SOLUTION: If 123y=83ten, find the value of y

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Question 1208439: If 123y=83ten, find the value of y
Found 4 solutions by josgarithmetic, ikleyn, math_tutor2020, greenestamps:
Answer by josgarithmetic(39618) About Me  (Show Source):
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123y=83ten
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Not sure how that is meant. It looks like instead of spelling a word, "ten", those are three separate variables.

123y%281%2F123%29=83ten%281%2F123%29
highlight%28y=%2883%2F123%29ten%29


But you asked for the "value" of y.
Could the "ten" really had been meant as factor, 10 ?

y=%2883%2F123%29%2A10=6.7479674796747967479674796... and the repeating digits continue...

Answer by ikleyn(52788) About Me  (Show Source):
You can put this solution on YOUR website!
.

83ten is an unknown and an unreadable mathematical hieroglyph.


Its meaning is dark,  turning the whole task into dust and making the solution impossible.

Be more accurate when posting to this forum.


Have a nice day  ( ! )



Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

If you are trying to solve this equation
123%5By%5D+=+83%5B10%5D
then it looks like you're trying to find the value of 123 base y when it is equal to 83 base 10.
y is an integer such that y+%3E=+2

123%5By%5D+=+1%2Ay%5E2%2B2%2Ay%5E1%2B3%2Ay%5E0+=+y%5E2%2B2y%2B3

123%5By%5D+=+83%5B10%5D

y%5E2%2B2y%2B3+=+83

y%5E2%2B2y%2B3-83+=+0

y%5E2%2B2y-80+=+0

%28y%2B10%29%28y-8%29+=+0

y%2B10+=+0 or y-8+=+0

y+=+-10 or y+=+8
Ignore the negative value since y+%3E=+2

y+=+8 is the only practical solution.

We go from 123%5By%5D+=+83%5B10%5D to 123%5B8%5D+=+83%5B10%5D
123 base 8 = 83 base 10

Verification using WolframAlpha
https://www.wolframalpha.com/input?i=123+base+8+%3D+83+base+10

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


123 base y = 83 base 10

You can solve the problem using formal algebra if you want, as tutor @Edwin did.

You can also solve the problem informally, using logical reasoning and basic arithmetic.

123 base y, with leading digit 1, is a bit more than 1*y^2 = y^2.
Since the numbers in both bases are whole numbers, we can assume the base y is a positive integer.
8^2 = 64 is less than 83; 9^2 = 81 is still less than 83, but it is too close to 81 to be a reasonable answer.

So make the logical guess that y is 8 and see if it works.

123 base 8 = 1(8^2)+2(8)+3 = 64+16+3 = 83

ANSWER: y = 8