algebra word problem

Question

Before last weekend's hiking trip, Juanita mixed 3 kg of peanuts and raisins as an energy snack. The peanuts cost $4.25 per kilogram and the raisins cost $3.50 per kilogram. The whole mix cost $12. How many kilograms of peanuts did Juanita have?

Answer 1

To answer this question, you first need to determine the two separate equations.

1) The total amount of trail mix made

1. peanuts + raisins = total
2. p + r = total (using variables)
3. p + r = 3 (based on the word problem)

2) The cost

1. peanuts + raisins = total
2. 4.25(p) + 3.5(r) = 12 (based on word problem)

Now you need to solve equation for a variable & substitutie into the other equation

1. p + r = 3 (subtract r from both sides)
2. p = 3 - r

Substitute this equation into the second one

1. 4.25(p) + 3.5(r) = 12
2. 4.25 (3 - r) + 3.5(r) = 12 (distribute the 4.25)
3. 12.75 - 4.25r + 3.5r = 12 (combine like terms)
4. 12.75 -.75r = 12 (subtract 12.75 from both sides)
5. -.75r = -.75 (divide each side by -.75)
6. r = 1 *We know that the trail mix had 1 kg of raisins.

Now to find the number of peanuts, we return to our first equation

1. p + r = 3 (r = 1)
2. p + 1 = 3 (subtract 1 from both sides)
3. p = 2 * There are 2 kgs of peanuts in the trail mix.

To check our work, we substitute p & r back into the second equation (This isn't required but a good way to check everything)

1. 4.25(p) + 3.5(r) = 12
2. 4.25(2) + 3.5(1) = 12 (multiple)
3. 8.5 + 3.5 = 12
4. 12 = 12 *this proves we did the math correctly!

Therefore, we are confident that there are 2 kg of peanuts in the trail mix that Juanita bought.