Find three mixed numbers so that the sum is 18 and the difference between the greatest number and the least number is 5 1/5.
Let's assign variables to the three mixed numbers. Let's call the greatest number G, the middle number M, and the least number L.
Based on the given information, we can set up the following equations:
1. G + M + L = 18 (The sum of the three numbers is 18)
2. G - L = 5 1/5 (The difference between the greatest number and the least number is 5 1/5)
To find the solution, we need to solve this system of equations.
Step 1: Solve equation 2 for G:
G - L = 5 1/5
G = L + 5 1/5
Step 2: Substitute the value of G into equation 1:
(L + 5 1/5) + M + L = 18
2L + M + 5 1/5 = 18
Step 3: Simplify the equation:
2L + M + 5 1/5 = 18
2L + M = 12 4/5
Step 4: We can assume a reasonable value for L and then solve for M and G using the equations.
Let's assume L = 3 (a reasonable value).
Step 5: Substitute L = 3 into equation 4:
2(3) + M = 12 4/5
6 + M = 12 4/5
Step 6: Subtract 6 from both sides:
M = 12 4/5 - 6
M = 6 4/5
Step 7: Substitute the values of L and M back into equation 2:
G - L = 5 1/5
G - 3 = 5 1/5
Step 8: Add 3 to both sides:
G = 5 1/5 + 3
G = 8 1/5
Therefore, the three mixed numbers that satisfy the conditions are:
G = 8 1/5, M = 6 4/5, L = 3.
To find three mixed numbers that satisfy these conditions, we can use a system of equations.
Let's assume the three mixed numbers as x, y, and z.
According to the problem, the sum of the three mixed numbers is 18. So we can write our first equation as:
x + y + z = 18 (Equation 1)
The second condition states that the difference between the greatest number and the least number is 5 1/5. We need to convert this mixed number to an improper fraction for easier calculations. The mixed number 5 1/5 can be written as 26/5. So we have:
Greatest number - Least number = 26/5
To make it simpler, let's assume that x is the greatest number, then z is the least number. So the equation becomes:
x - z = 26/5 (Equation 2)
Now we have a system of equations (Equations 1 and 2) that we can solve to find the numbers x, y, and z.
To solve this system, we'll use substitution.
Rearrange Equation 2 to solve for x:
x = z + 26/5 (Equation 3)
Substitute Equation 3 into Equation 1:
(z + 26/5) + y + z = 18
Simplify the equation by combining like terms:
2z + y + 26/5 = 18
To further simplify, multiply all terms by 5 to eliminate the fraction:
10z + 5y + 26 = 90 (Equation 4)
Now we have a system of equations:
2z + y = 64/5 (Equation 5)
10z + 5y = 90 (Equation 4)
We'll solve this system by substitution or elimination to find the values of z and y. Once we have z and y, we can substitute them back into Equation 3 to find x.
So, the next step is to solve Equations 4 and 5 for z and y using substitution or elimination method.
suppose y=0. That means
x+z = 18
x-z = 26/5
x = 58/5
z = 32/5
So, the largest x can be is 58/5
Now, suppose we toss in y. If we subtract y/2 from both x and z, then the difference between x and z is still 26/5, and the sum is
(x-y/2) + y + (z-y/2) = x+z = 18
For example, if y = 4,
x = 48/5
y = 20/5
z = 22/5
x-z is still 26/5, and the sum is still 18.
So, pick any value for y that you want, up to 32/5, which would make z=0.