Solve the word problems by writting it down as a system of equations and as a matrix.

Question

1)      To raise money a group of three friends is each trying to sell 100 candles each friend has a package containing the same number of tapers votives and tealights the tapers sell for $6.00 each the votives for $1.50 each and the tealights sell for $-.75 each. Maya sold all of the candles and raised $431.25 Lorrin sold all of the tapers and tealights to raise $378.75 while sara sold all the votives all the tealights and half the tapers to raise $251.25. How many of each type of candle did each package contain? 

 

2)      A health shop owner made trail mix containing dried fruit, nuts and carob chips. The dried fruit sells for $5.50 per pound, the nuts sell for $7.50 per pound and the carob chips sell for $8.50 a pound. The shop owner mixed 50 pound of trail mix and sells it for $6.70 per pund. If the amount of nuts is five pounds more than the amount of carob chips, how much of each item was used in the trail mix?

 

 

 

Answer 1

1)  Let's pick variables for each type of candle:

     t = tapers,  v = votives,  L = tea lights

The amount of money made from each type of candle sales will be

6t, 1.5v, .75L  since tapers are $6, votives are $1.50, and tea lights are $.75

 

Maya:    6t + 1.5v + .75L = 431.25   eqn 1

Lorrin:   6t            + .75L = 378.75   eqn 2

Sara:    1.5v         + .75L = 251.25   eqn 3

 

We have 3 equations with 3 unknowns.

 

From equation 3:  L = (251.25-1.5v)/.75

 

Substitute that value of L into equation 2:

6t + .75[(251.25-1.5v)/.75] = 378.75

6t + 251.25 - 1.5v = 378.75

6t - 1.5v = 127.5

t = (127.5+1.5v)/6

 

Substituting into equation 1:

6[(127.5+1.5v)/6] + 1.5v + .75[(251.25-1.5v)/.75] = 431.25

127.5 + 1.5v + 1.5v + 251.25 - 1.5v = 431.25

378.75 + 1.5v = 431.25

1.5v = 52.5

v = 35

 

t = (127.5 + 1.5(35))/6 = 30

 

L = (251.25-1.4(35))/.75 = 265

 

Each box contained 35 votive candles, 30 tapers, and 265 tea lights

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2.  Again, pick variables for each type of ingredient

    f = pounds of fruit, n = pounds of nuts, c = pounds of carob chips

 

    the money made from a 50 pound bag of mix is 50(6.7) = $335

 

eqn 1:  f + n + c = 50   the bag contained 50 pounds of trail mix

eqn 2:  n = c + 5          there were 5 more pounds of nuts than carob chips

 

Substitute c+5 in place of n inside equation 1:

f + c+5 + c = 50

f + 2c = 45

f = 45-2c

 

Given the cost of ingredients:

5.5f + 7.5n + 8.5c = 335

 

5.5(45-2c) + 7.5(c+5) + 8.5c = 335

 

247.5 - 11c + 7.5c + 37.5 + 8.5c = 335

5c + 285 = 335

5c = 50

c = 10

 

n = c+5 = 15

 

f = 45-2 = 25

 

The trail mix contained 25 founds of dried fruit, 15 pounds of nuts,

and 10 pounds of carob chips.

 

Hope this helps.  Good luck with your work.