Question

When 16 is subtracted from a three-digit number $abc$ (greater than 100) and the resulting difference is divided by 2, the result is a three-digit number (greater than 100) $cba$. If $a+b+c=20$, find $\left( 2a+3b+4c \right)$

Answer 1

Hint: To obtain the value of a given number we will use basic arithmetic operations. Firstly we will form an equation according to the condition given. Then we will use hit and trial method in such a way that it satisfies all the equations to solve the equation and get the number. Finally, according to the number we will put the values in the equation and get the desired answer.

Complete step by step solution:
The three-digit number is given as:
 $abc$……$\left( 1 \right)$
It is also given that:
$a+b+c=20$……$\left( 2 \right)$
Firstly we will form an equation using statement that when the number above is divided by 16 and the resulting difference is divided by 2 we get a three-digit number $cba$ as follows:
$\dfrac{abc-16}{2}=cba$…..$\left( 3 \right)$
Now using hit and trial let us assign our first variable the highest value possible i.e. 9 as follows:
 $a=9$
Now from equation (2)
$b+c=20-9$
$\Rightarrow b+c=11$….$\left( 4 \right)$
So next we will assign $b$ the second highest value i.e. 8 so from equation (4) c will be 3 as follows:
$\begin{align}
  & b=8 \\
 & c=3 \\
\end{align}$
Substitute all value in equation (3) and check whether they satisfy the equation as follows:
$\begin{align}
  & \dfrac{983-16}{2}=389 \\
 & \Rightarrow \dfrac{967}{2}=389 \\
 & \Rightarrow 483.5\ne 389 \\
\end{align}$
It doesn’t satisfy the equation so we will try different values.
So next we will keep $a=9$ and assign $b$ the third highest value i.e. 7 so $c$ will be 4 from equation (4) as follows:
$\begin{align}
  & a=9 \\
 & b=7 \\
 & c=4 \\
\end{align}$
So as we can see the above values satisfy equation (2).
Let us put the values in equation (3) as follows:
$\begin{align}
  & \Rightarrow \dfrac{974-16}{2}=479 \\
 & \Rightarrow \dfrac{958}{2}=479 \\
 & \Rightarrow 479=479 \\
\end{align}$
The equation (3) is also satisfied.
$\therefore abc=479$….$\left( 5 \right)$
Now we have to find the value of below number:
$\left( 2a+3b+4c \right)$
Using equation (5) value above we get,
$\begin{align}
  & \left( 2a+3b+4c \right)=2\times 4+3\times 7+4\times 9 \\
 & \Rightarrow \left( 2a+3b+4c \right)=8+21+36 \\
 & \therefore \left( 2a+3b+4c \right)=65 \\
\end{align}$
Hence we get $\left( 2a+3b+4c \right)$value as 65.

Note: The arithmetic operations are very easy to use as they are not complicated. One common mistake one can make in these types of questions is that they treat $abc$as a value that can be separated but it is a digit even if we want to expand the digit we have to multiply each value in it with 10 power something depending upon their position in the number.