Question

By using a Diophantine equation answer the following question. No credit will be given unless you show all your work. 11. 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 of each type did she sell? (10 points)

Answer 1

Let I and the boutique owner sells y number of pullovers ax no. of cardigans LEach " cardigan Pullover cost is $31 Cost is $28, Total revenue = $1460 Lie Sum of Cost of x no. of Cardigans - and cost of y no. of = $1460 - Pullovers 3ix + 28 y = 1460 We know that linear Diophantine egn axtby=c has solutions iff god (a, b) 1 c. Here a=31, b2 28 9 C2 1460 9.2.d (31,28) - 1 and 1/1460 By Euclidean Algorithm 9 31 z 28 X1 + 3x) 28 7 3x9 + IXI – 0 Now - 1 z 3X(-9 ) + 288 I (from ③ ) ( 28X(-1) + 31X1) X(-9) + 28X1 value of 3 from 1 28 X 10__+31 (9) 1 Multhlying by 1460 on both sides, we get 1460 = 28 X (14600)+ 31 X (1460X 1-9

1460 = 28 x (14600) +31X(-13140) - ☺ Comparing 6 by 6 we get 23-13140 , yz 14600 Other solm are given by 2= -13140 - 280 . 14600 + 318 rez JEZI Now a should be so $ y should be > 0 270 le -13140 - 2807 0 x < -410 y 70 le 14600 +310 70 & 87-470 when r=-470, x= -13140-28X(-470) - 20 4 z 14600 4318 (-470) = 30 Solis x = 20 gy=30 Batique owner sells 2o cardigans $ 30 pullovers,