Helen and Sam had some stamps. When Helen gave away 360 stamps and
Sam gave away 48 stamps, the number of stamps Helen had was 2/5 of the
number of stamps Sam had. How many stamps did Helen have at first if she had 114 more stamps than Sam?
H-360 = (2/5) (S-48) = (2 S - 96) / 5
S = H - 114
------------- substitute
5 H - 1800 = 2 S - 96
5 H - 1800 = 2 (H-114) - 96
5 H - 1800 = 2 H - 228 - 96
3 H = 1476
H = 492
Sam had x stamps
Helen had x+114 stamps.
after the Great Give-away:
Same had x-48
Helen had x+114 - 360 = x - 246
"Helen had was 2/5 of the number of stamps Sam had"
----> x - 246 = (2/5)(x - 48)
5x - 1230 = 2x - 96
3x = 1134
x = 378
Helen originally had 378+114 or 492 stamps
check:
Sam had 378
Helen had 492
after gifting:
Sam had 378 - 48 = 330
Helen had 492 - 360 = 132
and (2/5)(330) = 132 , looking good!
Whew :)
To solve this problem, let's create an equation to represent the given information.
Let's say the number of stamps Helen had originally is represented by 'H', and the number of stamps Sam had originally is represented by 'S'.
We are given two pieces of information:
1. "When Helen gave away 360 stamps and Sam gave away 48 stamps, the number of stamps Helen had was 2/5 of the number of stamps Sam had." This can be written as:
(H - 360) = (2/5)(S - 48)
2. "Helen had 114 more stamps than Sam." This can be written as:
H = S + 114
Now we can solve the system of equations to find the values of H and S.
Substituting the value of H from the second equation into the first equation:
(S + 114 - 360) = (2/5)(S - 48)
Simplifying:
(S - 246) = (2/5)(S - 48)
Multiplying both sides by 5 to clear the fraction:
5(S - 246) = 2(S - 48)
Expanding:
5S - 1230 = 2S - 96
Subtracting 2S from both sides:
3S - 1230 = -96
Adding 1230 to both sides:
3S = 1134
Dividing both sides by 3:
S = 378
Now, substituting the value of S back into the second equation to find H:
H = 378 + 114
H = 492
Therefore, Helen initially had 492 stamps.