anemone
Gold Member
MHB
POTW Director
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Hello MHB,
Recently I've come across a problem that jumped off the page at me (Find the first 3 figures after the decimal point in the decimal expression of the number $$\frac{0.12345678910\cdots495051}{0.515049\cdots987654321}$$).
I tried to approach it by making a table where I started to divide some smaller numbers but stick to the same pattern that is required by the aforementioned problem, i.e.
$$\frac{0.12}{0.21}=0.571...$$
$$\frac{0.123}{0.321}=0.383...$$
$$\frac{0.1234}{0.4321}=0.285...$$
$$\frac{0.12345}{0.54321}=0.227...$$
$$\frac{0.123456}{0.654321}=0.188...$$
$$\frac{0.1234567}{0.7654321}=0.161...$$
$$\frac{0.12345678}{0.87654321}=0.140...$$
$$\frac{0.123456789}{0.987654321}=0.124...$$
$$\frac{0.12345678910}{0.10987654321}=0.123...$$
$$\frac{0.1234567891011}{0.1110987654321}=0.111...$$
$$\frac{0.123456789101112}{0.121110987654321}=0.019...$$
$$\frac{0.12345678910111213}{0.13121110987654321}=0.940...$$
$$\frac{0.1234567891011121314}{0.1413121110987654321}=0.873...$$
$$\frac{0.123456789101112131415}{0.151413121110987654321}=0.815...$$
$$\frac{0.12345678910111213141516}{0.16151413121110987654321}=0.764...$$
$$\frac{0.1234567891011121314151617}{0.1716151413121110987654321}=0.719...$$
$$\frac{0.123456789101112131415161718}{0.181716151413121110987654321}=0.679...$$
$$\frac{0.12345678910111213141516171819}{0.19181716151413121110987654321}=0.643...$$
$$\frac{0.1234567891011121314151617181920}{0.2019181716151413121110987654321}=0.611...$$
and so on and so forth
but I failed to observe any pattern that's worth mentioning to help me to crack the problem.Could anyone help me with this particular problem? Thanks in advance.
Recently I've come across a problem that jumped off the page at me (Find the first 3 figures after the decimal point in the decimal expression of the number $$\frac{0.12345678910\cdots495051}{0.515049\cdots987654321}$$).
I tried to approach it by making a table where I started to divide some smaller numbers but stick to the same pattern that is required by the aforementioned problem, i.e.
$$\frac{0.12}{0.21}=0.571...$$
$$\frac{0.123}{0.321}=0.383...$$
$$\frac{0.1234}{0.4321}=0.285...$$
$$\frac{0.12345}{0.54321}=0.227...$$
$$\frac{0.123456}{0.654321}=0.188...$$
$$\frac{0.1234567}{0.7654321}=0.161...$$
$$\frac{0.12345678}{0.87654321}=0.140...$$
$$\frac{0.123456789}{0.987654321}=0.124...$$
$$\frac{0.12345678910}{0.10987654321}=0.123...$$
$$\frac{0.1234567891011}{0.1110987654321}=0.111...$$
$$\frac{0.123456789101112}{0.121110987654321}=0.019...$$
$$\frac{0.12345678910111213}{0.13121110987654321}=0.940...$$
$$\frac{0.1234567891011121314}{0.1413121110987654321}=0.873...$$
$$\frac{0.123456789101112131415}{0.151413121110987654321}=0.815...$$
$$\frac{0.12345678910111213141516}{0.16151413121110987654321}=0.764...$$
$$\frac{0.1234567891011121314151617}{0.1716151413121110987654321}=0.719...$$
$$\frac{0.123456789101112131415161718}{0.181716151413121110987654321}=0.679...$$
$$\frac{0.12345678910111213141516171819}{0.19181716151413121110987654321}=0.643...$$
$$\frac{0.1234567891011121314151617181920}{0.2019181716151413121110987654321}=0.611...$$
and so on and so forth
but I failed to observe any pattern that's worth mentioning to help me to crack the problem.Could anyone help me with this particular problem? Thanks in advance.
