Question 419129
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Another tutor has provided a standard formal algebraic solution to the problem.  But the numbers in this problem make it a good one for solving informally using logical reasoning instead of formal algebra.<br>
Since equal amounts of the glues worth $150 per barrel and $190 per barrel are to be used, the average cost of those two ingredients is halfway between those two prices, which is $170 per barrel.<br>
Then the problem becomes mixing glue worth $100 per barrel with glue worth $170 per barrel to get glue worth $120 per barrel.  The ratio in which those two ingredients must be mixed is exactly determined by where the $120 price lies between the $100 and $170 prices.<br>
The difference between $100 and $120 is $20; the difference between $120 and $170 is $50.  Those differences mean the two ingredients must be mixed in the ratio 20:50 = 2:5.<br>
Because $120 is closer to $100 than it is to $170, the larger portion must be the ingredient worth $100 per barrel.  So the mixture needs to be 5 parts of the glue worth $100 per barrel and 2 parts of the glue worth $170 per barrel.<br>
Given that there are 150 barrels of glue worth $100 per barrel, use a proportion to find that we need 60 barrels of the glue worth $170 per barrel:<br>
{{{150:5=x:2}}}
{{{5x=300}}}
{{{x=60}}}<br>
Then, since the glue worth $170 per barrel is equal amounts of the glues worth $150 and $190 per barrel, there must be 30 barrels each of the $150 per barrel glue and the $190 per barrel glue.<br>
ANSWERS:
(1) 30 barrels each of the $150 per barrel and $190 per barrel glue will be used
(2) The total number of barrels of glue will be 150+30+30 = 210<br>