When J gives A $6, they will both have the same amount. When A gives J $6, she will have 1/3 of J's amount. What are the amounts they both have at first?( please do not solve by algebra, i have not been taught algebra)
From the first sentence I can tell that J must have $12 more than A
Also from the second sentence A must have at least $6
so form two columns of possible amounts, starting with A having 6
A J
6 18
7 19
8 20
9 21
10 22
11 23
12 24
13 25
14 26
15 27
16 28
17 29
18 30
19 31
...
Now metally take 6 away from A and add 6 to J
when will the first value be 1/3 of the 2nd?
how about 18 and 30
(18-6) = (1/3)(30+6) ----> 12 = (1/3) of 36
I bet you can't wait to learn Algebra. You will be so surprised how really easy these questions become.
J-6 = A+6 ---> J-A = 12
A-6 = (1/3)(J+6)
3A - 18 = J+6
3A - J = 24
3A - J = 24
-A + J = 12
add them
2A = 36
A = 18, then J = A+12 = 30
ok. no algebra*
We know that J has $12 more than A. That's because when J is $6 less, it is the same as the new A, which is $6 more.
J can't have just $12, because that would mean that A started with 0, and could not give 6 to J.
A and J must both be multiples of $6, since we're juggling $6 chunks.
So, suppose A = 12. That would make J=24, but 12-6=6 is not 1/3 of 24+6=30.
If A=18, J=30 and 18-6=12 is 1/3 of 30+6=36.
*
But all of this reasoning is just doing algebra in your head. It's never to soon to learn a little algebra, so you might as well start now.
Algebra just gives you a way of writing down number facts so you don't have to keep track of everything in your head.
Let J and A represent the two starting amounts.
J gives A $6, so now we have
J-6 = A+6
If you have a true sentence (or equation), it stays true as long as, if you make changes, you make the same changes to both sides of the equation.
For example, if J is 5 years older than A, she will still be 5 years older a year from now. That is, if
J = A+5
next year,
J+1 = A+5+1
In our problem, we have
J-6 = A+6
If you add 6 to both sides, you get J by herself:
J-6+6 = A+6+6
J = A+12
Now, if we have two unknown numbers, we need to know two independent things about how they fit together. Our second thing is: If A gives J $6, she has 1/3 of J's new amount. Or, avoiding fractions, J will then have 3 times as much as A. In symbols,
J+6 = 3 * (A-6)
Here * means times. To get rid of the parentheses, you need to know about how multiplication distributes over addition. You already know this, but maybe not by name. If someone asks you to add 300 and 400, you have no trouble coming up with 700 as the answer. In your head, you probably say, well considering hundreds, 3+4=7. In symbols,
300 + 400 = 3*100 + 4*100 = (3+4)*100
In other words, you do what's in the parentheses first, then do the multiplication. In our case, we have
J+6 = 3*(A-6)
J+6 = 3A - 3*6
J+6 = 3A - 18
Add 18 to both sides:
J+24 = 3A
Now, we know J=A+12. Wherever we have J, we can just as well use A+12. We can make that substitution, and we now have
A+12 + 24 = 3A
A + 36 = 3A
Now, we need to get all the A's together, so subtract A from both sides.
A-A + 36 = 3A - A
But, A-A = 0, and 3A-A = 2A
So, now we have
36 = 2A
In words, 36 is twice as much as A.
so, A is just half of 36 = 18.
Since J=A+12, J=18+12 = 30.
Congratulations. While there are still some gaps to fill in, you've just covered half a semester of algebra.
To solve this problem without using algebra, we can use a trial and error approach. Let's start by assuming a possible amount for one person and then calculate the other person's amount using the given conditions.
Let's assume that J has $12 at first. According to the first condition, when J gives A $6, they will both have the same amount. So, after this transaction, A will have $12 as well.
Now, let's consider the second condition. When A gives J $6, she will have 1/3 of J's amount. If A has $12 at first, then 1/3 of J's amount should be $12. This means J should have $36.
However, this doesn't match the initial amount we assumed for J. So, let's try another assumption.
Assuming J has $18 at first, following the first condition, A would also have $18 after receiving $6 from J.
Now, let's consider the second condition. If A has $18 at first, then 1/3 of J's amount should be $18 after A gives $6 to J. This means J should have $54.
With the second assumption, we have a match. Therefore, the amounts they both have at first are $18 for A and $54 for J.