Question

How much is $– a – 2b – 3c$ less than a + 3b – 5c?
(a) $– 2a – 5b + 2c$
(b) $2a + 5b – 2c $
(c) $2a + 6b – 15c$
(d) $2a – 5b + 15c$

Answer 1

Hint: First write the first term given in the question. Next, write the second term given and after that find which of them is greater. Assume any variable as their difference. Now, find the value of the variable by using both the terms. This value of the variable is the required result.

Complete step-by-step solution
The first term which is given in the question can be written as $– a – 2b – 3c.$
The second term which is given in the question can be written as $a + 3b – 5c.$
It is given that the first term is less than the second term. So, we write, $- a -2b – 3c < a + 3b – 5c,$
Let us assume their difference to be a variable given by k. So, we can say that the second term is k more than the first term and also, we can say that the first term is k less than the second term given. By the above statement, we can write the equation in k, given by
\[-a-2b-3c+k=a+3b-5c\]
By adding ‘a’ on both the sides, we get the equation as,
\[-a-2b-3c+k+a=a+3b-5c+a\]
By grouping the similar terms, we get,
\[a-a-2b-3c+k=a+a+3b-5c\]
By simplifying the above equation, we get it as,
\[-2b-3c+k=2a+3b-5c\]
By adding 2b on both the sides, we get the equation as,
 \[-2b-3c+k+2b=2a+3b-5c+2b\]
By grouping similar terms together, we get,
\[2b-2b-3c+k=2a+3b+2b-5c\]
By simplifying the above equation, we get,
\[-3c+k=2a+5b-5c\]
By adding 3c on both the sides, we can write,
\[-3c+k+3c=2a+5b-5c+3c\]
By grouping the similar terms together, we get,
\[3c-3c+k=2a+5b+3c-5c\]
By simplifying the above equation, we get,
\[k=2a+5b-2c\]
So, we can say that $– a – 2b – 3c$ is $(2a + 5b – 2c)$ less than a + 3b – 5c.
Therefore, option (b) is the right answer.

Note: We have been asked to find out how much is $– a – 2b – 3c$ less than $a + 3b – 5c$, so some students might write the equation for this as $(– a – 2b – 3c) - k = (a + 3b – 5c)$ to get the answer, which is wrong. Do each and every step with utmost care. This is a simple algebraic difference question, students generally make mistakes in combining the same variables together, while combining they misplace the coefficients, thus resulting in wrong answers.