Q3 of 95 Page 68

Look at these:

1 × 4 = (2 × 3) – 2


2 × 5 = (3 × 4) – 2


3 × 6 = (4 × 5) – 2


4 × 7 = (5 × 6) – 2


i) Write the next two lines in this pattern.


ii) If we take four consecutive natural numbers, what is the relation between the products of the first and the last, and the product of the middle two?


iii) Write this as a general principle in algebra and explain it.

i) 5× 8 = (6× 7) – 2

6× 9 = (7× 8) – 2


ii) The above pattern is based on this statement only.


Let the four consecutive numbers be 1,2,3,4.


Product of first and last = (1× 4) = 4


Product of middle two terms = (2× 3) = 6


and 4 = 6 - 2


Let the four consecutive numbers be 4,5,6,7.


Product of first and last = (4× 7) = 28


Product of middle two terms = (5× 6) = 30


And 28 = 30 - 2


iii) General principle,


Let the four consecutive natural numbers be x , (x+1) , (x+2) , (x+3).


Product of first and last = x(x+3) = x2 + 3x


Product of middle two terms = (x+1)(x+2) = x(x+2) + 1(x+2)


= x2 + 2x + x + 2


= x2 + 3x + 2


Clearly, we can see that the difference between the product of middle two terms and the product of first and last term is equal to 2.


More from this chapter

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2

In the multiplication table made earlier, take a square of nine numbers, instead of four, and mark the numbers at the four corners:


What if we take a square of sixteen numbers?

 2

In the multiplication table made earlier, take a square of nine numbers, instead of four, and mark the numbers at the four corners:


What if we take a square of sixteen numbers?

 3

Look at these:

1 × 4 = (2 × 3) – 2


2 × 5 = (3 × 4) – 2


3 × 6 = (4 × 5) – 2


4 × 7 = (5 × 6) – 2


i) Write the next two lines in this pattern.


ii) If we take four consecutive natural numbers, what is the relation between the products of the first and the last, and the product of the middle two?


iii) Write this as a general principle in algebra and explain it.

 4

Shown below is a method to the product 46× 28.


i) Check this method for some other two-digit numbers.


ii) Explain why this is correct, using algebra. (Recall that any two-digit number can be written 10m + n, as seen in the section, Two-digit numbers of the lesson, Numbers and Algebra, in the class 7 textbook).