Question

Given that $a\left( a+b \right)=36$ and $b\left( a+b \right)=64,$ where $a$ and $b$ are positive, $\left( a-b \right)$ equals:
A. 2.8
B. 3.2
C. -2.8
D. -2.5

Answer 1

Hint: Add the equations $a\left( a+b \right)=36$ and $b\left( a+b \right)=64,$ to get the value of ${{\left( a+b \right)}^{2}}$ to get the value of $a+b$. Then, put it in the equation $a\left( a+b \right)=36$ and $b\left( a+b \right)=64,$ to get the values of $a$and $b$.

Complete step by step answer:
Before proceeding with the question, we must know how to solve linear equations in two variables. We solve the linear equation in two variables by finding the value of the first variable in terms of the second variable and then putting the value of the first variable in the second equation to get the value of the second variable. Then we can put the value of the second variable in the first equation to get the value of the first variable. In this way, we will get the required value of the variables.

In this question, we have been given that $a\left( a+b \right)=36$ and $b\left( a+b \right)=64,$ where $a$ and $b$ are positive. We have been asked to find the value of $\left( a-b \right)$.

Let us assume $a\left( a+b \right)=36.....(i)$ and $b\left( a+b \right)=64.....(ii)$.

We can write equation (i) and equation (ii) as

${{a}^{2}}+ab=36$ and $ab+{{b}^{2}}=64$

Adding two equation (i) and (ii) we get:

$\Rightarrow {{a}^{2}}+{{b}^{2}}+2ab=36+64$

$\Rightarrow {{\left( a+b \right)}^{2}}=100$

$\Rightarrow a+b=10.....(iii)$

Putting the value of $a+b$ in equation (i) we get:

$a\left( 10 \right)=36$

$\therefore a=3.6$

Putting the value of $a+b$ in equation (ii) we get:

$b\left( 10 \right)=64$

$\therefore b=6.4$

Therefore, putting the value of $a$ and $b$ in $\left( a-b \right)$ we get:

$a-b=6.4-3.6$

$\therefore a-b=2.8$

Hence, the correct answer is option A.


Note: Try to convert the equations in terms of the whole square. This will make the solution easy and we will get our answers very fast. After getting the values of $a$ and $b$, try to cross-check the answer by substituting these in the equations given in the question.