# Shopping Word Problem

#### dbmathis

How would I solve the following?

1 ) a b c d e went shopping
2 ) each had a whole dollar amount to spend
3 ) together they had 56 dollars
4 ) The absolute difference between what a and b had to spend was 19
5 ) The absolute difference between what b and c had to spend was 7
6 ) The absolute difference between what c and d had to spend was 5
7 ) The absolute difference between what d and e had to spend was 4
8 ) The absolute difference between what e and a had to spend was 11
9 ) How much did each have to spend?

I used a spreadsheet to find the solution, but wanted to know the algebra way:
a 21
b 2
c 9
d 14
e 10

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Math Team

#### Otis

Hi dbmathis. Denis' link didn't work for me, but I don't think Wolfram Alpha will answer your question, anyway (unless you pay to see their fancy steps).

When we consider absolute value statements like |a-b|=19, we need to consider both possibilities: Either a is the larger number or b is.

That is, a-b=19 OR b-a=19.

If we knew the increasing order were b,c,e,d,a, then we could write a system of five equations and solve it. In this exercise, we don't know the order. We're given only the distance between five pairs of values and that sum.

I started by noting (of the pairs listed) that a and b are farthest apart. Next, I noted that e is about half as far away from a than b is. So, it seemed natural to sketch e about halfway between a and b, as a first guess.

Next, I started taking cases, using facts that b is 7 units away from c AND d is 5 units away from c. My sketches quickly eliminated a number of possibilities because those results don't jive with what's known about the relative distances of a,b,e.

Eventually, I arrived at the following order and interstitial distances:

bâˆ™âˆ™âˆ™7âˆ™âˆ™âˆ™câˆ™âˆ™âˆ™1âˆ™âˆ™âˆ™eâˆ™âˆ™âˆ™4âˆ™âˆ™âˆ™dâˆ™âˆ™âˆ™7âˆ™âˆ™âˆ™a

I solved the following for b, and the rest followed:

5(b) + 4(7) + 3(1) + 2(4) + 1(7) = 56

~ Cheers

PS: I think this is what Denis was trying to do.