Question

Write a multiplication table of 9 using Vinculum Method and identify the number at fifth place.
  (a) 36
  (b) 45
  (c) 54
  (d) 90

Answer 1

HINT:- Before solving this question, let us know about the Vinculum Method:-
VINCULUM METHOD: The Vedic system uses a variety of methods for simplifying calculations. The vinculum is one of these as it allows us to remove some or all digits over five from a calculation so that only 0, 1, 2, 3, 4 and 5 are used.

Complete step-by-step solution -
Let us solve the question now.
In the Vinculum method, instead of multiplying the number with 9, it is multiplied with 10 and 1 and the results are subtracted.
We know that 9 is 1 less than 10.
So, we can write 9 as 10−1
Let us write the multiplication table for 9.
Table of 9 (9 = 10 – 1)
\[\begin{array}{*{35}{l}}
   \left( 10\text{ }-\text{ }1 \right)\times 1\text{ }=\text{ }9\text{ }\times \text{ }1\text{ }=\text{ }9 \\
   \left( 20\text{ }-\text{ }2 \right)\text{ }\times \text{ }1\text{ }=\text{ }18\text{ }\times \text{ }1\text{ }=\text{ }18 \\
   \left( 30\text{ }-\text{ }3 \right)\text{ }\times 1\text{ }=\text{ }27\text{ }\times \text{ }1\text{ }=\text{ }27 \\
   \left( 40\text{ }-\text{ }4 \right)\text{ }\times \text{ }1\text{ }=\text{ }36\text{ }\times \text{ }1\text{ }=\text{ }36 \\
   \left( 50\text{ }-\text{ }5 \right)\text{ }\times 1\text{ }=\text{ }45\text{ }\times \text{ }1\text{ }=\text{ }45 \\
   \left( 60\text{ }-\text{ }6 \right)\text{ }\times \text{ }1\text{ }=\text{ }54\text{ }\times \text{ }1\text{ }=\text{ }54 \\
   \left( 70\text{ }-\text{ }7 \right)\text{ }\times \text{ }1\text{ }=\text{ }63\text{ }\times \text{ }1\text{ }=\text{ }63 \\
   \left( 80\text{ }-\text{ }8 \right)\text{ }\times 1\text{ }=\text{ }72\text{ }\times \text{ }1\text{ }=\text{ }72 \\
   \left( 90\text{ }-\text{ }9 \right)\text{ }\times \text{ }1\text{ }=\text{ }81\text{ }\times \text{ }1\text{ }=\text{ }81 \\
   \left( 100\text{ }-\text{ }10 \right)\text{ }\times \text{ }1\text{ }=\text{ }90\text{ }\times \text{ }1\text{ }=\text{ }90 \\
\end{array}\]
Hence, this is the table of 9.
Let us find the number at fifth place by the following method:-
\[\left( 10\text{ }-\text{ }1 \right)\text{ }\times \text{ }5\text{ }=\text{ }10\text{ }\times \text{ }5\text{ }-\text{ }1\text{ }\times \text{ }5\text{ }=\text{ }50\text{ }-\text{ }5\text{ }=\text{ }45\]
So, the number at the fifth place in the table of 9 is 45.
Therefore, the correct option is (b) 45.

NOTE:- We can also find the multiplication table for 9 by the following method:-
\[\begin{array}{*{35}{l}}
   \left( 10\text{ }-\text{ }1 \right)\text{ }\times \text{ }1\text{ }=\text{ }10\text{ }\times \text{ }1\text{ }-\text{ }1\text{ }\times \text{ }1\text{ }=\text{ }10\text{ }-\text{ }1\text{ }=\text{ }9 \\
   \left( 10\text{ }-\text{ }1 \right)\text{ }\times \text{ }2\text{ }=\text{ }10\text{ }\times \text{ }2\text{ }-\text{ }1\text{ }\times \text{ }2\text{ }=\text{ }20\text{ }-\text{ }2\text{ }=\text{ }18 \\
   \left( 10\text{ }-\text{ }1 \right)\text{ }\times \text{ }3\text{ }=\text{ }10\text{ }\times 3\text{ }-\text{ }1\text{ }\times \text{ }3\text{ }=\text{ }30\text{ }-\text{ }3\text{ }=\text{ }27 \\
   \left( 10\text{ }-\text{ }1 \right)\text{ }\times \text{ }4\text{ }=\text{ }10\text{ }\times \text{ }4\text{ }-\text{ }1\text{ }\times \text{ }4\text{ }=\text{ }40\text{ }-\text{ }4\text{ }=\text{ }36 \\
   \left( 10\text{ }-\text{ }1 \right)\text{ }\times \text{ }5\text{ }=\text{ }10\text{ }\times 5\text{ }-\text{ }1\text{ }\times \text{ }5\text{ }=\text{ }50\text{ }-\text{ }5\text{ }=\text{ }45 \\
   \left( 10\text{ }-\text{ }1 \right)\text{ }\times \text{ }6\text{ }=\text{ }10\text{ }\times \text{ }6\text{ }-\text{ }1\text{ }\times \text{ }6\text{ }=\text{ }60\text{ }-\text{ }6\text{ }=\text{ }54 \\
   \left( 10\text{ }-\text{ }1 \right)\text{ }\times \text{ }7\text{ }=\text{ }10\text{ }\times \text{ }7\text{ }-\text{ }1\text{ }\times \text{ }7\text{ }=\text{ }70\text{ }-\text{ }7\text{ }=\text{ }63 \\
   \left( 10\text{ }-\text{ }1 \right)\text{ }\times \text{ }8\text{ }=\text{ }10\text{ }\times 8\text{ }-\text{ }1\text{ }\times \text{ }8\text{ }=\text{ }80\text{ }-\text{ }8\text{ }=\text{ }72 \\
   \left( 10\text{ }-\text{ }1 \right)\text{ }\times \text{ }9\text{ }=\text{ }10\text{ }\times \text{ }9\text{ }-\text{ }1\text{ }\times \text{ }9\text{ }=\text{ }90\text{ }-\text{ }9\text{ }=\text{ }81 \\
   \left( 10\text{ }-\text{ }1 \right)\text{ }\times \text{ }10\text{ }=\text{ }10\text{ }\times \text{ }10\text{ }-\text{ }1\text{ }\times \text{ }10\text{ }=\text{ }100\text{ }-\text{ }10\text{ }=\text{ }90 \\
\end{array}\]