Question

What number should be added to 57 to get $ - 75$?

Answer 1

Hint:
Here, we will use the required number to some variable. We will use the fact that we get a negative number as an answer by adding either two negative numbers or adding a greater negative number and smaller positive number. Thus, we will subtract the given positive number by negative number to find the required number.

Complete step by step solution:
Let $x$ be the number which should be added to 57 to get $ - 75$.
Mathematically, the above statement could be written as:
$57 + x = - 75$
Subtracting 57 from both sides, we get,
\[x = - 75 - 57 = \left( { - 75} \right) + \left( { - 57} \right)\] …………………………….$\left( 1 \right)$
Now, in order to add these numbers without using a number line, we should keep in mind the following rules of addition without using number line:
1) In order to add a positive number to a positive number, we simply add the two numbers and a positive i.e. the same sign is attached in front of the sum obtained.
2) In order to add a positive number to a negative number, we simply subtract the smaller number from the bigger one and the sign of the bigger number is attached with the answer obtained.
3) In order to add two negative numbers, we simply add both the numbers together and a negative, i.e. the same sign is attached in front of the answer obtained.
In the given question, we can notice that a negative number is being added to a negative number where the negative numbers are $ - 75$ and $ - 57$ respectively.
Hence, using the third rule of addition of numbers without using number line,
We will simply add both the numbers together.
Hence, we can write $\left( { - 75} \right) + \left( { - 57} \right)$as: $75 + 57 = 132$
But, since, in order to add two negative numbers, we simply add both the numbers together and a negative, i.e. the same sign is attached in front of the answer obtained.
 Therefore,
$x = - \left( {75 + 57} \right) = - 132$
Therefore $x = - 132$
Hence, $ - 132$ should be added to 57 to get $ - 75$.

Thus, $ - 132$ is the required answer.

Note:
We can use the number line to visualize how to add and subtract two numbers by thinking of the direction where we are required to move on the number line. But, since in this case, the numbers are bigger than usual then, using the number line would not be convenient. Drawing the number line would be very time consuming and would make the question unnecessarily lengthy. Hence, we should know how to use these rules and applying them will help us to solve the questions correctly and in less time.