Lesson INTRODUCTION - Mixture Questions
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<b></b> <b>Introduction</b> These types of "English into Algebrese" questions seem to cause students a lot of problems. In essence, the approach to use is: <b>INITIAL STATE + ADDITION = FINAL STATE</b> SO, lets look at an example. <b>EXAMPLE</b> <b>Q: </b>A vat contains 90kg of powdered milk and 10kg of sugar. How much more sugar must be added to the mixture to make a 25% sugar mix? <b>A: </b>Consider the sugar, as that is what the question is about. <b>Initially: </b>the total amount in the mixture is (10+90) --> 100kg. We therefore get that (10/100) is the fraction of sugar, initially. Now add some sugar, Let this be x. This will increase the amount of sugar and also the total amount of the mixture, giving: <b>Finally: </b>(10+x)/(100+x) is the new fraction of sugar. We are told that the final fraction of sugar is 0.25 (--> the 25%: ie 25 out of 100) So, we have {{{(10+x)/(100+x) = 0.25}}} {{{10 + x = 0.25(100+x)}}} {{{10 + x = 25 + 0.25x}}} {{{10 + 0.75x = 25}}} {{{0.75x = 15}}} {{{x = 15/0.75}}} --> x = 20kg and now <b>CHECK</b> that it makes sense: final mix is {{{(10+x)/(100+x)}}}, which becomes {{{(10+20)/(100+20)}}} {{{(30)/(120)}}} --> 1/4 --> 25% so therefore correct. <b>EXAMPLE</b> <b>Q: </b>How many kilogram of fine glucose must be added to 25 kilograms of a 12% sweetened juice concentrate to prepare instead a light syrup 20% sweet? <b>A: </b>Consider the glucose, as that is what the question is about. <b>Initially: </b>there is 12% glucose in the mixture. This is 12% of 25kg. In other words, (12/100)*25 --> 3kg of glucose. So, we have 3/25 as the initial fraction of glucose --> the 12%. We then add some more. Let this be xkg. <b>Finally: </b>we end up with {{{(3+x)/(25+x)}}} as the fraction of glucose. We are told that this needs to be 20% ie 0.20. So we have {{{(3+x)/(25+x) = 0.20}}} {{{3 + x = 0.20(25+x)}}} {{{3 + x = 5 + 0.20x}}} {{{3 + 0.80x = 5}}} {{{0.80x = 2}}} --> x = 2/0.80 --> x = 2.5kg Now, do the <b>CHECK</b> we had {{{(3+x)/(25+x)}}}, which becomes {{{(3+2.5)/(25+2.5)}}} {{{(5.5)/(27.5)}}} --> 0.20 which is 20%.
