Three even numbers sum up to 104. The smallest is half the largest and the middle number is
2
3
the largest. What are the three numbers?
If The middle number is
2
3
the largest.
mean:
The middle number is 2 / 3 the largest.
then:
a = smalest number
b = middle number
c = largest number
a = c / 2
b = ( 2 / 3 ) c
a + b + c = 104
c / 2 + ( 2 / 3 ) c + c = 104
3 c / 6 + 4 c / 6 + 6 c / 6 = 104
13 c / 6 = 104 Multiply both sides by 6
13 c = 104 * 6
13 c = 624 Divide both sides by 13
c = 624 / 13 = 48
a = c / 2 = 48 / 2 = 24
b = ( 2 / 3 ) c = 2 * 48 / 3 = 96 / 3 = 32
a = 24
b = 32
c = 48
To solve this problem, we are given three pieces of information:
1. The sum of the three even numbers is 104.
2. The smallest number is half the largest.
3. The middle number is 2 less than the largest.
Step 1: Represent the numbers algebraically.
Let's represent the smallest number as "x", the middle number as "y", and the largest number as "z".
According to the given information:
The smallest number is half the largest: x = z/2
The middle number is 2 less than the largest: y = z - 2
Step 2: Form an equation with the given information.
The sum of the three numbers is 104: x + y + z = 104
Step 3: Substitute the expressions from step 1 into the equation from step 2.
Substitute x = z/2 and y = z - 2 in the equation x + y + z = 104:
(z/2) + (z - 2) + z = 104
Step 4: Solve the equation.
Multiply through by 2 to eliminate the fraction:
z + 2z - 4 + 2z = 208
Combine like terms:
5z - 4 = 208
Add 4 to both sides:
5z = 212
Divide both sides by 5:
z = 42.4
Since we are dealing with even numbers, the largest number cannot be a decimal. Therefore, we need to round down 42.4 to the nearest even number, which is 42.
Step 5: Substitute the value of z back into the expressions from step 1 to find x and y.
x = z/2 = 42/2 = 21
y = z - 2 = 42 - 2 = 40
Therefore, the three numbers are 21, 40, and 42.