Question 610676: Do not understand age problems when we have fractions. WHat do we multiply?
Roberta is three years younger than Rachel. Eight years ago, Roberta was ONE HALF Rachel's age. How old is each girl?
Found 2 solutions by lwsshak3, bucky:
Answer by lwsshak3(11628)   (Show Source):
You can put this solution on YOUR website! Roberta is three years younger than Rachel. Eight years ago, Roberta was ONE HALF Rachel's age. How old is each girl?
**
let x=Rachel's present age
x-3=Roberta's present age
8 years ago:
Rachel's age=x-8
Roberta's age=x-3-8=x-11
x-11=(1/2)(x-8)
2x-22=x-8
x=14
x-3=11
ans:
Rachel's present age=14
Roberta's present age=11
Answer by bucky(2189) (Show Source):
You can put this solution on YOUR website! Don't get too lost trying to understand numbers. A lot of it is just relaxing and using your common sense. Let's try it on this problem.
.
First we recognize that we don't know how old Roberta and Rachel are at the present time. So, let's just say that currently Roberta is X years old and Rachel is Y years old.
.
From math we need to know one thing. If we have two unknowns (which X and Y are) then we will need two separate equations to find these two unknowns. That being the case, let's see what the problem tells us that will lead us into writing two equations.
.
First the problem tells us that Roberta (who is presently X years old) is three years younger than Rachel (who is presently Y years old). Since Roberta is the younger by three years, we can find Rachel's age by adding 3 years to Roberta's age and that answer will equal Rachel's age. In other words:
.
X (which is Roberta's age) plus 3 years equals Rachel's age (which is Y)
.
Or in shorter form: X + 3 = Y <----- this is one of the two equations
.
Next the problems talks about 8 years ago. What do we know about that? Presently one of the girls (Roberta) is X years old. How old would she have been 8 years ago? Think about how old you were 8 years ago. You would find that by taking your present age and subtracting 8 from it. Same thing for Roberta. We take her present age (X) and subtract 8 from it. In shortened form it is X - 8. Same thing for Rachel. Her present age is Y so 8 years ago she would have been Y - 8 years old.
.
So eight years ago Roberta's age was X - 8 and Rachel's age was Y - 8. But Roberta was half of Rachel's age at that time. Suppose you had a friend and you were half her age. How would you find her age? You could double your age. Or if you knew her age, you could take half of it and get your age. Roberta is the younger, So we could double her age and get Rachel's age. Or we could take half of Rachel's age and know that it would equal Roberta's age. Either way ... it will make no difference.
.
So we could double Roberta's age (X - 8) and it would equal Rachel's age (Y - 8). Or we could take half of Rachel's age and it would equal Roberta's age. In equation form these would be:
.
Doubling Roberta's age to equal Rachel's age ... 2*(X - 8) = Y - 8 or
.
Taking half of Rachel's age to equal Roberta's age ... (1/2)*(Y - 8) = X - 8
.
We can use either one of these equations as the second equation needed to solve the problem. (If we worked on simplifying both of these equations, we would find that they turn out to be identical. They are just different ways of saying the same thing.)
.
Let's use the first of these two equations ... the one doubling Roberta's age ... just because it doesn't contain a fraction to work with. The other equation (taking half of Rachel's age) contains that fraction 1/2 we would have to deal with. (But it would work out to give us the same results.)
.
So our two equations are:
.
X + 3 = Y and
2*(X - 8) = Y - 8
.
Let's use the substitution method of solving these two. From the first equation we see that Y equals X + 3. So we can go to the second equation and in it replace Y by X + 3 to make it become:
.
2*(X - 8) = X + 3 - 8
.
Do the distributed multiplication on the left side by multiplying 2 times each of the terms in the parentheses. When we do that the left side becomes 2X - 16 and the equation then is:
.
2X - 16 = X + 3 - 8
,
On the right side combine the +3 and the -8 to get -5 and this reduces the equation to:
.
2X - 16 = X - 5
.
Now we can get rid of the -16 on the left side by adding 16 to both sides. When we add 16 to the left side, it combines with the -16 to cancel each other out. And when we add 16 to the right side it combines with the -5 to give +11. The effect is that the equation becomes:
.
2X = X + 11
.
Then we can get rid of the X on the right side by subtracting X from both sides. When we do that the equation reduces to:
.
X = 11
.
This tells us that X (Roberta's present age) is 11. Since Roberta is 3 years younger than Rachel, we then know (from our first equation) that Rachel must be 14. And 8 years ago, Roberta would have been 3 and Rachel would have been 6. So Rachel would have been double Roberta's age (or Roberta would have been half of Rachel's age).
.
See how things work out?
.
I hope this helps you to understand the problem and gives you a little insight into dealing with whether you double or take 1/2 of the ages 8 years ago. You can do it either way as long as you keep track of whose age you are doubling or whose age you are taking half of.
|
|
|
