Question

If we are given an expression as $63.9805 = 6A + \dfrac{3}{B} + 9C + \dfrac{8}{D} + 5E$ , then what is the value of $4A + 7B + 6C + D + 3E$ ?
A). $47.603$
B). $4.7603$
C). $147.6003$
D). $47.6003$

Answer 1

Hint: In this type of problem we need to compare the L.H.S (i.e., Left Hand Side) and the R.H.S (i.e., Right Hand Side) of the equation to get the result. In this case the L.H.S of the equation contains a number which can be written as the sum of $5$ different values and R.H.S contains $5$ unknowns i.e., $A,{\text{ B, C, D, E}}$ . By modifying L.H.S as the sum of five different values and then individually comparing them with terms in the R.H.S gives the required result.

Complete step-by-step solution:
Given that $63.9805 = 6A + \dfrac{3}{B} + 9C + \dfrac{8}{D} + 5E$ $ \cdot \cdot \cdot \cdot \cdot (1)$
We can write the L.H.S (Left Hand Side) of the equation $63.9805$ as $60 + 3 + 0.9 + 0.08 + 0.0005$ i.e.,
$60 + 3 + 0.9 + 0.08 + 0.0005 = 63.9805$ $ \cdot \cdot \cdot \cdot \cdot (2)$
Now on comparing the Equation $(1)$ and Equation $(2)$ we get
$6A + \dfrac{3}{B} + 9C + \dfrac{8}{D} + 5E = 60 + 3 + 0.9 + 0.08 + 0.0005$
Now let us assume in the following way
$6A = 60$
$ \Rightarrow A = 10$
On equating $6A$ to $60$ we got $A = 10$
$\dfrac{3}{B} = 3$
$ \Rightarrow B = 1$
On equating $\dfrac{3}{B}$ to $3$ we got $B = 1$
$9C = 0.9$
$ \Rightarrow C = 0.1$
On equating $9C$ to $0.9$ we got $C = 0.1$
$\dfrac{8}{D} = 0.08$
$ \Rightarrow D = 100$
On equating $\dfrac{8}{D}$ to $0.08$ we got $D = 0.01$
$5E = 0.0005$
$ \Rightarrow E = 0.0001$
On equating $5E$ to $0.0005$ we got $E = 0.0001$
Now we have the values of $A,{\text{ B, C, D, E}}$ . By substituting these values in the required sum we can get the required result.
$4A + 7B + 6C + D + 3E = 4 \times 10 + 7 \times 1 + 6 \times 0.1 + 100 + 3 \times 0.0001$
$ = 40 + 7 + 0.6 + 100 + 0.0003$
$ = 147 + 0.6 + 0.0003$
$ = 147.6003$
$4A + 7B + 6C + D + 3E = 147.6003$
Therefore, Option (C) is correct.

Note: It is important to strictly follow the order on both the sides while comparing and equating the terms individually. After modifying the L.H.S as the sum of values we need to equate the terms in the order of values from the largest to the smallest. $63.9805$ is taken as $60 + 3 + 0.9 + 0.08 + 0.0005$ which follows the descending order from left to right. Comparing the terms from the largest to smallest with the alphabetical order gives the correct result.