Columnar Multiplication?

Number Theory Level 1

$\large{\begin{array}{cccccc} &0.& 1 & 6& 6 & 6&6& \ldots\\ \times&0. & 3 & 3& 3 & 3&3&\ldots\\ \hline &0. &0 &\bigstar &\bigstar& \bigstar& \bigstar&\ldots\\ \hline \end{array}}$

The above shows the product of the two fractions $\dfrac16$ and $\dfrac13$ when written in decimal representation.

Find the value that represents the symbol $\bigstar$ .

3 4 5 6

3 solutions

Chung Kevin
Apr 15, 2016

The product of the fractions $\dfrac16$ and $\dfrac13$ is simply $\dfrac{1\times1}{6\times3} = \dfrac1{18}$ .

The long multiplication given in question tells us that $0.0\bigstar\bigstar\bigstar\bigstar\ldots = \dfrac1{18}$ .

Before we substitute them, let us convert this non-terminating decimal into a simple form first.

$0.0\bigstar\bigstar\bigstar\bigstar\ldots = 0.0\bigstar + 0.00\bigstar + 0.000\bigstar + 0.0000\bigstar + \cdots$

The expression above represents a geometric progression sum with first term $a = 0.0\bigstar = \dfrac{\bigstar}{100}$ and common ratio $r = \dfrac1{10}$ . This sum is equal to $\dfrac a{1-r} = \dfrac\bigstar{90}$ .

We have $\dfrac\bigstar{90} = \dfrac1{18} = \dfrac{1\times5}{18\times5} = \dfrac5{90}$ , so $\bigstar =\boxed5$ .

I mean, sure, you can convert the decimal to a "simple" form and then use a geometric progression sum and common ratios and divide by 90... Or you could use long division.

Alex Li - 5 years, 1 month ago

Jesse Nieminen
May 5, 2016

$\frac{1}{3} \times \frac{1}{6} = \frac{1}{2} \times \frac{1}{9} = 0.5 \times 0.\overline{1} = 0.0\overline{5} \Rightarrow \bigstar = \boxed{5}$

Moderator note:

Nice approach, converting into fraction form. That is often the best way of interpreting these repeating decimals.

Munem Shahriar
Dec 19, 2017

$\dfrac 16 \times \dfrac 13 = \dfrac 1{18} \approx 0.0\overline{5}$

$\bigstar = 5$

Columnar Multiplication?

3 solutions

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