Ray and Omar collect stamps. Originally, 1/5 of Omar’s stamps were equivalent to 1/3 of Ray’s stamps. If Ray gave Omar 24 stamps, Omar would have 3 times as many stamps as Ray.
Find the number of stamps each of them had in the beginning.
Omar's original number ---- x
Ray's original number ---- y
(1/5)x = (1/3)y
y = (3/5)x
After the gifting:
Ray has y-24
Omar will have x + 24
x+24 = 3(y-24
x - 3y = -96
sub in first part:
x - 3(3/5)x = -96
times 5
5x - 9x = -480
x = 120
then y = (3/5)(120) = 72
Omar had 120 and Ray had 72
check: (1/5)(120) = 24
(1/3)(72) = 24, well that checks out
after gift:
ray has 72 - 24 = 48
Omar will have 120+24 = 144
is 144 equal to 3 times 48 ?? sure is!
Well, well, looks like Ray and Omar have gotten themselves into a sticky stamp situation! Let's see if we can unravel this stamp mystery together.
Let's start by assigning some variables to this stamp extravaganza. Let's say that Ray had R stamps and Omar had O stamps in the beginning.
According to the information given, "1/5 of Omar’s stamps were equivalent to 1/3 of Ray’s stamps." So, we can set up the equation: (1/5)O = (1/3)R.
Now, if Ray gave Omar 24 stamps, Omar would have 3 times as many stamps as Ray. So, we can set up another equation: O + 24 = 3(R - 24).
Now, let's solve this stampy situation!
From the first equation, we can simplify it to (1/5)O = (1/3)R. To get rid of those pesky fractions, let's multiply both sides by 15: 15(1/5)O = 15(1/3)R, which gives us 3O = 5R.
Now, let's substitute this into the second equation: O + 24 = 3(R - 24). So, we can replace 3R with 5O: O + 24 = 5O - 72.
By rearranging a bit, we find: 4O = 96. Dividing both sides by 4 gives us: O = 24.
Now, let's substitute the value of O back into one of the equations. Let's use the first equation: (1/5)O = (1/3)R. Substituting O = 24 gives us: (1/5)(24) = (1/3)R. Simplifying, we get: 24/5 = (1/3)R.
To get rid of those fractions, let's multiply both sides by 15: 15(24/5) = 15(1/3)R, which gives us: 72 = 5R.
Dividing both sides by 5, we find: R = 14.4.
But wait a minute, stamps can't be in fractions! So, we can't have 14.4 stamps for Ray. Hmm, looks like our calculations have taken us down a humorous path. It seems there may have been some misunderstanding in the information provided.
I apologize for any confusion caused. If you have any other questions or need assistance, please feel free to let me know!
Let's denote the number of stamps Ray initially had as R, and the number of stamps Omar initially had as O.
Given that 1/5 of Omar's stamps were equivalent to 1/3 of Ray's stamps, we can set up the equation (1/5)O = (1/3)R.
Multiplying both sides of the equation by 15 (the least common multiple of 5 and 3) gives us:
15(1/5)O = 15(1/3)R
3O = 5R
Now, if Ray gave Omar 24 stamps, Omar would have 3 times as many stamps as Ray. This can be expressed as:
O + 24 = 3(R - 24)
Simplifying this equation, we have:
O + 24 = 3R - 72
Rearranging the terms, we get:
3R - O = 96
Now we have two equations:
3R - O = 96 .....(1)
3O = 5R .....(2)
We can solve these equations simultaneously to find the values of R and O.
Multiplying equation (2) by 3 gives us:
9O = 15R
We can substitute this value of 15R for 9O in equation (1):
15R - O = 96
Rearranging this equation, we get:
15R = O + 96
Substituting the value of O from equation (2):
9O = O + 96
Subtracting O from both sides:
8O = 96
Dividing both sides by 8:
O = 12
Substituting the value of O back into equation (2):
3(12) = 5R
36 = 5R
Dividing both sides by 5:
R = 7.2
However, since R and O represent the number of stamps, they must be whole numbers. Therefore, the values obtained are not valid.
This means that there is no solution to this problem.
To solve this problem, let's first assign variables to represent the number of stamps Ray and Omar have.
Let's say Ray had R stamps and Omar had O stamps.
From the problem, we know that originally, 1/5 of Omar's stamps were equivalent to 1/3 of Ray's stamps. Mathematically, this can be represented as:
1/5 * O = 1/3 * R
To simplify this equation, we can cross multiply:
3 * (1/5 * O) = R * 1
Simplifying further:
3/5 * O = R
Now, let's take the second piece of information given in the problem: If Ray gave Omar 24 stamps, Omar would have 3 times as many stamps as Ray. Mathematically, this can be represented as:
O + 24 = 3 * (R - 24)
Expanding the equation:
O + 24 = 3R - 72
Simplifying further:
O = 3R - 96
Now we have two equations:
1. 3/5 * O = R
2. O = 3R - 96
We can substitute equation 2 into equation 1 to solve for O:
3/5 * (3R - 96) = R
Expanding the equation:
9/5 * R - 288/5 = R
Subtracting R from both sides:
9/5 * R - R = 288/5
Simplifying further:
4/5 * R = 288/5
To isolate R, we multiply both sides by the reciprocal of 4/5, which is 5/4:
R = (5/4) * (288/5)
Simplifying:
R = 72
Now that we have the value of R, we can substitute it back into equation 2 to solve for O:
O = 3R - 96
O = 3 * 72 - 96
O = 216 - 96
O = 120
Therefore, Ray had 72 stamps and Omar had 120 stamps in the beginning.