bought in 3 kg boxes that cost Rf 4.50 or in 5 kg boxes that cost Rf 6.58. She needs exactly 17 kg of seed. How many boxes of each size should purchase ?

Question

need answer polyas method

Answer 1

Is there a budget that she must follow of a maximum amount she can spend?

Let's call the budget  $D, the max amount she is allowed to spend, with D>0

 

Let X be the # of 3kg boxes and Y be the number of 5 kg boxes.

 

3X + 5Y = 17

4.50X + 6.58Y = D

 

We now have two equations and two unknown quantities

 

Next step, is solving the first equation for Y and multiplying everything

in the second equation by 100 to clear away the decimals. The system

now looks like this:

 

Y = (17 - 3x)/5

450X + 658Y = 100D

 

Now, it substitutes the first equation into the second

to solve for x in terms of D:

 

450X + 658( 17 - 3x)/5 = 100D

 

Multiplies everything by 5 to clear the fraction

 

2250X + 658( 17 - 3x) = 500D

 

2250X + 11186 - 1974x = 500D

 

276x = 500D - 11186

 

x = (500D - 11186)/276

 

 

 

   = (250D - 5593)/138  <-- reducing by a factor of 2

 

  

Prime factoring 138 = 2 * 69 = 2 * 3 * 23.

So it cannot be reduced any further. In order 

for this value to be positive, D > 22.372

 

We now need to find Y = (17-3x)/5 for this X.

 

3x = (250D - 5593)/46

 

Negating:  -3x = (5593 - 250D)/46 

 

Adding 17 = 17*46/46 = 782/46

 

(782 + 5593 - 250D)/46

 

so Y = (6375 - 250D)/46

 

Prime factoring 46 = 2 * 23, so it cannot be reducid any further

 

In order for this value to be positive, D < 25.5

 

So Y = (6375 - 250D)/46

 

  and X = (250D - 5593)/138

 

 

  where 22.372 < D < 25.5 is the budget amount

 

The following table shows the purchase options and prices for various budget amounts

increasing by a quarter or 25 cents:

 

D                X      Y      cost of X      cost of Y      total cost
22.372         0     17           0             111.86        111.86
22.5     0.231884058 16.30434783 1.043478261 107.2826087 108.326087
22.75    0.684782609 14.94565217 3.081521739 98.3423913 101.423913
23        1.137681159 13.58695652 5.119565217 89.40217391 94.52173913
23.25   1.59057971 12.22826087 7.157608696 80.46195652 87.61956522
23.5     2.043478261 10.86956522 9.195652174 71.52173913 80.7173913
23.75   2.496376812 9.510869565 11.23369565 62.58152174 73.81521739
24        2.949275362 8.152173913 13.27173913 53.64130435 66.91304348
24.25    3.402173913 6.793478261 15.30978261 44.70108696 60.01086957
24.5      3.855072464 5.434782609 17.34782609 35.76086957 53.10869565
24.75    4.307971014 4.076086957 19.38586957 26.82065217 46.20652174
25         4.760869565 2.717391304 21.42391304 17.88043478 39.30434783
25.25     5.213768116 1.358695652 23.46195652 8.940217391 32.40217391
25.5      5.666666667       0                 25.5               0                25.5

 

 

 

 

 

 

 

Answer 2

How about some mathematical reasoning?

4(3 oz) = 12

1(5 oz) =5

12 + 5 = 17