## Averages, Ratio and Proportion

Theory

"Averages, Percentages and Ratio and Proportion" is one of the easiest topics in quantitative section in CAT. The theory involved in this section is just an extension of high school mathematics. It is for this reason that questions from these topics are often asked in conjunction with other topics. The fundamentals of this topic are hence important not just from a stand-alone perspective, but also to answer questions from other topics.

Formula Questions
• The average of n terms equals $$\frac{x_1+x_2+. . .+x_n}{n}$$
Formula Questions
• The weighted average of n terms equals $$\frac{w_1*x_1+ w_2*x_2+. . .+ w_n*x_n}{ w_1+ w_2+ . . . w_n}$$
Formula Questions
• Percentage Change = (Final Value â€“ Initial  Value)/Initial Value * 100
• x % of T = $$\dfrac{x}{100}\times\ T$$
Formula Questions
• For two successive changes of of x% and y%, the total % change is (x + y + xy/100)%
Formula Questions
• Alligation Rule: This is used to find the ratio of individual components in a mixture. If two components A and B, costing Rs. X and Rs. Y individually, are mixed and the resultant mixture has an average price of Rs. Z, then the ratio of A and B in the mixture is $$\frac{Z-Y}{X-Z}$$
Formula Questions
• If a, b, c, d and x are positive integers such that $$\frac{a}{b}=\frac{c}{d}$$

1. If $$a < b$$, $$\frac{a+x}{b+x} > \frac{a}{b}$$
2. If $$a > b$$, $$\frac{a+x}{b+x} < \frac{a}{b}$$
3. If $$a > b$$, then $$\frac{a-x}{b-x} > \frac{a}{b}$$
4. If $$a < b$$, then $$\frac{a-x}{b-x} < \frac{a}{b}$$
5. $$\frac{a+c}{b+d}=\frac{a}{b}=\frac{c}{d}$$
6. $$\frac{a+b}{a-b}=\frac{c+d}{c-d}$$
Formula Questions

If a container has 'a' liters of liquid A and if 'b' liters of solution is withdrawn and is replaced with an equal volume of another liquid B and the operation is repeated for 'n' times, then after nth operation,

The final quantity of Liquid A in the container = $$\left(\ \frac{\ a-b}{a}\right)^{^n}\times\ a$$

Formula Questions

In N liters mixture of a solution C, if the amount of liquid A is 'a' liters then the concentration of A in the solution is a*100/N %

Formula Questions

If x is directly proportional to y => x = ky

If x is directly proportional to square of y => x = $$ky^2$$

If x is directly proportional to cube of y => x = $$ky^3$$

Here, 'k' is a proportionality constant.

When x is indirectly proportional to y => $$x=\dfrac{k}{y}$$

Formula

In a mixture of mixtures, two quantities of some mixtures are mixed to get a mixture of mixtures.

Let Mixture 1 have ingredients A and B in ratio a: b, and Mixture 2 have ingredients A and B in ratio x : y.

Now, the M unit of mixture 1 and N unit of mixture 2 are mixed to form a resultant mixture. Then, in the resultant mixture, the ratio of A and B is

$$\dfrac{Q_a}{Q_b}=\ \dfrac{M\left(\frac{a}{a+b}\right)+N\left(\frac{x}{x+y}\right)\ }{M\left(\frac{b}{a+b}\right)+N\left(\frac{x}{x+y}\right)}$$