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Question

Dear Mitch,

Since it's getting close to the last day of school before the Thanksgiving break, could you post the "explanatory answer" that you alluded to in your earlier post about the Thanksgiving balance-scale problem ?

Thank you,

Mr. Willis 

Answer

Dear Mr. Willis,

Sure, here it is!

First, the answer: 

ANSWER:  9 pieces of Pumpkin Pie.

Next, it's worth reiterating what the question was:

Three balance scales were shown, scale A, scale B, and scale C.  Scale A was balanced, and scale B was balanced, but scale C was not...

Scale C showed one turkey and one gravy boat on its left-hand side, and nothing on its right-hand side. 

The question asked us to balance out the left-hand side of scale C by placing only pieces of pumpkin pie on the right-hand side.

Altogether, there were three kinds of items on the scales, pieces of pie, turkeys, and gravy boats.  

Using letters instead of words, to simplify things and represent items in the way we try to do in mathematics, let:

T = 1 turkey

G = 1 gravy boat

P = 1 piece of pie 

Scale A showed 1 T + 1 P equaling 1 G.  (By 'balanced', we mean "equal" in math, just as 3 + 4 is 'balanced' with 7.

Scale B showed 3 T's equaling 2 G's + 2 P's. 

These balance-scale problems involving more than one scale always represent what we call a 'system of equations' in math, which means we can use any information that we can grasp from one scale to help us figure out what we need by using the arrangements on the other scales.  

So, since scale A tells us that (1 T + 1 P) = 1 G, we can replace each G on scale B with (1 T + 1 P).  We get:

3 T's = P + P + (T + P) + (T + P)

Simplifying that by grouping together all the P's and then grouping together all the T's, we get: 

3 T's = 4 P's + 2 T's

Next, we can remove two T's from each side of that balance, because if one combination is balanced with another combination on a balance scale, then we can keep things in balance by either adding the same items to both sides, or by subtracting the same items from both sides.  S we get:

T = 4 P's

So on scale C we can balance the T with 4 P's.

Since scale A shows us that 1 G = T + P, and we know that T = 4 P's, then replacing the T in scale A with 4 P's we learn that 5 P's = 1 G.

So, to balance the T and G on the left-hand side of scale C with only P's, we need 5 P's for the G, and we need 4 P's for the T, which means we would need 9 P's altogether to balance the 1 T and the 1 G, which is the symbolic answer to what the question asked!

Now, translating that answer back to the world of words, we get: 

ANSWER: 9  pieces of pie.

 

I hope this explanation helped!

Happy holidays,

Mitch