Question 1197826
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p = number of pears moved from B to A
m = number of mangoes moved from B to A
These variables represent positive whole numbers {1,2,3,...}


After moving p of the pears and m mangoes from crate B to crate A, we have 207+p pears out of 207+176+p+m = 383+p+m total fruit in crate A.
The ratio (207+p)/(383+p+m) is set equal to 0.40 since the new percentage of pears in crate A is 40%.


Let's solve for the variable m.
(207+p)/(383+p+m) = 0.40 
207+p = 0.40*383+0.40*p+0.40*m 
207+p = 153.2+0.40p+0.40m
0.40m = 207+p-153.2-0.40p 
0.40m = 0.6p + 53.8
m = (0.6p + 53.8)/0.40
m = (0.6p)/0.40 + (53.8)/0.40
m = 1.5p + 134.5
We'll use this in a substitution step later.


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After moving p of the pears and m of the mangoes from B to A we have 293-p pears left in crate B
This is out of 293+274-p-m = 567-p-m total fruit in crate B.


We want this ratio
(293-p)/(567-p-m)
to be set equal to 0.70 to reflect the fact 70% of the fruit left in crate B are pears.


So we form this equation
(293-p)/(567-p-m) = 0.70


Let's replace m with 1.5p+134.5
This is what I referred to in the substitution step.
(293-p)/(567-p-m) = 0.70
(293-p)/(567-p-(1.5p+134.5)) = 0.70
(293-p)/(567-p-1.5p-134.5) = 0.70
(293-p)/(-2.5p+432.5) = 0.70


We're left with one variable in which we can solve for p.
(293-p)/(-2.5p+432.5) = 0.70
293-p = 0.70(-2.5p+432.5)
293-p = 0.70(-2.5p)+0.70(432.5)
293-p = -1.75p+302.75
-p+1.75p = 302.75-293
0.75p = 9.75
p = 9.75/0.75
<font color=red>p = 13</font>
Rob moved <font color=red>13 pears</font> from crate B to crate A.


Use this value of p to find m.
m = 1.5p + 134.5
m = 1.5*13 + 134.5
m = 19.5 + 134.5
<font color=red>m = 154</font>
He also moved <font color=red>154 mangoes</font> from crate B to crate A.


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Let's check the answers.<table border = "1" cellpadding = "5"><tr><td></td><td colspan=2>Crate A</td><td colspan=2>Crate B</td></tr><tr><td></td><td># of pears</td><td># of mangoes</td><td># of pears</td><td># of mangoes</td></tr><tr><td>Before moving</td><td>207</td><td>176</td><td>293</td><td>274</td></tr><tr><td>After moving</td><td>207+13 = 220</td><td>176+154 = 330</td><td>293-13 = 280</td><td>274-154 = 120</td></tr></table>
which cleans up a bit to this<table border = "1" cellpadding = "5"><tr><td></td><td colspan=2>Crate A</td><td colspan=2>Crate B</td></tr><tr><td></td><td># of pears</td><td># of mangoes</td><td># of pears</td><td># of mangoes</td></tr><tr><td>Before moving</td><td>207</td><td>176</td><td>293</td><td>274</td></tr><tr><td>After moving</td><td>220</td><td>330</td><td>280</td><td>120</td></tr></table>The table format is optional, but I find it helps to keep all the numbers organized. 


After moving 13 pears and 154 mangoes from B to A, there are:
220 pears out of 220+330 = 550 fruit in crate A.
Then note how 220/550 = 0.40 = 40% to show that 40% of the fruit in crate A are pears.


So far so good.


After moving 13 pears and 154 mangoes from B to A, there are:
280 pears left out of 280+120 = 400 fruit left in crate B.
280/400 = 0.70 = 70%
Showing that 70% of the fruit left in crate B are pears.


The answers are fully confirmed.



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Answers:
He moved <font color=red>13 pears</font> and <font color=red>154 mangoes</font> from crate B to crate A.
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