** Shortcuts in Quantitative Aptitude for Competitive Exams – Ratio & Proportion**

Shortcuts in Quantitative AptitudeReasoningEnglish

**Ratio & Proportion**

**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.

In general, the ratio of a number x to a number y is defined as the quotient of the numbers x and y.

**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 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 or 2 : 5, 4: 3, 4: 5 according to magnitude.

**REMEMBER**

- In the ratio of two quantities 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 numerator and the denominator are multiplied or divided by the same number.

**COMPOUND RATIO**

Ratios are compounded by multiplying together the numerators for a new denominator and the denominator for a new denominator.

**Shortcut Approach**

**PROPERTIES OF RATIOS**

**Shortcut Approach**

**Shortcut Approach**

- In any 2-dimensional figures, if the corresponding sides are in the ratio x : y, then their areas are in the ratio x2 : y2.
- In any two 3-dimensional similar figures, if the corresponding sides are in the ratio x : y, then their volumes are in the ratio x3 : y3.

**Shortcut Approach**

- If the ratio between two numbers is a : b and if each number is increased by x, the ratio becomes c : d. Then, two numbers are

- If the sum of two numbers is A and their difference is a, then the ratio of numbers is given by A + a : A – a.

**Shortcut Approach**

- Let a vessel contains Q unit of mixture of ingredients A and B. From this, R unit of mixture is taken out and replaced by an equal amount of ingredient B only.

If this process is repeated n times, then after n operations

- In a container, milk and water are present in the ratio a : b. If x L of water is added to this mixture, the ratio becomes a : c. Then,
- Quantity ofmilk in mixture = and quantity in original mixture =
- A container has milk and water in the ratio a : b, a second container has milk and water in the ratio c : d. Ifboth the mixtures are emptied into a third container, then the ratio of milk of water in third container is given by

**PROPORTION**

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

This is expressed by saying that ‘a’ isto‘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 the means.

**REMEMBER**

- If four quantities are in proportion, the product of the extremes is equal to the product of the means.

Let a, b, c, d be in proportion, then

- If three quantities a, b and c are in continued proportion, then a : b = b : c

ac = b^{2}is called mean proportional.

**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 decreases to the same extent, then we say that the given two quantities are directly proportional. If A and B are directly proportional then we denote it by A ∝ B.

**Some Examples :**

- Work done ∝ number of men
- Cost ∝ number of Articles
- 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. If A and B are indirectly proportional then we

**Some Examples :**

- More men, less time
- Less men, more 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 same 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.

The complete procedure can be understand by the examples.

**PARTNERSHIP **

A partnership is an association of two or more persons who invest their money in order to carry on a certain business.

A partner who manages the business is called the working partner and the one who simply invests the money is called the sleeping partner. Partnership is of two kinds :

- Simple
- Compound.

**Simple partnership :**

If the capitals is of the partners are invested for the same period, the partnership is called simple.

**Compound partnership :**

If the capitals of the partners are invested for different lengths of time, the partnership is called compound.

**Shortcut Approach**

- If the period of investment is the same for each partner, then the profit or loss is divided in the ratio of their investments. If A and B are partners in a business, then

- If A, B and C are partners in a business, then

- When the amount of capital invested by different partners is same
- Ratio of profit/loss = Ratio of time period for which the capital is invested

**MONTHLY EQUIVALENT INVESTMENT**

It is the product of the capital invested and the period for which it is invested.

If the period of investment is different, then the profit or loss is divided in the ratio of their Monthly Equivalent Investment.

**Shortcut Approach**

**MIXTURE**

**Simple Mixture :** When two different ingredients are mixed together, it is known as a simple mixture.

**Compound Mixture :** When two or more simple mixtures are mixed together to form another mixture, it is known as a compound mixture.

**Alligation :** Alligation is nothing but a faster technique of solving problems based on the weighted average situation as applied to the case of two groups being mixed together.

The word ‘ Alligation ’ literally means‘ linking ’.

**ALLIGATION RULE**

It states that when different quantities of the same or different ingredients of different costs are mixed together to produce a mixture of a mean cost, the ratio of their quantities is inversely proportional to the difference in their cost from the mean cost.

**Graphical representation of Alligation Rule :**

**Applications of Alligation Rule :**

To find the mean value of a mixture when the prices of two or more ingredients, which are mixed together and the proportion in which they are mixed are given.

To find the proportion in which the ingredients at given prices must be mixed to produce a mixture at a given price.

**Shortcut Approach**

**Price of the Mixture:**

**STRAIGHT LINE APPROACH OF ALLIGATION**

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