5/12 of the mass of nuts Jeff had is cashew nuts and the rest is peanuts. After he bought another 1/4 kg of cashew nuts and gave away 3/8 kg of peanuts, he had an equal mass of cashew nuts and peanuts. Find the mass of the cashew nuts he had at first.
c = 5m / 12 ... p = 7m / 12
5m/12 + 1/4 = 7m/12 - 3/8
10m/24 + 6/24 = 14m/24 - 9/24 ... 10m + 6 = 14m - 9 ... 15 = 4m ... m = 15/4
c = 75/4 / 12 = 75 / 48 = 25/16 kg
To solve this problem, we can set up an equation based on the given information. Let's denote the mass of cashew nuts Jeff had at first as C and the mass of peanuts as P.
According to the problem, 5/12 of the total mass of nuts Jeff had is cashew nuts. This means that C = 5/12 * (C + P).
After buying another 1/4 kg of cashew nuts, the total mass of cashew nuts becomes C + 1/4.
After giving away 3/8 kg of peanuts, the total mass of peanuts becomes P - 3/8.
The problem states that Jeff had an equal mass of cashew nuts and peanuts after these transactions. Therefore, we can set up the equation:
C + 1/4 = P - 3/8.
Now, let's solve this equation step by step:
Multiply through the equation by 8 to eliminate the denominators:
8C + 2 = 8P - 3.
Rearrange the equation:
8C - 8P = -5.
Using the equation C = 5/12 * (C + P), we can simplify it as:
12C = 5C + 5P.
Rearrange the equation:
12C - 5C = 5P.
Simplify further:
7C = 5P.
Now we have a system of equations:
8C - 8P = -5,
7C = 5P.
To solve this system of equations, we can use either substitution or elimination method.
Let's employ the substitution method:
Solve the second equation for P:
P = (7C) / 5.
Substitute this value of P into the first equation:
8C - 8((7C) / 5) = -5.
Simplify:
8C - (56C / 5) = -5.
Multiply through the equation by 5 to eliminate the fraction:
40C - 56C = -25.
Combine like terms:
-16C = -25.
Divide through the equation by -16:
C = -25 / -16.
Simplify:
C = 25/16.
Therefore, the mass of cashew nuts Jeff had at first is 25/16 kg.
Let's assume the total mass of nuts Jeff had at first is M kg.
According to the given information, 5/12 of the mass are cashew nuts, and the remaining (1 - 5/12) = 7/12 of the mass is peanuts.
So, the mass of cashew nuts at first is (5/12)M kg, and the mass of peanuts is (7/12)M kg.
After Jeff bought another 1/4 kg of cashew nuts, the mass of cashew nuts became (5/12)M + 1/4 kg.
After giving away 3/8 kg of peanuts, the mass of peanuts became (7/12)M - 3/8 kg.
According to the given condition, the mass of cashew nuts became equal to the mass of peanuts:
(5/12)M + 1/4 = (7/12)M - 3/8
Multiply through by 24 to clear the fractions:
10M + 6 = 14M - 9
Subtract 10M from both sides and add 9 to both sides:
6 + 9 = 14M - 10M
15 = 4M
Divide both sides by 4:
M = 15/4
Therefore, the mass of the cashew nuts Jeff had at first was (5/12) * (15/4) kg = (25/8) kg = 3.125 kg.