Shawn took in $69.15 in two hours work on Saturday selling Belts for $8.05 and earrings for $4.50. How many of each did Shawn sell?
I'm not too sure how you are expected to solve this problem.
You will end up with a Diophantine Equation (one involving integer coefficients and integer solutions only) If you have never heard of Diophantus, then the precise mathematical solution later on should be ignored. Instead ...
The (Diophatine) equation is
161B + 90E = 1363
You should be able to show that Shawn can only sell a maximum of 8 belts (i.e. if he sells no earrings), and a maximum of 15 earrings (i.e. if he sells no belts).
So start with the smaller number (8). Shawn sells up to 8 belts. So put B = 1 in the (Diophantine ) equation and see if B is an integer. If not, then B=1 cannot be a solution, So now try again with B = 2, and so on ..
My precise mathematical solution now follows.
Let B be the number of belts sold @ $8.05 per belt.
Let E be the number of earrings sold @ $4.50 per pair.
Total money made is P = $69.15
Total money made is: B*$8.05 + E*$4.50
Equating the two expressions,
8.05B + 4.50E = 69.15
Making the coefficients as integers we end up with a Diophantine equation.
161B + 90E = 1383
We now manipulate the coefficients 161 and 90. We should end up with a relation between the two that should produce the value 1.
This manipulation is in two parts. In the first line, we are writing the larger coefft, 161, as a multiple of the smaller coefft plus a remainder.
1st Part
161 = 1x90 + 71
90 = 1x71 + 19 now we write the 1st multiple as a multiple of the 1st remainder + remainder
71 = 3x19 + 14 we write the 2nd multiple as a multiple of the 2nd remainder + remainder
19 = 1x14 + 5
14 = 2x5 + 4
5 = 1x4 + 1
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2nd Part
We rewrite that last equation so that 1 is on the left hand side
1 = 5 – 1x4
We now use the remainder from the next equation up.
1 = 5 – 1x(14 – 2x5)
1 = 3x5 – 1x14
And again we use the remainder in the next equation up
1 = 3(19 – 1x14) – 1x14
1 = 3x19 – 4x14
1 = 3x19 – 4(71 – 3x19)
1 = 15x19 – 4x71
1 = 15(90 – 1x71) – 4x71
1 = 15x90 – 19x71
1 = 15x90 – 19(161 – 1x90)
1 = 34x90 – 19x161
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Therefore, 1383 = (-19x1383).161 + (34x1383).90
1383 = -26,277*161 + 57,022*90
Which implies
B = -26,277
E = 47,022
which is obviously incorrect!
But all is not lost. These two values for B and E are simply two values that happen to satisfy the original Diophantine equation.
What we need is the general solution, which now follows.
In general,
B = -26,277 + 90k (using the coefft of E)
E = 47,022 – 161k (using the coefft of B)
If you substitute these expressions into the original Diophantine equation, the k-terms will cancel out, leaving you with the original eqn.
We know that B and E must be smallish numbers and that neither can be negative.
For example, if no earrings were sold, Shawn would need to sell over 8 belts to clear $69, and if no belts were sold, then Shawn would need to sell over 15 earrings.
So Shawn needs to sell between 0 and 8 belts and between 0 and 15 earrings.
Setting k = 292,
B = -26,277 + 90*292 = 3
E = 47,022 – 161*292 = 10
B = 3, E = 10
If k is greater than or less than 292 by 1, or more, then either B will be negative or E will be negative. So this is the answer.
Check
3*8.05 + 10*4.50 = 25.15 + 45.00 = 69.15 – Correct!
Answer: Shawn sells 3 belts and 10 earrings
