Question 202132: 1.Ben is 3 years younger than Dan. The sum of their ages is 53. How old is each?
Please if you can explain how to solve age problems it can help too thanks in advance.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! let x = Ben's age.
let y = Dan's age.
The sum of their ages is 53.
x+y = 53
Ben is 3 years younger than Dan.
x = y-3
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you now have 2 equation that need to be solved simultaneously.
solving equations simultaneously means that the same answer applies to both equations.
one way to solve them simultaneously is to do a substitution in order to eliminate one of the unknowns. you can then solve one of the equations for the remaining unknown.
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since you know that x + y = 53 and you know that x = y-3, just substitute y-3 for x in the x + y = 53 equation.
substituting y-3 for x you get
y - 3 + y = 53
you now have one unknown in one equation which can be solved for that unknown.
to solve the y - 3 + y = 53 equation, do the following:
add 3 to both sides of the equation to get
y + y = 56
simplify by combining like terms to get
2y = 56
divide both sides of the equation by 2 to get
y = 28
you now know the value of y (which is Dan's age).
you can now solve for the value of x (which is Ben's age).
take either equation and substitute 28 for y and solve.
take x = y - 3
this becomes
x = 28 - 3
which becomes
x = 25
you now have
x = 25 which is Ben's age.
y = 28 which is Dan's age.
The sum of their ages is 25 + 28 = 53 which is what you want to have happen because it was one of the up front requirements.
Ben is 3 years younger than Dan which is also what you want to have happen because that was also one of the up front requirements (y - x = 28 - 25 = 3).
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You need to assign a variable to their age now and then manipulate the variables to conform to the requirements of the problem.
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THAT'S THE END OF YOUR PROBLEM. IF YOU WANT TO LEARN A LITTLE MORE ABOUT HOW TO SOLVE THESE PROBLEMS, CONTINUE BELOW.
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Here's another problem that is more complicated, but should help you to understand better.
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Before I show you the problem, however, let me tell you what you can do to learn more about word problems. You go to google, or yahoo, and do a search on "math age problems" or "algebra age problems". The results of the search will lead you to various websites that specialize in helping you do math.
Try one of them and see if it is easy for you to understand. If not, then try another one. Keep trying until you get one that's easy for you to understand, or at least easier than the others.
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One that I got the following problem was .....
http://www.purplemath.com/modules/ageprobs.htm
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here's the problem
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In three more years, Miguel's grandfather will be six times as old as Miguel was last year. When Miguel's present age is added to his grandfather's present age, the total is 68. How old is each one now?
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You have to start somewhere and the best place to start is usually to represent their ages now.
let x = miguel's age now.
let y = miguel's grandfather's age now.
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The first statement says that in 3 more years, Miguel's grandfather will be 6 times as old as Migual was last year.
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If his grandfather's age is now y years old, in 3 years his grandfather's age will be y + 3.
if miguel's age is now x years old, 1 year ago miguel's age was x - 1.
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In 3 years, miguel's grandfather will be 6 times as old as miguel was a year ago means that
(y+3) = 6 * (x-1)
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you have translated the first statement into an algebraic equation.
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The second statement says that miguel's age now and his father's age now totals 68.
this means that
x + y = 68
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you have translated the second statement into an algebraic equation.
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you now have 2 equations in 2 unknowns that need to be solved simultaneously.
the easiest way to do that unless you know other ways would be to use one of the equations to establish a relationship between the two unknowns and then use that relationship to eliminate one of the unknowns and solve for the other.
here's how that's done.
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the 2 equations are:
x + y = 68
(y + 3) = 6 * (x - 1)
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looks like x + y = 68 is the easier equation to work with so use that to establish the relationship.
x + y = 68
subtract y from both sides of the equation to get
x = 68 - y
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you have just established the relationship between x and y. you can now substitute (68 - y) for x in the other equation.
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(y + 3) = 6 * (x - 1)
substitute (68 - y) for x to get
(y + 3) = 6 * ((68 - y) - 1)
solve for y
equation becomes:
y + 3 = 6 * (68 - y - 1)
which becomes:
y + 3 = 6 * 68 - 6 * y - 6 * 1
which becomes
y + 3 = 408 - 6*y - 6
which becomes
y + 3 = 402 - 6*y
subtract 3 from both sides of the equation and add 6y to both sides of the equation to get
y + 6*y = 402 - 3
which becomes
7*y = 399
divide both sides by 7 to get
y = 57
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you now have the value of y which is miguel's grandfather's age now.
use that to solve for x in either one of the equations.
take x + y = 68 since it's easier to work with.
it becomes
x + 57 = 68
subtract 57 from both sides to get
x = 68 - 57 = 11
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x = 11 = miguel's age now.
y = 57 = miguel's grandfather's age now.
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to prove the answer is good, substitute these numbers in the 2 statements to see if they're true.
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the first statement says that in 3 years miguel's grandfather will be 6 times as old as miguel was a year ago.
57 + 3 = 60 = miguel's grandfather's age 3 years from now.
11 - 1 = 10 = miguel's age 1 year ago.
60 = 6 * 10 is true so the numbers hold up there.
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the second statement says that miguel's age now and miguel's grandfather's age now total 68.
57 + 11 = 68 is true so the numbers hold up there as well.
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you have verified that your answer is good so the last step is to answer the question.
the question was what is miguel's age now and what is his grandfather's age now.
the answer should state.
miguel is 11 years old now and his grandfather is 57 years old now.
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