# Types of Number System Questions

In this article, we will try to cover all the types of aptitude questions that are framed on the concepts of Number System.

## Type 1

##### Q. Two statements S1 and S2 are given below followed by a question.

S1: $(a + b)^2 < (a - b)^2$
S2: a = b

Is ab positive?

(a) S1 alone is sufficient to answer the Question.
(b) S2 alone is sufficient to answer the Question.
(c) S1 and S2 together are sufficient to answer the Question, but neither S1 alone nor S2 alone is sufficient to answer the Question.
(d) Either S1 alone or S2 alone is sufficient to answer the Question.

Explanation:

Evaluating Statement 1:
$(a + b)^2 < (a - b)^2$
Expanding both sides of the inequality, we get $a^2 + b^2 + 2ab < a^2 + b^2 - 2ab$
Simplifying we get, 4ab < 0 or ab < 0.
So, we can conclude that ab is not positive.
Thus, Statement 1 ALONE is sufficient.

Evaluating Statement 2:
a = b
Product of two positive numbers or two negative numbers will be always positive. But if both a and b are equal to zero, then ab will be zero too, which is not a positive number.
Thus, Statement 2 ALONE is not sufficient.

## Type 2

##### Q. Two statements S1 and S2 are given below followed by a question.

S1: Sum of the digits is even.
S2: The number, when multiplied by 2, fetches an even number.

Is the two-digit number odd?

(a) S1 alone is sufficient to answer the Question.
(b) S2 alone is sufficient to answer the Question.
(c) S1 and S2 together are sufficient to answer the Question, but neither S1 alone nor S2 alone is sufficient to answer the Question.
(d) S1 and S2 together are not sufficient to answer the Question.

Explanation:

The two-digit odd number means unit digit contains 1, 3, 5, 7, or 9.

Statement 1:
If the sum of digits is even, then both the digits are odd, such as 33, 31, 75, 77, 99, etc.
Or both the digits are even, such as 24, 46, 64, etc.
So, Statement 1 alone is not sufficient

Statement 2:
When even number or odd number is multiplied by 2, then we always get an even number.
So, Statement 2 alone is not sufficient.

Even by using both statements together, we do not get to know whether the number is even or odd.
∴ S1 and S2 together are not sufficient to answer the Question.

## Type 3

##### Q. Two statements S1 and S2 are given below followed by a question.

S1: x > 0
S2: x < 1

Is $x^3 > x^2$?

(a) S1 alone is sufficient to answer the Question.
(b) S2 alone is sufficient to answer the Question.
(c) S1 and S2 together are sufficient to answer the Question, but neither S1 alone nor S2 alone is sufficient to answer the Question.
(d) S1 and S2 together are not sufficient to answer the Question.

Explanation:

Statement 1: x > 0
We know that x is a positive number.

Interval 1: If 0 < x < 1, then $x^3 < x^2$
For example, $(0.5)^3$ = 0.125, $(0.6)^2$ = 0.36

Interval 2: If x > 1, then $x^3 > x^2$
For example, $4^3$ = 64, $4^2$ = 16
We do NOT have a DEFINITE answer using statement 1. So, Statement 1 ALONE is NOT sufficient.

Statement 2: x < 1
Interval 1: For positive values of x, i.e., 0 < x < 1, we know that $x^3 < x^2$
Interval 2: For negative values of x, $x^3$ will be a negative number and $x^2$ will be a positive number.
Hence, $x^3 < x^2$

Lastly, what about x = 0?
When x = 0, $x^3 = x^2$
Hence, if we know that x < 1, then we can conclude that $x^3$ is NOT GREATER THAN $x^2$.