x(1-3%2F7)x(1-3%2F10)x(1-3%2F13)x(1-3%2F97)x(1-3%2F100).jpeg "(1-3/4)x(1-3/7)x(1-3/10)x(1-3/13)x(1-3/97)x(1-3/100)")
Unraveling the Pattern: A Step-by-Step Exploration of a Unique Multiplication
Let's delve into the fascinating multiplication:
(1-3/4) x (1-3/7) x (1-3/10) x (1-3/13) x (1-3/97) x (1-3/100)
At first glance, this expression might seem daunting. However, closer inspection reveals a beautiful pattern and an elegant solution.
Simplifying the Expression
The key to understanding this multiplication lies in simplifying each individual term. Let's break it down:
- (1-3/4) = 1/4
- (1-3/7) = 4/7
- (1-3/10) = 7/10
- (1-3/13) = 10/13
- (1-3/97) = 94/97
- (1-3/100) = 97/100
Notice the emerging pattern: the numerator of each fraction is 3 more than the denominator of the previous fraction. This pattern continues throughout the sequence.
The Power of Cancellation
Now, let's rewrite the entire multiplication using the simplified terms:
(1/4) x (4/7) x (7/10) x (10/13) x (94/97) x (97/100)
Look what happens when we multiply these fractions. Notice the cancelling effect:
- The 4 in the numerator of the second fraction cancels with the 4 in the denominator of the first fraction.
- The 7 in the numerator of the third fraction cancels with the 7 in the denominator of the second fraction.
- This cancellation continues throughout the entire multiplication.
After cancellation, we are left with:
1/100
Conclusion
The seemingly complex multiplication (1-3/4) x (1-3/7) x (1-3/10) x (1-3/13) x (1-3/97) x (1-3/100) simplifies to 1/100 due to a beautiful pattern and the power of cancellation. This example demonstrates the elegance and efficiency of mathematical patterns and the art of simplifying complex expressions.