Question

Add without using the number line: \[\left( { - 250} \right) + \left( { + 150} \right)\].

Answer 1

Hint:
Here, we need to find the sum of the given numbers. We will rearrange the numbers in the sum, and rewrite the expression. Then, we will convert the subtraction to addition using parentheses, and simplify to get the sum of the given numbers.

Complete step by step solution:
First, we will rewrite the negative integer in the given sum.
The number \[ - x\] can be written as the product of the negative integer \[ - 1\], and the positive integer \[x\].
Therefore, rewriting the number \[ - 250\], we get
\[ - 250 = \left( { - 1 \times 250} \right)\]
Next, we will rewrite the positive integer in the sum.
The number \[ + x\] can be written as the positive integer \[x\].
Therefore, rewriting the number \[ + 150\], we get
\[ + 150 = 150\]
Now, we will find the sum of the given numbers.
The sum of the numbers \[ - 250\] and \[ + 150\] is given by the expression \[\left( { - 250} \right) + \left( { + 150} \right)\].
Rewriting the expression by using the commutative property of addition, we get
\[ \Rightarrow \left( { - 250} \right) + \left( { + 150} \right) = \left( { + 150} \right) + \left( { - 250} \right)\]
Substituting \[ - 250 = \left( { - 1 \times 250} \right)\] and \[ + 150 = 150\] in the expression, we get
\[ \Rightarrow \left( { - 250} \right) + \left( { + 150} \right) = \left( {150} \right) + \left( { - 1 \times 250} \right)\]
Rewriting the expression, we get
\[ \Rightarrow \left( { - 250} \right) + \left( { + 150} \right) = \left( {150} \right) - 1\left( {250} \right)\]
Factoring out \[ - 1\], we get
\[ \Rightarrow \left( { - 250} \right) + \left( { + 150} \right) = - 1\left( {250 - 150} \right)\]
Subtracting the terms of the expression, we get
\[ \Rightarrow \left( { - 250} \right) + \left( { + 150} \right) = - 1\left( {100} \right)\]
Therefore, we get
\[ \Rightarrow \left( { - 250} \right) + \left( { + 150} \right) = - 100\]

\[\therefore \] We get the value of the given expression \[\left( { - 250} \right) + \left( { + 150} \right)\] as \[ - 100\].

Note:
We used the term ‘integer’ in the solution. An integer is a rational number that is not a fraction. For example: 1, \[ - 1\], 3, \[ - 7\], are integers. Integers can be positive or negative. Positive integers are the numbers 1, 3, 6, etc. Negative integers are the numbers \[ - 1\], \[ - 5\], \[ - 92\], \[ - 41\], etc.
We used the commutative property of addition to get the equation \[\left( { - 250} \right) + \left( { + 150} \right) = \left( { + 150} \right) + \left( { - 250} \right)\]. The commutative property of addition states that the sum of two or more numbers remains the same, irrespective of the order in which they are added. This can be written as \[a + b = b + a\].