SOLUTION: To be exact, this word problem is all about Systems of Linear Equation in Three Variables. There are three boys, Cyril, Ryan, and Guye. The sum of their ages is twenty years. Cy

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Question 1192437: To be exact, this word problem is all about Systems of Linear Equation in Three Variables.
There are three boys, Cyril, Ryan, and Guye. The sum of their ages is twenty years. Cyril’s age added to Guye’s is four-time Ryan’s age, and Ryan’s age added to Cyril’s is four times Guye’s. How old are three boys?
Your answer would mean a lot. Thank you in advance!
Found 2 solutions by Edwin McCravy, ikleyn:
Answer by Edwin McCravy(20056)   (Show Source):
You can put this solution on YOUR website!

Ikelyn thinks you've already finished the section on 3 equations in 3 unknowns, and that now you are studying clever ways of getting answers without using the methods of solving 3 equations in 3 unknowns. Or -- She might be trying to show us how smart she is (or how dumb I am for not using her "clever" method). -------------------------------------------------------------------- Anyway, Instead of doing your homework for you, I'll do one exactly like yours but change all the numbers: I'll do this one:

There are three boys, Cyril, Ryan, and Guye. The sum of their ages
is sixty years. Cyril’s age added to Guye’s is five times Ryan’s age,
and Ryan’s age added to Cyril’s is five times Guye’s. How old are
the three boys?

The sum of their ages is sixty years.
C + R + G = 60

Cyril’s age added to Guye’s is five times Ryan’s age,
G+C = 5R

and Ryan’s age added to Cyril’s is five times Guye’s.
C + R = 5G Get the terms in order and lined up vertically Eliminate C and G by multiplying the first equation by -1 and adding the second equation to it: -C - R - G = -60 C - 5R + G = 0 ----------------- 0 -6R + 0 = -60 -6R = -60 R = 10 Substitute 10 for R in all three equations: Simplify Simplify further: The first two equations are the same so eliminate one of them: Solve the first equation for C C = 50 - G Substitute (50 - G) for C in the second equation: 50 - G - 5G = -10 50 - 6G = -10 -6G = -60 G = 10 Substitute 10 for R and 10 for G in C + R + G = 60 C + 10 + 10 = 60 C + 20 = 60 C = 40 <--Cyril is a 40-year-old "BOY". lol Now do yours the same way. Cyril will be a BOY in your problem. lol Edwin

Answer by ikleyn(52787)   (Show Source):
You can put this solution on YOUR website!
.
There are three boys, Cyril, Ryan, and Guye. The sum of their ages is twenty years.
Cyril’s age added to Guye’s is four-time Ryan’s age, and Ryan’s age added to Cyril’s is four times Guye’s.
How old are three boys?
~~~~~~~~~~~~~~

            This problem can be solved MENTALLY without using any equations.

            Common sense and mental reasoning is / (are) just enough.

Indeed, we are given that Cyril’s age added to Guye’s is four-time Ryan’s age. Add to it the Ryan's age. You will get five times the Ryan's age. From the other side, the sum of ages of three boys is 20. It means that five times the Ryan's age is 20 years - - - hence, Ryan's age is 20/5 = 4 years. Similar reasoning gives 20/5 = 4 years for the Guye's age. It leaves 20-4-4 = 12 years for the Cyril's age.

Solved.