Question 1208603
14 is added to 2\3 of a number. The result is 1 whole number 1\4 times the
original number.find the numbers.
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That is not grammatical. Something is wrong or missing.

Tutor Ikleyn assumed "1 whole number 1\4" meant "1 PLUS 1/4" or 5/4.
IOW, she interpreted it to mean this:

<b>14 is added to 2/3 of a number. The result is {{{1&1/4}}} times the original number. Find the number.</b>

With her interpretation, she obtained the answer as 24.

Notice that the student wrote "find the numbers" (plural) but she 
only found only one number, and changed "numbers" to "number".

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Tutor josgarithmetic(39489) interpreted it as though the "1\4 times" was
reversed and should be "times 1/4".  He interpreted it as:

<b>14 is added to 2/3 of a number. The result is 1 whole number times 1/4 the
original number. Find the numbers.</b>  

He gave an example where the original number is -84 and the whole number is 2.
There are 6 solutions in all for his interpretation:

(-84,2), (-21,0), (6,12), (24,5), (42,4), (168,3)

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I will assume that the student inadvertently omitted the word PLUS.
So I'll do it with this interpretation:

<b>14 is added to 2/3 of a number. The result is 1 whole number PLUS 1/4 times
the original number. Find the numbers.</b>

Let x be the original number and let y be the 1 whole number.

{{{expr(2/3)x+14=y+expr(1/4)x}}}

Solve for y, which must be a whole number:

{{{expr(2/3)x+14-expr(1/4)x=y}}} 

{{{(2/3-1/4)x+14=y}}}

{{{(8/12-3/12)x+14=y}}}

{{{expr(5/12)x+14=y}}}

Since y must be a whole number, x must be a multiple of 12
to cancel the denominator in the first term.

Let x = 12k

{{{expr(5/12)(12k)+14=y}}}

{{{5k+14=y}}}

Whole numbers are non-negative integers, so

{{{y=5k+14>=0}}}

{{{5k>=-14}}}

{{{k>=-14/5=-2.8}}}

k is an integer, so {{{k>=-2}}}

So the infinite set of solutions is 

<b>{ (12k, 5k+14) | k <u>></u> -2 }</b>.

Some examples of solutions are:

(-24,4), (-12,9), (0,14), (12,19), (24,24), (36,29), (48,34), ...

Edwin</pre>