.

**TO FIND THE FOURTH PROPORTIONAL**

**Example:** **Find the fourth proportional to p^2-pq+q^2,p^3+q^3,p-q**

**USING THEOREM ON EQUAL PROPORTION**

**DIRECT PROPORTION**

If on the increase of one quantity, the other quantity increases to the same extent or on the decrease of one, the other decrease to the same extent, then we say that the given two quantities are directly proportional.

**Some Examples:**

- Work done number of men
- Cost number of men
- Work wages
- Working hour of a machine fuel consumed
- Speed distance to be covered

**INDIRECT PROPORTION (OR INVERSE PROPORTION)**

If on the increase of one quantity, the other quantity decreases to the same extent or vice versa, then we say that the given two quantities are indirectly proportional.

**Some Examples:**

- More men, less time
- Less men, more hours
- More speed, less time
- More speed, less taken time to be covered distance

**RULE OF THREE**

In a problem on simple proportion, usually three terms are given and we have to find the fourth term, which we can solve by using Rule of three. In such problems, two of given terms are of same kind and the third term is of same kind as the required fourth term. First of all we have to find whether given problem is a case of direct proportion or indirect proportion. For this, write the given quantities under their respective headings and then mark the arrow in increasing direction. If both arrows are in some direction then the relation between them is direct otherwise it is indirect or inverse proportion. Proportion will be made by either head to tail or tail to head.

**Example: A man completes 5/8 of a job in 10 days. At this rate, how many more days will it takes him to finish the job?**