Sam and Jill had 576 stamps in all. Sam gave 1/7 of his stamps to Jill. Then Jill gave 1/4 of the total number of stamps she now had to Sam. In the end, both of them had the same number of stamps. What was the difference in the number of stamps they each had at first?
Sam has X stamps.
Jill has (576-X) Stamps.
X-x/7 = 6x/7 = Sam's bal.
576-x + x/7 = 576 - 6x/7 = Jill's bal.
1/4(576-6x/7) = 144 - 6x/28
3/4(576-6x/7) = 432 - 9x/14 = Jill's new
bal.
6x/7 + 144-6x/28 = 144 + 24x/28-6x/28 =
144 + 9x/14 = Sam's new bal.
Jill's new bal = Sam's new bal
432 - 9x/14 = 144 + 9x/14
18x/14 = 144-432 = 288
18x = 4032
X = 224
576-X = 576-224 = 352.
Difference = 352 - 224 128
Let's solve this step-by-step:
Step 1: Find the number of stamps Sam gave to Jill.
- Sam gave 1/7 of his stamps to Jill.
- So, the number of stamps Sam gave to Jill is (1/7) * 576 = 82 stamps.
Step 2: Find the number of stamps Jill now has.
- Jill initially had 576 stamps.
- After Sam gave her 82 stamps, Jill now has 576 + 82 = 658 stamps.
Step 3: Find the number of stamps Jill gave to Sam.
- Jill gave 1/4 of the total number of stamps she had to Sam.
- So, the number of stamps Jill gave to Sam is (1/4) * 658 = 164.5 stamps.
Step 4: Find the number of stamps Sam now has.
- Sam initially had 576 stamps.
- After Jill gave him 164.5 stamps, Sam now has 576 + 164.5 = 740.5 stamps.
Step 5: Determine the final number of stamps each person has.
- After the exchange, Sam and Jill both have the same number of stamps.
- Let's assume that they each have x stamps.
- So, Sam has 740.5 - x stamps and Jill has 658 - x stamps.
Step 6: Set up an equation.
- Since Sam and Jill have the same number of stamps, we can set up an equation:
740.5 - x = 658 - x.
Step 7: Solve the equation.
- Solving the equation, we get:
740.5 - 658 = x - x
82.5 = 0
Step 8: Interpret the result.
- The equation has no solutions. Therefore, there is no difference in the number of stamps they each had at first.
Thus, the difference in the number of stamps they each had at first is 0.
To find the difference in the number of stamps Sam and Jill each had at first, we need to break down and solve the problem step by step.
Let's start by determining how many stamps Sam had at first.
Let's assume that Sam had x stamps at first. Since Jill had 576 stamps in total, we can conclude that Jill had 576 - x stamps at first.
Next, Sam gives 1/7 of his stamps to Jill. This means Sam gives (1/7) * x stamps to Jill. Subsequently, Sam is left with (6/7) * x stamps.
After this, Jill gives 1/4 of the total number of stamps she now had (which is 576 - x) to Sam. We can calculate this as (1/4) * (576 - x). As a result, Jill has (3/4) * (576 - x) stamps remaining.
According to the problem, both of them end up with the same number of stamps. Therefore, we can equate the remaining stamps for Sam and Jill:
(6/7) * x = (3/4) * (576 - x)
To solve for x, we can simplify the equation:
(6/7) * x = (3/4) * 576 - (3/4) * x
Multiply both sides of the equation by 7 and 4 to get rid of the fractions:
24x = 21 * 576 - 21x
Combine like terms:
24x + 21x = 21 * 576
45x = 21 * 576
Divide both sides of the equation by 45:
x = (21 * 576) / 45
Now we can calculate the value of x:
x = 268.8
Since we can't have a fraction of a stamp, we can round the number to the nearest whole number:
x ≈ 269
Therefore, Sam had approximately 269 stamps at first.
Now we can determine the number of stamps Jill had at first:
Jill's initial number of stamps = 576 - x
Jill's initial number of stamps = 576 - 269
Jill's initial number of stamps = 307
So, Jill had 307 stamps at first.
To find the difference between the number of stamps they each had at first, we subtract Jill's initial number of stamps from Sam's initial number of stamps:
Difference = Sam's initial number of stamps - Jill's initial number of stamps
Difference = 269 - 307
Difference ≈ -38
The difference in the number of stamps they each had at first is approximately -38. This indicates that Jill had 38 more stamps than Sam at the beginning.