Ex 2.1 Q3 Fractions and Decimals Solution-NCERT Maths Class 7

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Question

 In a “magic square”, the sum of the numbers in each row, in each column and along the diagonals is the same. Is this a magic square?

\[\begin{align} \frac{4}{11} &\frac{9}{11} \frac{2}{11} \\ \frac{3}{11} & \frac{5}{11} \frac{7}{11} \\ \frac{8}{11} & \frac{1}{11} \frac{6}{11} \\ \end{align}\]

Text Solution

What is Known?

A square with three rows and three columns

What is unknown?

If the square is a magic square or not.

Reasoning:

As stated in the question, in a magic square, the sum of the numbers in each row, in each column and along the diagonals is the same. We can add the fractions in all rows, columns and diagonals to see if sum is the same or not.

Steps:

Sum of rows:

Sum of first row 

\(\begin{align}& =\frac{4}{11}+\frac{9}{11}+\frac{2}{11} \\ {} & =\frac{4+9+2}{11}=\frac{15}{11} \\ \end{align}\)

Sum of second row 

\(\begin{align} & =\frac{3}{11}+\frac{5}{11}+\frac{7}{11} \\ {} & =\frac{3+5+7}{11}=\frac{15}{11} \\ \end{align}\)

Sum of third row

\(\begin{align} & =\frac{8}{11}+\frac{1}{11}+\frac{6}{11} \\ {} & =\frac{8+1+6}{11}=\frac{15}{11} \\ \end{align}\)

Sum of columns:

 Sum of first column 

\(\begin{align} & =\frac{4}{11}+\frac{3}{11}+\frac{8}{11} \\ {} & =\frac{4+3+8}{11}=\frac{15}{11} \\ \end{align}\)

 Sum of second column 

\(\begin{align} & =\frac{9}{11}+\frac{5}{11}+\frac{1}{11} \\ {} & =\frac{9+5+1}{11}=\frac{15}{11} \\ \end{align}\)

Sum of third column 

\(\begin{align} & =\frac{2}{11}+\frac{7}{11}+\frac{6}{11} \\ & =\frac{2+7+6}{11}=\frac{15}{11} \\ \end{align}\)

Sum of diagonals:

 Sum of first diagonal 

\(\begin{align} & =\frac{4}{11}+\frac{5}{11}+\frac{6}{11} \\ & =\frac{4+5+6}{11}=\frac{15}{11} \\ \end{align}\)

 Sum of second diagonal 

\(\begin{align} & =\frac{2}{11}+\frac{5}{11}+\frac{8}{11} \\ & =\frac{2+5+8}{11}=\frac{15}{11} \\ \end{align}\)

Since, the sum of fractions in each row, column, and along the diagonal is the same, therefore, the given square is a magic square.

  
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