Question 464882
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Let *[tex \Large f] represent the father's age.  Let *[tex \Large m] represent the mothers age, and let *[tex \Large c_1,\ \ ]*[tex \Large c_2,\ \ ]*[tex \Large c_3,\ \ ]and *[tex \Large c_4] represent the ages of child 1, 2, 3, and 4.


The first thing we know is that:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f\ +\ m\ +\ c_1\ +\ c_2\ +\ c_3\ +\ c_4\ =\ 140] (Eq 1)


The next thing we are told is something about the situation 10 years ago:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f\ -\ 10\ +\ m\ -\ 10\ +\ c_1\ -\ 10\ +\ c_2\ -\ 10\ =\ 140\ -\ 55]


Which can be simplified to:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f\ +\ m\ +\ c_1\ +\ c_2\ =\ 125] (Eq 2)


Then we are told that


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f\ =\ 10c_4] (Eq 3)


and


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ m\ =\ 8c_4] (Eq 4)


and


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ c_2\ -\ (c_3\ +\ c_4)\ =\ 0] (Eq 5)


and finally:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ c_1\ -\ (c_2\ +\ c_4)\ =\ 0]  (Eq 6)


Notice the handy (and very necessary) fact that we have 6 equations to match our 6 variables.


Start by combining Equations 1 and 2.  Multiply (2) by -1 and add the result to (1) with the following result:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ c_3\ +\ c_4\ =\ 15] (Eq 7)


Which can be written as:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ c_3\ =\ 15\ -\ c_4] (Eq 8)


Substituting the value of *[tex \Large c_3\ +\ c_4] from Equation 7 into Equation 5, we can deduce that:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ c_2\ =\ 15] (Eq 9)


Using the value of *[tex \Large c_2] from Equation 9 in Equation 6, we can deduce that


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ c_1\ =\ c_4\ +\ 15] (Eq 10)


Notice that we now have an expression for each of the variables *[tex \Large f,\ \ ]*[tex \Large m,\ \ ]*[tex \Large c_1,\ \ ]*[tex \Large c_2,\ \ ]and *[tex \Large c_3] in terms of *[tex \Large c_4], (Equations 3, 4, 10, 9, and 8) so make all of the appropriate substitutions:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 10c_4\ +\ 8c_4\ +\ c_4\ +\ 15\ +\ 15\ +\ 15\ -\ c_4\ +\ c_4\ =\ 140]


The rest is simply solving the single variable equation for the value of *[tex \Large c_4] to get the youngest child's age and then multiplying by 10 to get the father's age.


John
*[tex \LARGE e^{i\pi} + 1 = 0]
My calculator said it, I believe it, that settles it
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