Question

Solve the equation: $\dfrac{{0.5\left( {x - 0.4} \right)}}{{0.35}} - \dfrac{{0.6\left( {x - 2.71} \right)}}{{0.42}} = x + 61$

Answer 1

Hint:
For this type of question we should follow the BODMAS rule. In this question first do the least common factor(LCM) and then find the value of x using the BODMAS rule. We can check the answer whether the answer is correct or not by substituting the answer in the place of x.

Complete Step by Step Solution:
The objective of the problem is to find the value of x by simplifying the given equation.
Given equation $\dfrac{{0.5\left( {x - 0.4} \right)}}{{0.35}} - \dfrac{{0.6\left( {x - 2.71} \right)}}{{0.42}} = x + 61$
First let us find the LCM of $0.35,0.42$
Divide both the numbers with 100 to make the numbers decimal free .
Now $0.35 = \dfrac{{35}}{{100}},0.42 = \dfrac{{42}}{{100}}$
LCM of $\left( {0.35,0.42} \right)$is equal to LCM of $\left( {\dfrac{{35}}{{100}},\dfrac{{42}}{{100}}} \right)$
It is easier to find the LCM of fractions than finding the LCM of decimal numbers. We can easily find the LCM of fractions by using the simple formula. That is
LCM of $\left( {\dfrac{a}{b},\dfrac{c}{d}} \right) = \dfrac{\text{LCM of numerators}}{\text{HCF of denominators}}$
Now by using above formula we get LCM of $\left( {\dfrac{{35}}{{100}},\dfrac{{42}}{{100}}} \right)$=\[\dfrac{{LCM\,of\,\left( {35,42} \right)}}{{HCF\,of\,\left( {100,100} \right)}}\]
 Now find the LCM of 35 and 42.
For this find the prime factors of 35 and 42. The prime factors of 35 are 5 and 7. And the prime factors of 42 are 2,3 and 7.now multiply the prime factors that occur more number of times.
LCM of $\left( {35,42} \right) = 5 \times 7 \times 2 \times 3 = 210$
Now find the HCF of 100 and 100
For this find the prime factors of 100 and 100. The prime factors of 100 are 2,2,5,5.now multiply the prime factors common to both.
HCF of $\left( {100,100} \right) = 2 \times 2 \times 5 \times 5 = 100$
Now LCM of $\left( {\dfrac{{35}}{{100}},\dfrac{{42}}{{100}}} \right)$$ = \dfrac{{210}}{{100}} = \dfrac{{21}}{{10}} = 2.1$
Now we have , $\dfrac{{0.5\left( {x - 0.4} \right)}}{{0.35}} - \dfrac{{0.6\left( {x - 2.71} \right)}}{{0.42}} = x + 61$
By using the BODMAS rule ,first calculate the bracket terms. That is
$
  \dfrac{{\left( {0.5x - 0.4 \times 0.5} \right)}}{{0.35}} - \dfrac{{\left( {0.6x - 2.71 \times 0.6} \right)}}{{0.42}} = x + 61 \\
  \dfrac{{\left( {0.5x - 0.2} \right)}}{{0.35}} - \dfrac{{\left( {0.6x - 1.626} \right)}}{{0.42}} = x + 61 \\
 $
Now let us take the LCM and solve .
$\Rightarrow \dfrac{{6\left( {0.5x - 0.2} \right) - 5\left( {0.6x - 1.626} \right)}}{{2.1}} = x + 61$
On expanding the numerator by multiplying we get
$\Rightarrow \dfrac{{\left( {3x - 1.2} \right) - \left( {3x - 8.13} \right)}}{{2.1}} = x + 61$
On solving numerator we get
$
\Rightarrow \dfrac{{3x - 1.2 - 3x + 8.13}}{{2.1}} = x + 61 \\
\Rightarrow \dfrac{{6.93}}{{2.1}} = x + 61 \\
\Rightarrow 3.3 = x + 61 \\
 $
Simplify the above equation we get
$
\Rightarrow x = 3.3 - 61 \\
\Rightarrow x = - 57.7 \\
 $

Note:
We can check the answer by substituting the value in the given equation. The BODMAS rule is required for this type of question which stands for B means brackets O stands for orders D stands for division M stands for multiplication A stands for addition S stands for subtraction.