Huge multiplication

Algebra Level 1

9914 × 9916 + 1 991 5 2 = ? \large \dfrac{9914 \times 9916 +1}{9915^{2}}= \ ? \


The answer is 1.

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Akshay Yadav by Akshay Yadav , 20, India

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3 solutions

Isaac Buckley
Jun 5, 2015

We use the difference of two squares.

9914 × 9916 + 1 991 5 2 = ( 9915 − 1 ) × ( 9915 + 1 ) + 1 991 5 2 = 991 5 2 − 1 2 + 1 991 5 2 = 991 5 2 991 5 2 = 1 \frac{9914 \times 9916 +1}{9915^{2}}= \frac{(9915-1) \times (9915+1) +1}{9915^{2}}=\frac{9915^2-1^2 +1}{9915^{2}}=\frac{9915^2}{9915^{2}}=1

9 Helpful 0 Interesting 0 Brilliant 0 Confused
Pedro Henrique
Jan 9, 2016

Transform the smaller number near to the others in a constant and just do the math...

⇒ { 9914 = a } ⇒ a ( 2 + a ) + 1 ( 1 + a ) 2 ⇒ 2 a + a 2 + 1 2 a + a 2 + 1 ⇒ 1 \Rightarrow \{ 9914=a\} \\ \\ \Rightarrow \frac { a(2+a)+1 }{ { (1+a) }^{ 2 } } \\ \\ \Rightarrow \frac { 2a+{ a }^{ 2 }+1 }{ 2a+{ a }^{ 2 }+1 } \\ \\ \Rightarrow 1

0 Helpful 0 Interesting 0 Brilliant 0 Confused

But it is not exactly 1
It is 1.0001008369

0 Helpful 0 Interesting 0 Brilliant 0 Confused

It is exactly 1.

rushi panmand - 5 years, 5 months ago

You must have done 9914 × ( 9916 + 1 ) 9914\times(9916+1) while typing it in your calculator instead of ( 9914 × 9916 ) + 1 (9914\times9916)+1 . That is why you got the wrong answer.

Akshay Yadav - 5 years, 5 months ago

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