Question

By using a Diophantine equation answer the following question. No credit will be given unless you show all your work. ll . A boutique owner sells two styles of sweaters: cardigan and pullover. She sells cardigans for $31 each and pullovers for $28 each. If her total revenue from a day is $1460, how many ofeach type did she sell

Answer 1

Answer:

• 20 cardigans
• 30 pullovers

Step-by-step explanation:

Let c and p represent the numbers of cardigans and pullovers sold, respectively. Then the revenue equation is ...

  31c +28p = 1460

Solving for p, we have ...

  p = (1460 -31c)/28 = 52 -c +(4 -3c)/28

Now, define ...

  a = (4 -3c)/28

and solve for c:

  (4-28a)/3 = c = 1 -9a +(1 -a)/3

At this point, we can define ...

  b = (1 -a)/3

Working backward, we can find c and p.

  a = 1 -3b

  c = 1 -9a +b = 1 -9(1 -3b) +b = -8 +28b

  p = 52 -c +a = 52 -(-8 +28b) +(1 -3b) = 61 -31b

That is, for some integer b, solutions to the equation will be ...

  c = 28b -8

  p = 61 -31b

The only value of b that gives positive solutions for both c and p is b=1. For that value of b, we have ...

  c = 20, p = 30

The boutique owner sold 20 cardigans and 30 pullovers.

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Comment on the solution

This is a variation of the Extended Euclidean Algorithm. Here, we have defined intermediate variables (a and b) that might not show up in a tableau solution form.