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Difference between revisions of "2017 AIME I Problems/Problem 5"

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==Solution 2==
 
==Solution 2==
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The parts before the decimal points must be equal as must the parts after. Therefore 8<math>a</math> + <math>b</math> = 12<math>b</math> + <math>b</math> and <math>c</math>/8 + <math>d</math>/64 = <math>b</math>/12 + <math>a</math>/144.  Simplifying the first equation gives: <math>a</math> = 3/2<math>b</math>.  Plugging this into the second equation gives 3<math>b</math>/32 = <math>c</math>/8 + <math>d</math>/64.  Multiply by 64: 6<math>b</math> = 8<math>c</math> + <math>d</math>.  <math>a</math> and <math>b</math> are both digits between 1 and 7 (they must be a single digit in base eight) so using <math>a</math> = 3/2<math>b</math>, <math>a</math>,<math>b</math> = (3,2) or (6,4).  Testing these gives that (6,4) doesn't work, and (3,2) gives <math>a</math> = 3, <math>b</math> = 2, <math>c</math> = 1, and <math>d</math> = 4.  Therefore <math>abc</math> = <math>\boxed{321}</math>
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The parts before the decimal points must be equal as must the parts after. Therefore 8<math>a</math> + <math>b</math> = 12<math>b</math> + <math>b</math> and <math>c</math>/8 + <math>d</math>/64 = <math>b</math>/12 + <math>a</math>/144.  Simplifying the first equation gives: <math>a</math> = 3/2<math>b</math>.  Plugging this into the second equation gives 3<math>b</math>/32 = <math>c</math>/8 + <math>d</math>/64.  Multiply by 64: 6<math>b</math> = 8<math>c</math> + <math>d</math>.  <math>a</math> and <math>b</math> are both digits between 1 and 7 (they must be a single non-zero digit in base eight) so using <math>a</math> = 3/2<math>b</math>, <math>a</math>,<math>b</math> = (3,2) or (6,4).  Testing these gives that (6,4) doesn't work, and (3,2) gives <math>a</math> = 3, <math>b</math> = 2, <math>c</math> = 1, and <math>d</math> = 4.  Therefore <math>abc</math> = <math>\boxed{321}</math>

Revision as of 18:25, 8 March 2017

Problem 5

A rational number written in base eight is $\underline{ab} . \underline{cd}$, where all digits are nonzero. The same number in base twelve is $\underline{bb} . \underline{ba}$. Find the base-ten number $\underline{abc}$.

Solution 1

First, note that the first two digits will always be a positive number. We will start with base twelve because of its repetition. List all the positive numbers in base twelve that have equal twelves and ones digits in base 8.

$11_{12}=15_8$

$22_{12}=32_8$

$33_{12}=47_8$

$44_{12}=64_8$

$55_{12}=101_8$

We stop because we only can have two-digit numbers in base 8 and 101 is not a 2 digit number. Compare the ones places to check if they are equal. We find that they are equal if $b=2$ or $b=4$. Evaluating the places to the right side of the decimal point gives us $22.23_{12}$ or $44.46_{12}$. When the numbers are converted into base 8, we get $32.14_8$ and $64.30_8$. Since $d\neq0$, the first value is correct. Compiling the necessary digits leaves us a final answer of $\boxed{321}$


Solution 2

The parts before the decimal points must be equal as must the parts after. Therefore 8$a$ + $b$ = 12$b$ + $b$ and $c$/8 + $d$/64 = $b$/12 + $a$/144. Simplifying the first equation gives: $a$ = 3/2$b$. Plugging this into the second equation gives 3$b$/32 = $c$/8 + $d$/64. Multiply by 64: 6$b$ = 8$c$ + $d$. $a$ and $b$ are both digits between 1 and 7 (they must be a single non-zero digit in base eight) so using $a$ = 3/2$b$, $a$,$b$ = (3,2) or (6,4). Testing these gives that (6,4) doesn't work, and (3,2) gives $a$ = 3, $b$ = 2, $c$ = 1, and $d$ = 4. Therefore $abc$ = $\boxed{321}$

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