SOLUTION: Bianca and her father picked 109 apples. Bianca's father picked 13 more apples than triple the apples Bianca picked. How many apples did each pick?

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Question 1173010: Bianca and her father picked 109 apples. Bianca's father picked 13 more apples than triple the apples Bianca picked. How many apples did each pick?
Found 2 solutions by mccravyedwin, ikleyn:
Answer by mccravyedwin(407) About Me  (Show Source):
You can put this solution on YOUR website!
Ikleyn below solved the problem using one unknown.  I solve it below using
two unknowns. Both ways are equally correct and give the same answers.

As a rule, using 1 unknown is usually SHORTER, but using 2 unknowns is EASIER
to set up.  So often it is a "give-and-take" situation.

Bianca and her father picked 109 apples. 
Bianca picked x apples.
Her father picked y apples. So,

x + y = 109
Bianca's father picked 13 more apples than triple the apples Bianca picked.
So,

y = 3x + 13

The system of equations to solve is

system%28x%2By=109%2Cy=3x%2B13%29

Substituting (3x+13) for y in the first equation:

x%2B%283x%2B13%29=109

x%2B3x%2B13=109

4x%2B13=109

4x=96

x=24   <--so Bianca picked 24 apples

Substitute 24 for x in y=3x+13

y=3%2824%29%2B13

y=85   <--so Bianca's father picked 85 apples.

Edwin

Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.

    Bianca  +  Father    = 109    (in total)


      B     +  (3B + 13) = 109


      B     + 3B   + 13  = 109


         4B              = 109 - 13

         4B              = 96

          B              = 96/4 = 24.


ANSWER.  Bianca picked 24 apples.  The father picked 3*24 + 13 = 85 apples.


CHECK.  24 + 85 = 109, in total.   ! Precisely correct !

Solved.

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Which approach to use - one equation or two equations ? ? ?

It mostly depends on the age, level/grade and readiness of the student . . .

Young students may not know systems of two equations - - - then only one equation approach does work.

As the student becomes older, he (or she) is able to adopt the two equations approach.

I always try to guess from the context, which age and which level the student is - and react accordingly.

A mature student, ideally, should know both approaches; freely manipulate with either approach
and understand when and why they both are equivalent.

Then the issue on which approach to use is only the question of taste and traditions . . .


With two small additions. 

1.  One equation approach gives the opportunity for earlier education and 

    earlier involvement of a young student to problems solving - - - comparing with the two-equation approach.



2.  There is a class of problems, which are, from the first glance, for 3-equation approach,

    but actually can be easily solved using 1-equation approach.


    For this class of problems, it is CONCEPTUALLY IMPORTANT to teach young students to make right setup.

    It is the moment, when right teaching really helps to built young minds in a right way.