Jumbo Bag
April 23rd, 2010
12. A mathematical market trader packages fruit in three sizes. An Individual bag holds 1 apple and 2 bananas;?
12. A mathematical market trader packages fruit in three sizes. An Individual bag holds 1 apple and 2 bananas; a Jumbo bag holds 4 apples and 3 bananas; and a King-Size bag holds 8 apples and 7 bananas. She draws two vector arrows a and b to represent an apple and a banana respectively, and then represents the three sizes of bag by vectors I=a+2b, J=4a+3b and K=8a+7b. Find numbers s and t such that K=sI+tJ.
By midday she has sold all her King-size bags, but she has plenty of Individual and Jumbo bags left. She decides to make up some more King-size bags by using contents of the other bags. How can she do this so that she has no loose fruit left over?
s * (1,2) + t * (4,3) = u * (8, 7)
Then we have:
s + 4t = 8u
2s + 3t = 7u
Adding those two we get: 3s + 7t = 15u
Subtracting, we get: -s + t = u
-3s + 3t = 3u (multiply second one by 3)
then add those two:
10t = 18u
t / u = 9 / 5
So try u = 5, t = 9.
Then:
s + 4t = 8u –> s + 36 = 40 –> s = 4
2s + 3t = 7u –> 2s + 27 = 35 –> s = 4
So we have s = 4, t = 9, u = 5 (or any multiple thereof).
4 individual bags = 4 apples + 8 bananas
9 jumbo bags = 36 apples + 27 bananas
5 king bags = 40 apples + 35 bananas
How to make plastic jumbo bag.
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