Question

What is the value of ? in the following: \[4+1=5\text{ , }5+2=12\text{ , }3+6=21\text{ , }8+11=?\]

Answer 1

Hint: In this type of question we have to use the concept of identifying a pattern in the set of equations and applying it to the unknown. In this pattern we take two numbers and have to find a hidden equation that involves simple operations like addition, subtraction, multiplication and division.

Complete step by step solution:
Now, we have given that,
\[4+1=5\text{ , }5+2=12\text{ , }3+6=21\]
And we need to find \[\text{8+11=?}\]
The given sets of addition follow a series of logical thinking.
We can notice that \[4+1=5\] can also be written as \[\left( 4\times 1 \right)+1=5\]
By applying same logic we can rewrite the remaining two equations as follows:
\[\begin{align}
  & \Rightarrow 5+2=12\Rightarrow \left( 5\times 2 \right)+2=12 \\
 & \Rightarrow 3+6=21\Rightarrow 3+\left( 3\times 6 \right)=21 \\
\end{align}\]
That is we add the small number to the product of the two numbers to get the answer. In general, we can express it as \[a+b\] means \[a+\left( a\times b \right)\] where \[a\] is small compared to \[b\].
Applying the pattern to the last equation gives
\[\Rightarrow 8+11=8+\left( 8\times 11 \right)=96\]
Hence, if \[4+1=5\text{ , }5+2=12\text{ , }3+6=21\] then \[8+11=96\].

Note: In this type of question students may use another pattern and arrive at an answer. Some students solve this question by using running total that means by adding the result in the previous line to the new numbers to get new answer which works as follows:
First we rewrite all the equations, so we get,
\[\begin{align}
  & \Rightarrow 1+4=5 \\
 & \Rightarrow 2+5=12 \\
 & \Rightarrow 3+6=21 \\
\end{align}\]
Here, we can observe that, in each line the new numbers are incremented 1 more from the previous line.
We can continue this pattern so the next lines would be \[4+7,5+8,6+9,7+10\And 8+11\]
Now, the first equation is valid mathematically.
\[\Rightarrow 1+4=5\]
For the next line we take this result of 5 and add it to the new numbers to get the new answer.
\[\Rightarrow 5+\left( 2+5 \right)=12\]
By doing same for the next equation we get
\[\Rightarrow 12+\left( 3+6 \right)=21\]
Continuing in this way we can write,
\[\begin{align}
  & \Rightarrow 21+\left( 4+7 \right)=32 \\
 & \Rightarrow 32+\left( 5+8 \right)=45 \\
 & \Rightarrow 45+\left( 6+9 \right)=60 \\
 & \Rightarrow 60+\left( 7+10 \right)=77 \\
 & \Rightarrow 77+\left( 8+11 \right)=96 \\
\end{align}\]
Hence, if \[4+1=5\text{ , }5+2=12\text{ , }3+6=21\] then \[8+11=96\].