Please Help Me With This Word Problem.

Question

A store is having a sale on trail mix and jelly beans. For 3 pounds of trail mix and 5 pounds of jelly beans, the total cost is $22. For 12 pounds of trail mix and 2 pounds of jelly beans, the total cost is $25. Find the cost for each pound of trail mix and each pound of jelly beans.

Answer 1

let TM = price of trail mix per pound

let JB = price of jelly bean per pound

so the price of 3 pounds of trail mix will cost 3*cost of trail mix per pound which equals 3TM

the price of 5 pounds of jelly beans will cost 5*cost of jelly beans per pound which equals 5JB

therefore, from the first given information the total cost of 3 pounds of trail mix and 5 pounds of jelly beans will be 22. The equation will be:

3TM + 5JB = 22

Following the same logic, the second information can be translated to an equation:

12TM + 2JB = 25

Now you get two simultaneous eqations

EQ 1: 3TM + 5JB = 22

EQ2: 12TM + 2JB = 25

since we have two unknowns and two equations, we have to solve one variable interms of the other and substitute that in the second equation. As follows:

Lets take EQ. 1:

3TM + 5JB = 22

3TM + 5JB -5JB = 22- 5JB................................subtracting 5JB from both sides to solve for TM

3TM = 22-5JB

1/3 * (3TM) = 1/3(22-5JB).........................Dividing both sides by 3 to solve for TM

TM = 1/3 (22-5JB).......................Now we have solved for TM interms of JB

Next step is to plug this value instead of TM. We use EQ.2

originally it was 12TM + 2JB = 25

Plug in 1/3* (22-5JB) instead of TM

12*(1/3*(22-5JB)) + 2JB =25

simplify

4*(22-5JB) + 2JB = 25

88-20JB +2JB = 25

88-18JB=25

88-18JB +18JB = 25 +18JB ......................adding 18JB to both sides

88 = 25 +18JB

88-25 = 25-25 +18JB..................subtracting 25 from both sides

63 = 18JB

simplify

JB =3.5

The price of the Jelly beans is $3.5 per pound.

Now we use this value of JB to solve for TM.

Earlier we have solved for TM interms of JB so we use that.

TM = 1/3* (22-5JB)

JB =3.5

TM = 1/3*(22-(5*3.5))

TM = 1/3* (22-17.5)

TM = 1/3* (4.5)

TM = 1.5

The Price of Trail Mix is $1.5 per pound.

To Check plug these value to EQ 1 and EQ 2

EQ 1: 3TM + 5JB = 22

3*(1.5) + 5*3.5) = 22

4.5 + 17.5 =22

22.0 = 22

EQ 2 : 12TM + 2JB = 25

12*(1.5) +2 *(3.5) = 25

18.0 + 7.0 + 25

25.0 = 25