Don't forget the carry over

Algebra Level 2

0. 2 2 2 2 … × 0. 5 5 5 5 … 0. 1 2 3 4 … \begin{array}{cccccccc} && & 0. & 2 & 2 & 2&2\ \ldots \\ \times && & 0. & 5 & 5 & 5&5\ \ldots \\ \hline && & 0. & 1& 2 &3&4\ \ldots \\ \end{array}

The above shows a long multiplication, where only the first 4 decimal places are shown.

As you can see, the first 4 digits after the decimal place of the final product is 1234.

But what is the first 8 digits after the decimal place of the final product?


The answer is 12345679.

Pi Han Goh by Pi Han Goh , 30, Malaysia

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

2 solutions

Naren Bhandari
Oct 1, 2018

Note that { 0.222 ⋯ = 0.2 + 0.02 + 0.002 + ⋯ = ∑ n = 1 ∞ 2 1 0 n 0.555 ⋯ = 0.5 + 0.05 + 0.005 + ⋯ = ∑ n = 1 ∞ 5 1 0 n \begin{cases} 0.222\cdots =0.2 + 0.02 +0.002+\cdots = \displaystyle \sum_{n=1}^{\infty}\dfrac{2}{10^n} \\ 0.555\cdots =0.5+ 0.05+0.005+\cdots = \displaystyle \sum_{n=1}^{\infty}\dfrac{5}{10^n} \end{cases} Thus above result gives us ∑ n = 1 ∞ 2 1 0 n = 2 9 ∑ n = 1 ∞ 5 1 0 n = 5 9 \sum_{n=1}^{\infty}\dfrac{2}{10^n} = \dfrac{2}{9}\qquad \sum_{n=1}^{\infty}\dfrac{5}{10^n}=\dfrac{5}{9} Making the answer ∑ n = 1 ∞ 2 1 0 n ⋅ ∑ n = 1 ∞ 5 1 0 n = 10 81 = 0.123456790 ⋯ \sum_{n=1}^{\infty}\dfrac{2}{10^n}\cdot \sum_{n=1}^{\infty}\dfrac{5}{10^n}= \dfrac{10}{81}=0.123456790\cdots Thus, our required answer is 12345679 12345679 .

2 Helpful 0 Interesting 0 Brilliant 0 Confused

We know 0.2222 ⋯ = 2 × 0.1111 ⋯ = 2 × 1 9 = 2 9 0.2222\dots=2 \times 0.1111\dots=2 \times \frac{1}{9}=\frac{2}{9} & 0.5555 ⋯ = 5 × 0.1111 ⋯ = 5 × 1 9 = 5 9 0.5555\dots=5 \times 0.1111\dots=5 \times \frac{1}{9}=\frac{5}{9}

So the answer is 2 9 × 5 9 = 10 81 ≈ 0.12345679 \frac{2}{9} \times \frac{5}{9}=\frac{10}{81} \approx \boxed{\large{0.12345679}}

0 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...