SOLUTION: Tony and Peter both had some marbles at first. After Tony gave Peter 4/14 of his marbles, the ratio of Tony’s marbles to Peter’s marbles became 5:3. What was the ratio of Peter

Algebra ->  Percentage-and-ratio-word-problems -> SOLUTION: Tony and Peter both had some marbles at first. After Tony gave Peter 4/14 of his marbles, the ratio of Tony’s marbles to Peter’s marbles became 5:3. What was the ratio of Peter      Log On


   

Question 1189654: Tony and Peter both had some marbles at first. After Tony gave Peter 4/14 of his marbles, the ratio of Tony’s marbles to Peter’s marbles became 5:3. What was the ratio of Peter’s marbles to Tony’s marbles at first?
Found 3 solutions by math_tutor2020, greenestamps, ikleyn:
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

T = amount of marbles Tony has initially
P = amount of marbles Peter has initially

Tony gives 4/14 = 2/7 of his marbles to Peter.
That means Tony gives (2/7)T marbles to Peter.
Tony's count T becomes T-(2/7)T = (5/7)T
Peter's count P becomes P+(2/7)T

Let's say
M = Tony's new count of marbles
N = Peter's new count of marbles

To connect back to the previous variables, we would have:
M = (5/7)T
N = P+(2/7)T

After the gift is made, we have
M:N = 5:3
which is the same as writing
M%2FN+=+5%2F3

Let's cross multiply to get
M%2FN+=+5%2F3

3M+=+5N

Now apply substitution
3M+=+5N

3%285%2F7%29T+=+5%28P%2B%282%2F7%29T%29

%2815%2F7%29T+=+5P%2B%2810%2F7%29T

Fractions are a bit of a pain to work with, but we can multiply both sides by 7 to clear them out.
%2815%2F7%29T+=+5P%2B%2810%2F7%29T

7%2A%2815%2F7%29T+=+7%2A%285P%2B%2810%2F7%29T%29

15T+=+35P%2B10T

The ultimate goal is to find the ratio P:T where P and T will be replaced with actual numbers. That ratio is effectively the same as the fraction P/T

Let's see if we can somehow isolate P/T from that previous equation above
15T+=+35P%2B10T

15T-10T+=+35P

5T+=+35P

5T%2F5+=+35P%2F5

1T+=+7P

T%2FP+=+7%2F1

P%2FT+=+1%2F7

This leads to the ratio P:T = 1:7 telling us that Tony has 7 times more marbles compared to Peter when comparing the initial counts.


For example, let's say Peter starts with 10 marbles. That leads to Tony having 7*10 = 70 marbles.
Next, Tony gives 4/14 of his count to Peter. He gives (4/14)*70 = 20 marbles.
Tony's count becomes 70-20 = 50
Peter's count becomes 10+20 = 30
Then notice that the ratio Tony:Peter becomes 50:30 which reduces fully to 5:3
This example helps confirm the answer.

As another example, if Peter starts with 5 marbles then Tony starts with 7*5 = 35 marbles.
4/14 of 35 = 10 marbles are given
Tony = 35-10 = 25
Peter = 5+10 = 15
The ratio Tony:Peter = 25:15 reduces to 5:3
So as you can see, there are infinitely many possibilities for the values of P and T; however, their ratio P:T is fixed at 1:7

I'll let you try out other examples.


Answer is the ratio 1:7

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Before plunging into a "standard" algebraic solution, take the time to analyze the problem to see if there is an easier way to set the problem up.

At the start of the problem, Tony gave Peter 4/14=2/7 of his marbles; that means the number of marbles Tony started with was a multiple of 7. So

Let 7x = # of marbles Tony started with.

The number of marbles he gave to Peter was then 2x; the number Tony was left with was 5x.

After Tony gave those marbles to Peter, the ratio of marbles the two of them had was 5:3. So, since Tony finished with 5x marbles, Peter finished with 3x marbles.

2x of the marbles Peter finished with were what Tony gave him; since Peter finished with 3x marbles, he started with x marbles.

So Tony started with 7x marbles and Peter started with x marbles.

ANSWER: The ratio of Peter's marbles to Tony's marbles at first was x:7x = 1:7

Answer by ikleyn(52803) About Me  (Show Source):
You can put this solution on YOUR website!
.
Tony and Peter both had some marbles at first. After Tony gave Peter 4/14 of his marbles,
the ratio of Tony’s marbles to Peter’s marbles became 5:3.
What was the ratio of Peter’s marbles to Tony’s marbles at first?.
~~~~~~~~~~~~~~~~


            For me,  it is psychologically difficult to see  (to read)  so long solution for so simple problem.

            So, I came to bring a shorter solution  (thinking that longer solution does not help understanding). 


Finally, Tony has 5x marbles, while Peter has 3x marbles, where x is the common measure.


Let P = # of marbles Peter had initially;

    T = # of marbles Tony  had initially.


Then we have these equations

    3x = P + %284%2F14%29%2AT    (1)

    5x = %2810%2F14%29%2AT       (2)


From equation (2),  x = %285%2F7%29%2A%281%2F5%29%2AT = %281%2F7%29%2AT.


Substitute it into equation (1) to get

    %283%2F7%29T = P + %282%2F7%29%2AT.


So,  P = %283%2F7%29%2AT - %282%2F7%29%2AT = %281%2F7%29%2AT.


It means that  P%2FT = 1%2F7.     ANSWER

Solved.


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My conception is this: everything which is longer than 5 lines of standard text, is not a Math problem.

If a standard/regular school Math problem is given, than its solution should be no longer than 10 - 15 lines of the text; maximum 20 lines.

If a problem is exceptional, its solution may require "more lines" - it depends and it can be justified.

But if the solution to a regular/standard school Math problem is longer than 20 lines, then nobody even will read it . . .