<PRE><B>Question 1209504</B>
&lt;br&gt;
Another bizarre and incorrect AI solution from that pseudo-tutor....&lt;br&gt;
&quot;Since they had the same amount of money left, the fractions of money they spent must be equal&quot;&lt;br&gt;
That makes no logical sense.  And, besides, the statement of the problem tells us the fractions of their money that each spent, and they are not equal.&lt;br&gt;
Following is a logical and correct solution from a human tutor....&lt;br&gt;
Let J and G respectively represent the amounts of money Jamie and Grace started with.&lt;br&gt;
Let x be the cost of each diary; then the cost of each book was x+$3.30.&lt;br&gt;
Jamie spent 3/4 of her money, so she finished with 1/4 of what she started with.&lt;br&gt;  Grace spent 5/9 of her money, so she finished with 4/9 of what she started with.&lt;br&gt;
They finished with the same amounts of money:&lt;br&gt;
[1] {{{(1/4)J=(4/9)G}}}&lt;br&gt;
Jamie spent 3/4 of her money on 3 books:&lt;br&gt;
[2] {{{(3/4)J=3(x+3.30)}}}&lt;br&gt;
Grace spent 5/9 of her money on 2 diaries:&lt;br&gt;
[3] {{{(5/9)G=2(x)}}}&lt;br&gt;
The amount of money Jamie had left was {{{(1/4)/(3/4)=1/3}}} of what she spent:&lt;br&gt;
[4] {{{(1/3)(3(x+3.30))=x+3.30}}}&lt;br&gt;
The amount of money Grace had left was {{{(4/9)/(5/9)=4/5}}} of what she spent:&lt;br&gt;
[5] {{{(4/5)(2(x))=(8/5)x}}}&lt;br&gt;
The amounts they ended up with were the same:&lt;br&gt;
[6] {{{x+3.30=(8/5)x}}}
{{{3.30=(3/5)x}}}
{{{x=5.50}}}&lt;br&gt;
The cost of each diary was x=$5.50; the cost of each book was x+$3.30 = $8.80.&lt;br&gt;
Jamie ended up with x+$3.30 = $8.80; that was 1/4 of what she started with, so she started with 4($8.80) = $35.20.&lt;br&gt;
Grace ended up with was (8/5)x = $8.80; that was 4/9 of what she started with, so she started with (9/4)($8.80) = 9($2.20) = $19.80.&lt;br&gt;
The total the two of them started with together was $35.20+$19.80 = $55.&lt;br&gt;
ANSWER: $55&lt;br&gt;
Summary....
Jamie started with $35.20 and spent 3/4 of it ($26.40) on 3 books costing $8.80 each, finishing with 1/4 of $35.20, or $8.80.
Grace started with $19.80 and spent 5/9 of it ($11.00) on 2 diaries costing $5.50 each, finishing with 4/9 of $19.80, or $8.80.
The two of them finished with equal amounts.&lt;br&gt;
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