Tips and tricks to solve problems from Time and Work, Problems on Trains, and Profit and Loss is already discussed in the previous article (Part I). This section (Part II) includes another three very important section from quantitative aptitudes including *“Problems on Ages”, “Average”, and “Permutation and Combination”*.

**Shortcut methods and tricks to solve questions from ‘Problems on Ages’ are given below:**

Consider the letter ‘A’ to denote the age of a person. Now the following rules are very important to solve any related aptitudes:

If the current age is a, then x times the age = ax.

If the current age is a, then age x years later = a + x.

If the current age is a, then age x years ago = a – x.

The ages in a ratio p : q will be pa and qa.

If current age is a, then 1 / n of the age is a / n.

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**Shortcut methods and tricks to solve questions from ‘Average’ are given below:**

This part has two sections: *1. Average and 2. Average Speed*

1. Average = (Sum of observations/Number of observations)

2. Average Speed

Suppose you are covering a certain distance at p kmph and the equal distance at q kmph.

The calculation of average speed during the whole journey is [(2pq / (p + q)]

Let’s we suppose a natural number n. The average of consecutive numbers will be

(n + 1) / 2

The average of 1st n even numbers is (n + 1)

The average of natural even numbers upto n is (n + 2) + 1

The average of natural odd numbers upto n is (n + 1) / 2

The average of n multiples of x will be [x(n+1)]/2

If the average of x1 items is a1 and the average of x2 item is a2, then the average of total items (x1 + x2) will be [(x1a1 + x2a2) / (x1 + x2)]

In a family, let the average ages of p members is q years. Another member joins the family and the average age of all the existing family members become r. The average age of the new member in the family is:

[r – p(r – q)] years.

In a family, let the average ages of p members is q years. A member leaves the family and the average age of all the existing family members become r. The average age of the new member in the family is:

[r + p(q – r)] years.

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**Shortcut methods and tricks to solve questions from ‘Permutation and Combination’ are given below:**

This is very important part of** quantitative aptitude questions for banking exam**. You must see some questions from this part, so it’s very important for you to be an expert in math to solve the problems regarding permutation and combination.

Let’s consider n be the positive integer. Then factorial of n is denoted as ˪n or n!

The definition of n! Is as given below:

n! = n(n-1)(n-2)……………………………3.2.1.

1! = 1, 0! = 1

Permutation is different type of arrangements of a given number by taking some or all together.

If you take three letters a, b, c; then the permutation formula will be (ab, ba, bc, cb, ac, ca) considering two letters at a time.

If you take three letters a, b, c; then the permutation formula will be (abc, bac, bca, cba, acb, cab) considering all three letters at a time.

Number of all permutation of n things, taken r at a time is given as

n p r = n(n-1) (n-2)………(n-r+1) = !n / (n-r)!

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