Question

Find the sum of the given expression: 105 + (-162) + (-162) + (-240).

Answer 1

Hint: Use the fundamental concept of multiplying the signs (+ and -). It is given as
Product of (+) and (+) = (+)
Product of (+) and (-) = (-)
Product of (-) and (+) = (-)
Product of (-) and (-) = (+)
If we subtract two numbers, then subtract the lower magnitude number from the higher magnitude number and put the sign the same to the number with higher magnitude. Here, magnitude means the value of a number without using ‘+’ or ‘- ‘sign. Value of the expression ‘-a-b’ can be calculated by adding the magnitudes of ‘a’ and ‘b’ and putting ‘-sign to the result.

Complete step-by-step solution -
Here, we need to calculate the value of the expression 105+(-162) +(-162) +(-290).
Now, as we know the product of two positive numbers will give a positive number, and the product of two negative numbers will give a positive number, but if we multiply any positive number to a negative number, the result will be a negative value. It is happening because of the multiplication of signs + and -.
So, rules of multiplication of signs are given as
Product of + and + = +
Product of + and - = -
Product of - and + = -
Product of - and - = +
So, now, observe the given relation in the question i.e. 105+(-162) +(-162) +(-240). As, there are ‘+’ signs in front of terms written in brackets and all terms written in brackets have negative signs to them. It means if we open the bracket then the ‘+’ sign will get multiplied to ‘- ‘sign. So, as we have already discussed that multiplication of ‘+’ and ‘- ‘signs are ‘- ‘. So, we rewrite the expression as
$105-162-162-240.......(i)$
Now, we know the numbers with the ‘- ‘signs will get added and put the ‘-‘sign to its summation .It means, we need to add numbers 162, 162 and 240 and hence have to put ‘-‘ sign in front of the result.
So, adding the numbers 162,162 and 240 from the equation (i) as
\[\begin{align}
  & \underline{\begin{matrix}
   {} & 1 & 6 & 2 \\
   {} & 1 & 6 & 2 \\
   + & 2 & 4 & 0 \\
\end{matrix}} \\
 & \underline{\begin{matrix}
   {} & 5 & 6 & 4 \\
\end{matrix}} \\
\end{align}\]
Hence, we can replace the term ‘-162-162-240- by -564 as sum of 162,162,240 is 564 as calculated above, So, we get equation (i) as
$105-564.......(ii)$
Now, we know that two numbers with opposite signs will get subtracted from a bigger number to a lower number and need to put the sign of the number which is greater between them. So, we can observe 564 is greater than 105; So, we need to subtract 105 from 564 and need to put ‘- ‘sign to the result as 564(greater) has ‘- ‘sign and 105 (lower) has ‘+’ sign. So, we get
\[\begin{align}
  & \underline{\begin{matrix}
   {} & 5 & 6 & 4 \\
   - & 1 & 0 & 5 \\
\end{matrix}} \\
 & \underline{\begin{matrix}
   {} & 4 & 5 & 9 \\
\end{matrix}} \\
\end{align}\]
Hence, the value of equation (ii) will be – 459 as per the rules discussed above. So, the value of the expression 105 + (-162) + (-162) + (-240) is -459.

Note: Another approach for the question would be that we can simplify the expression by solving the two terms taken at a time. As the expression is 105 – 162 – 162 – 240. So, solve 105 – 162 at first step, then subtract the given result to 162 and hence, subtract the further result to 240.So, it can be another approach as well.
Follow the sign rule very carefully. One may put multiplication of ‘+’ and ‘- ‘by ‘+’ sign which is wrong. As, multiplication of ‘+’ and ‘- ‘will be ‘- ‘. So, be clear with the properties of positive and negative signs.