Solving a Mixture Problem Intuitively and with Algebra

Date: 07/06/2007 at 10:59:49
From: Rick
Subject: mixtures (algebra)

There are 7 litres of turpentine in a mixture of 21 litres of water 
and turpentine.  How many litres of turpentine must be added to make 
a mixture of 75% turpentine?

Date: 07/12/2007 at 10:14:03
From: Doctor Roboto
Subject: Re: mixtures (algebra)

Hello Rick:

Thanks for writing to Dr. Math.  Let's solve this problem in two ways,
first using intuition and second using algebra.

Initially we know that the volume of water in the mixture is 14 liters
(21-7).  Also, we know that the fraction of water in the final mixture
is to be 1/4 of the total volume.  Because the volume of water does
not change, 1/4 of the final volume must equal 14 liters.  This means
the total final volume is (4 x 14) or 56 liters.  The amount of
turpentine to be added can be easily calculated by subtraction.

  amount of turpentine added = 56 liters - 7 liters of initial
  turpentine - 14 liters water

  amount of turpentine added = 56 - 7 - 14 = 35 liters

Problems like this cannot always be solved using intuition.  Let's do
it using algebra.

Start by giving variable names to the unknowns.

  T = the amount of turpentine to add
  V = the final volume after adding turpentine

Now, write down what you know in terms of these unknowns.  We know
that 3/4 of the final volume is equal to the initial volume of
turpentine plus the amount added:

  3/4 V = 7 + T

We also know that the final volume is equal to the initial volume plus
the amount of turpentine added:

  V = 21 + T

Now we have two equations in two unknowns.  Solve these to verify the
answer that we obtained using intuition.

Let me know if you would like any further clarification.

- Doctor Roboto, The Math Forum
  http://mathforum.org/dr.math/