SOLUTION: Brian and Rayan shared a bag of marbles. 1/6 of Brian's share was equal to 1/4 of Rayyan's marbles. Brian lost 42 of his marbles while Rayyan bought another 60 marbles. They then h

Algebra ->  Percentage-and-ratio-word-problems -> SOLUTION: Brian and Rayan shared a bag of marbles. 1/6 of Brian's share was equal to 1/4 of Rayyan's marbles. Brian lost 42 of his marbles while Rayyan bought another 60 marbles. They then h      Log On


   

Question 1205459: Brian and Rayan shared a bag of marbles. 1/6 of Brian's share was equal to 1/4 of Rayyan's marbles. Brian lost 42 of his marbles while Rayyan bought another 60 marbles. They then had an equal number of marbles.
(a) How many marbles did Brian have at first?
(b) What was the total number of marbles they had in the end?
Found 3 solutions by josgarithmetic, math_tutor2020, greenestamps:
Answer by josgarithmetic(39630) About Me  (Show Source):
You can put this solution on YOUR website!
b, how many Brian started with
r, how many Rayyan started with

1/6 of Brian's share was equal to 1/4 of Rayyan's marbles.
b%2F6=r%2F4

Brian lost 42 of his marbles while Rayyan bought another 60 marbles.
Then b-42 for Brian and r%2B60 for Rayyan

They then had an equal number of marbles.
b-42=r%2B60


The first and second equations can become the system
highlight_green%28system%282b-3r=0%2Cb-r=102%29%29.

You can choose Elimination Method to find b and r.

system%282b-3r=0%2C2b-2r=204%29
Subtract...
highlight_green%28r=204%29------how many Rayyan initially had


system%282b-3r=0%2C3b-3r=306%29
Subtract....
highlight_green%28b=306%29------how many Brian had initially

TOTAL AMOUNT MARBLES BOTH FINALLY HAD:
%28b-42%29%2B%28r%2B60%29
b%2Br%2B18
306%2B204%2B18
highlight%28528%29

Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

b = Brian's starting marble count
r = Rayyan's starting marble count
These are positive whole numbers.
It's not clear if the spelling should be Rayan or Rayyan. I'll go with the 2nd version since it pops up more frequently in the instructions.

b%2F6+=+r%2F4 since "1/6 of Brian's share was equal to 1/4 of Rayyan's marbles"

Cross multiply to get 4b+=+6r
Then divide both sides by 4 to isolate b.
You should end up with b+=+6r%2F4+=+1.5r or simply b+=+1.5r
Whatever starting marble count Rayyan has, multiply by 1.5 (aka 3/2) to get Brian's starting marble count.

You probably can spot right away that the value of r must be even.
Otherwise b would land on some non-integer value.
Eg: r = 11 means b = 1.5r = 1.5*11 = 16.5 which makes no sense.
However, r = 12 would be valid since b = 1.5r = 1.5*12 = 18

Next we will turn to these sentences
"Brian lost 42 of his marbles while Rayyan bought another 60 marbles. They then had an equal number of marbles."
to help us form the equation
b+-+42+=+r+%2B+60

But since b = 1.5r, we can replace each copy of b with 1.5r
That will eliminate one of the variables so we can solve for the other variable.
b+-+42+=+r+%2B+60

1.5r+-+42+=+r+%2B+60

1.5r-r+=+60%2B42

0.5r+=+102

r+=+102%2F%280.5%29 dividing by half is the same as doubling

r+=+204
Rayyan starts off with 204 marbles.
Plug that value into b = 1.5r to find how many marbles Brian started with.
b+=+1.5r

b+=+1.5%2A204

b+=+306
Brian starts off with 306 marbles which is the answer to part (a).

Return to this sentence
"Brian lost 42 of his marbles while Rayyan bought another 60 marbles."
It will mean there's a net gain of -42+60 = 18 marbles.
Add this net gain to the initial marble counts we found earlier.
r+b+18 = 204+306+18 = 528 is the number of marbles that both people have at the end.

Alternatively,
b-42 = amount Brian ends up with
r+60 = amount Rayyan ends up with
(b-42)+(r+60) = b+r+(-42+60) = b+r+18 = total amount at end
Plug in r = 204 and b = 306 to get 528 which is the answer to part (b).

A thing to notice: If somehow you were able to calculate part (b) first (or perhaps you saw this answer on the answer sheet), then you'll know that both people end up with 528/2 = 264 marbles. We can divide by 2 since each person has the same marble count at the end.
From there just reverse the "Brian lost 42 marbles" to rewind the clock. He gains those 42 marbles back to get 264+42 = 306 which is the answer to part (a).

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Answers:

(a) 306
(b) 528

Answer by greenestamps(13216) About Me  (Show Source):
You can put this solution on YOUR website!


You have received two responses showing solutions that are pretty much the same except for the details. Let's look at a different solution method that uses a different starting point.

The information that they end up with an equal number of marbles suggests that we start our solution using that information.

Let x be the number of marbles that each of them ends up with.

Brian lost 42 marbles, so he started with x+42 marbles.

Rayan bought 60 marbles, so he started with x-60 marbles.

In the beginning, 1/6 of Brian's marbles was equal to 1/4 of Rayan's marbles:

%28x%2B42%29%2F6=%28x-60%29%2F4
6%28x-60%29=4%28x%2B42%29
6x-360=4x%2B168
2x=528
x=528%2F2=264

We have answered the second question already: the total number of marbles they had at the end was 2x = 528.

For the first question, the number Brian had at the start was x+42 = 264+42 = 306.

The problem doesn't ask for the number Rayan had at first... but it was x-60 = 264-60 = 204.

ANSWERS:
(a) Brian had 306 at first (and Rayan had 204)
(b) They had a total of 528 at the end

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The solutions from the other two tutors are fine; but the algebra needed to solve the problem by the above method is considerably easier.

Always be open to the idea of looking for ways to set up a problem that makes the work required to solve the problem as easy as possible.