Question 1176918: Hi
Serene and Tina have black and white beads. The ratio of white to black beads Serene has is 5:1. The ratio of white to black beads Tina has is 2:3. Serene has 3 times as many beads as Tina
Find the ratio of white beads Serene has to the number of black beads Tina has.
If Tina gives 78 black beads to Serene the ratio of white to black beads Serene has becomes 3:1.
How many beads do both girls have.
Thanks
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the ratio of white beads to black beads for serene is 5 to 1.
this means that, for every 6 beads she has, 5 of them are white and 1 of them are black.
this also means that the total amount of beads she has must be a multiple of 6 in order to preserve the overall ratio of 5 to 1.
the ratio of white beads to black beads for tina is 2 to 3.
this means that, for every 5 beads she has, 2 of them are white and 3 of them are black.
this also means that the total amount of beads she has must be a multiple of 5 in order to preserve the overall ratio of 2 to 3.
you are given that serene has 3 times as many beads as tina.
you are also given that, if tina gives 78 black beads to serene, the ratio of white beads to black beads for serena becomes 3 to 1.
that last statement will serve to fix the number of beads that serene can have.
the original ratio for serene is 5 white to 1 black.
the changed ratio for serene is 3 white to 1 black, after serene is given 78 black beads.
the formula for these two given conditions are:
W / B = 5/1
W / (B + 78) = 3/1
from the first equation, solve for W to get W = 5 * B.
from the second equation, solve for W to get W = 3 * (B + 78).
simplify this to get W = 3 * B + 3 * 78
simplify further to get W = 3 * B + 234
you have two equations that need to be solved simultaneously.
they are:
W = 5B
W = 3B + 234
subtract the second equation from the first to get 0 = 2B - 234.
solve for B to get B = 234 / 2 = 117.
you now know the value that B has to be for serene.
since the ratio of white to black beads for serene is 5 to 1, this means that the number of white beads that serene has must be 5 * 117 = 585.
we have the original number of white beads for serene is 585 and the original number of black beads for serene is 117.
when we add 78 beads that tina gave serene, we get the number of white beads for serene is still 585 and the number of black beads for serene is now 117 + 78 = 195.
the ratio of white beads to black breads for serene is now 585 to 195.
that's a ratio of 3 to 1, which it is supposed to be.
this indicates the arithmetic is correct and that the number of of white beads that serene has is 585 and the number of black beads that serene has is 117 before the transfer of 78 breads from tina to serene.
that means that serene originally had a total of 585 + 117 = 702 beads.
since serene originally had 3 times as many beads as tina had, then tina must have had 702 / 3 = 234 beads.
since the ratio of white beads to black beads that tina originally had was 2 to 3, then the total number of beads that tina had must be a multiple of 5 in order to preserve that ratio.
234 is not a multiple of 5, therefore the number of beads that tina originally had couldn't have been 234.
this means that there is no possible solution that this problem could possibly have that would satisfy all the given requirements.
because of that, i'm assuming this problem can't be solved based on the given set of requirements.
i graphed the problem and noticed something that i wasn't able to see before.
the graph looks like this:
what i noticed is that the number of WHITE beads that serene has is 3 times the TOTAL number of beads that tina has.
this changes the problem.
one of the given requirements now becomes that the number of white beads that serene has is 3 times the number of total beads that tina has.
the number of white and black beads that serene has remains at 585 white and 117 black.
the total number of beads that tina has becomes equal to 585 / 3 = 195.
since the ratio of white to black for tina is 2 to 3, then we get 2/5 * 195 for white and 3/5 * 195 for black to get 78 white beads and 117 black beads for tina.
195 total for tina is divisible by 5, so the ratio of 2 to 3 is preserved.
the numbers for serene and tina are now:
serene has 585 white and 117 black.
tina has 78 white and 117 black.
the ratio of serene white beads to tina black beads is 5 to 1.
this is the same ratio of serene white beads to tina black beads because tina has the same number of black beads as serene has.
the answers to the questions now become:
the ratio of white beads serene has to the number of black beads tina has is 5 to 1.
when tina give serene 78 black beads, the ratio of white to black beads serene has becomes 3 to 1.
the number of beads both girls have is 585 + 117 + 78 + 117 = 897.
but .....,
this solution is only possible if you assume that serene had 3 times as many white beads as tina had total, rather than serene has 3 times as many total beads as tina has.
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