Who Speaks In Different Bases?

Number Theory Level 2

Jam, Jim, and John count in binary, hexadecimal and decimal respectively.

Jam says "I have 101 trading cards."
Jim says "Me too!"
And John says "So do I!"

In decimal representation, how many do they have altogether?

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The answer is 363.

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Geoff Pilling by Geoff Pilling , 53, USA

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2 solutions

Chung Kevin
Apr 14, 2016

The total number of trading cards Jam has is 10 1 2 101_2 .
The total number of trading cards Jim has is 10 1 16 101_{16} .
The total number of trading cards Jim has is 10 1 10 101_{10} .

Let's convert all of these numbers to decimal numbers.

10 1 2 = 1 ( 2 2 ) + 0 ( 2 1 ) + 1 ( 2 0 ) = 5 10 1 16 = 1 ( 1 6 2 ) + 0 ( 1 6 1 ) + 1 ( 1 6 0 ) = 257 10 1 10 = 1 ( 1 0 2 ) + 0 ( 1 0 1 ) + 1 ( 1 0 0 ) = 101 \begin{array} {l c l l } 101_2 &=& 1(2^2) + 0(2^1) + 1(2^0) &= 5 \\ 101_{16} &=& 1(16^2) + 0(16^1) + 1(16^0) &= 257 \\ 101_{10} &=& 1(10^2) + 0(10^1) + 1(10^0) &= 101 \end{array}

Thus, in decimal representation, the total number of trading cards Jam, Jim and John have is

5 + 257 + 101 = 363 . 5 + 257 + 101 = \boxed{363} \; .

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Geoff Pilling
Apr 12, 2016

101 equals 257 in hexadecimal, 101 in decimal, and 5 in binary. So all together they have 257+101+5 = 363 trading cards.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

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