The price of 10 citrons and 11 fragrant wood apples is 62 units. The price of 11 citrons and 10 fragrant wood apples is 64 units. Find the price of a citron and the price of a wood apple.
citron: 4 units
wood apple: 2 units
To solve this problem, we can set up a system of equations based on the given information.
Let's assume the price of a citron is "c" units, and the price of a fragrant wood apple is "w" units.
From the first sentence, we know that the price of 10 citrons and 11 fragrant wood apples is 62 units. Using the variables we defined, this can be represented as:
10c + 11w = 62 ----(Equation 1)
From the second sentence, we know that the price of 11 citrons and 10 fragrant wood apples is 64 units:
11c + 10w = 64 ----(Equation 2)
Now, we have a system of two equations with two variables. We can solve these equations simultaneously to find the values of "c" and "w".
Method 1: Substitution Method
We can solve this system of equations using the substitution method:
1. Solve Equation 1 for c in terms of w:
10c = 62 - 11w
c = (62 - 11w)/10
2. Substitute this value of c into Equation 2:
11(62 - 11w)/10 + 10w = 64
Simplify:
(682 - 121w)/10 + 10w = 64
Multiply both sides by 10 to get rid of the fractions:
682 - 121w + 100w = 640
Combine like terms:
-21w = -42
Divide both sides by -21:
w = 2
3. Substitute the value of w = 2 into either Equation 1 or 2. Let's use Equation 1:
10c + 11(2) = 62
Simplify:
10c + 22 = 62
Subtract 22 from both sides:
10c = 40
Divide both sides by 10:
c = 4
Therefore, the price of a citron is 4 units and the price of a fragrant wood apple is 2 units.
Method 2: Elimination Method
We can also solve this system using the elimination method:
1. Multiply Equation 1 by 11 and Equation 2 by 10 to eliminate w:
110c + 121w = 682 ----(Equation 1')
110c + 100w = 640 ----(Equation 2')
2. Subtract Equation 2' from Equation 1' to eliminate c:
(110c + 121w) - (110c + 100w) = 682 - 640
Simplify:
21w = 42
Divide both sides by 21:
w = 2
3. Substitute the value of w = 2 into either Equation 1 or 2. Let's use Equation 1:
10c + 11(2) = 62
Simplify:
10c + 22 = 62
Subtract 22 from both sides:
10c = 40
Divide both sides by 10:
c = 4
Thus, the price of a citron is 4 units, and the price of a fragrant wood apple is 2 units.
Both methods yield the same result: the price of a citron is 4 units, and the price of a fragrant wood apple is 2 units.
10x+11y = 62
11x+10y = 64
Multiply Eq1 by 11 and Eq2 by 10 and sub. Eq2 from Eq1.
110x+121y = 682
110x+100y = 640
Diff.: 21y = 42
Y = 2 fragrants.
In Eq1, replace y with 2 and solve for x:
10x+11*2 = 62
X =