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Question 285793: The sum of three numbers is 98. The first number is 2/3 of the second, and the second is 5/8 of the third. What is the second number?
A. 18 B. 20 C. 30 D. 48 E. None of these.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! Sum of 3 number is 98.
x = first number
y = second number
z = third number
x = (2/3)*y
y = (5/8)*z
x + y + z = 98
Since y = (5/8) * z, the equation becomes:
x + (5/8)*z + z = 98
Since x = (2/3) * y, and y = (5/8) * z, this means that:
x = (2/3)*(5/8)*z.
Substitute that value into the equation, and the equation becomes:
(2/3)*(5/8)*z + (5/8)*z + z = 98
You can factor out the z to get:
z * ((2/3)*(5/8) + (5/8) + 1) = 98
Simplify this equation to get:
z * ((10/24) + (5/8) + (8/8)) = 98
Simplify further to get:
z * ((10/24) + (15/24) + (24/24)) = 98
Combine like terms to get:
z * (49/24) = 98
Divide both sides of this equation by (49/24) to get:
z = 98 / (49/24) which is the same as:
z = 98 * (24/49).
Simplify this to get:
z = 2*24 = 48
Since z = 48, and y = (5/8)*z, this means that y = (5/8)*48 = 5*6 = 30.
You have:
z = 48
y = 30
Since y = 30, and x = (2/3)*y, this means that x = (2/3)*30 = 20.
You have:
z = 48
y = 30
x = 20
To confirm these are accurate, substitute in your original equation to see if that equation is true using these values.
Your original equation is:
x + y + z = 98
That becomes:
20 + 30 + 48 = 98 which becomes 98 = 98 which is true.
These values are good for the sum equation.
Substitute in the piece part equations to see if they still hold up.
The first piece part is x = 2/3 * y.
Since 20 = 2/3 * 30 is true, this part is good.
The second piece part is y = (5/8) * z.
Since 30 = 5/8 * 48, this part is also good.
The solutions of:
x = 20
y = 30
z = 48
are all good.
The answer to your question is:
The second number is 30.
That would be selection C.
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