Answer:
- n+d=40; 5n+10d=235; (n, d) = (33, 7)
- x+y=10; 0.4x+0.12y=0.32(10); (x, y) = (7.14, 2.86)
see below for variable definitions
Step-by-step explanation:
You want a system of equations and its solution for each of the given scenarios.
Variables
In general, we can define a variable for each of the values that the problem statement is requesting. It is often convenient to choose a variable name that reminds you what the variable stands for.
In problem 1, you want the number of nickels and the number of dimes. It is convenient to use the variables n and d to represent those numbers, respectively.
In problem 2, you want the numbers of cubic feet of two different grades of soil. In our answer below, we use x and y for the two soil types. We use x for the first type mentioned (40% clay) and y for the second type mentioned (12% clay). These have no obvious relation to the soil types, so we need to be careful in our interpretation of their values.
System of equations
The system of equations you write will reflect the relationships given in the problem statement. Here, those relations involve (a) the total amount represented by the two variables, and (b) the result of the contributions of each.
1. Coins
The equations can be ...
n + d = 40 . . . . . . . the total number of coins
5n +10d = 235 . . . the number of cents in their combined value
Using substitution for n, we have ...
n = 40 -d . . . . . . . . . . . use the first equation to write an expression for n
5(40 -d) +10d = 235 . . . substitute for n
200 +5d = 235 . . . . . . eliminate parentheses
5d = 35 . . . . . . . . . . . subtract 200
d = 7 . . . . . . . . . . . . divide by 5
n = 40 -7 = 33 . . . find n
Isabel has 33 nickels and 7 dimes.
2. Garden
The equations can be ...
x +y = 10 . . . . . . . . . total cubic feet of soil
0.40x +0.12y = 0.32(10) . . . . . . cubic feet of clay in the mix
Using substitution for y, we have ...
y = 10 -x . . . . . . . . . . . . . . . . use the first equation to find y
0.40x +0.12(10 -x) = 3.20 . . . substitute for y
0.28x +1.20 = 3.20 . . . . . . . . eliminate parentheses
0.28x = 2.00 . . . . . . . . . . . . subtract 1.20
x = 2.00/0.28 ≈ 7.14 . . . . . divide by 0.28
y = 10 -7.14 = 2.86 . . . . . . find y
Lucie needs about 7.14 cubic feet of 40% clay soil and 2.86 cubic feet of 12% clay soil.
Conclusion
The coin problem must have an integer solution, as we can only have whole numbers of coins.
The garden problem has a fractional solution. The exact solution is 7 1/7 ft³ and 2 6/7 ft³. We have rounded to 2 decimal places on the assumption that will be sufficient accuracy to determine what to buy at the garden center. We don't know what the garden center might do when we ask for a fraction of a cubic foot. (The required proportion is 5:2, so any mixing method that gives that proportion will work.)
These findings demonstrate that mixture problems can be solved fairly quickly and easily, so working out the details of a mixture does not need to be intimidating. That, perhaps, is the importance in real life.
