Question 1168227: a man leaves 60000$to his children and grandchildren, seven in all. the children receive 1/3 of it, which is 2000 more a piece than the grandchildren get. how much is the amount received by the grandchildren
Found 2 solutions by Theo, MathTherapy:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the answer is 2 children and 5 grandchildren.
each child received 10,000.
each grandchild received 8,000.
2 * 10,000 = 20,000
5 * 8,000 = 40,000
total is 60,000
there may be different ways of getting it, but here's how i got it.
let x = number of children.
let y = number of grandchildren
let c = amount that each child got.
let g = amount that each grandchild got.
then xc = amount that x children got.
then yg = amount that y grandchildren got.
you know that x + y = 7
you know that xc = 20,000
you know that xc + yg = 60,000
from that, you determined that yg = 40,000
you know that c = g + 2000
first thing i did was divide all the money by 1000 to i didn't have to carry all those zeroes.
what i knew became.
x + y = 7
xc + yg = 60
xc = 20
yg = 40
c = g + 2
the goal was to reduce the number of unknowns as much as possible so i only had to solve for one unknown at a time.
from xc = 20, i replaced c with g + 2 to get x(g+2) = 20
from yg = 40, i replaced y with (7 - x) to get (7-x)g = 40
from x(g+2) = 20, i got xg + 2x = 20
from (7-x)g = 40, i got 7g - xg = 40
i added these two equations together to get 7g + 2x + xg - xg = 60
the xg canceled out and i was left with 7g + 2x = 60
i needed to put the equation in terms of x or g, not both.
i chose to solve for g.
from yg = 40, i got g = 40/y
from x+y=7, i got y=7-x
g = 40/y became g = 40/(7-x)
in the equation of 7g + 2x = 60, i replaced g with 40/(7-x) to get:
7(40/(7-x)) + 2x = 60
i multiplied both sides of the equation by (7-x) to get:
7*40 + (7-x)(2x) = 60(7-x)
i simplified to get:
280 + 14x - 2x^2 = 420 - 60x
i subtracted 420 from both sides of the equation and added 60x to both sides of the equation to get:
280 - 420 + 14x + 60x - 2x^2 = 0
i combined like terms to get:
-140 + 74x - 2x^2 = 0
i multiplied both sides of the equation by -1 and ordered the terms in descending order of degree to get:
2x^2 - 74x + 140 = 0
i divided both sides of the equation by 2 to get:
x^2 - 37x + 70 = 0
i solved this quadratic equation to get x = 2 or x = 35
since x + y = 7, x couldn't be 35, so i tried x = 2
when x = 2, y = 5,
my original equation of x + y = 7 became 7 = 7 which is true.
my original equation of xc + yg = 60 became 2c + 5g = 60
since i know that c = g + 2, i replaced x with g + 2 to get:
2*(g+2) + 5g = 60
i simplified to get:
2g + 4 + 5g = 60
i combined like terms and subtracted 4 from both sides of the equation to get:
7g = 56
i solved for g to get:
g = 8
since c = g + 2, i got:
c = 10
i multiplied the values of c and g by 1000 to get back to the original numbers.
i got c = 10,000 and g = 8,000
xc + yg = 60,000 became 2*10,000 + 5*8,000 = 60,000 which became 20,000 + 40,000 = 60,000 which became 60,000 = 60,000 which is true.
everything checked out and so my solution was that x = 2, y = 5, c = 10,000 and g = 8,000.
this means that:
2 children received 10,000 each for a total of 20,000
5 grandchildren received 8,000 each for a total of 40,000
totalmoney given to children and grandchildren was 60,000
if you were asked how much was the amount received by the grandchildren, then you could have skipped all of these calculations and just said 40,000 because you knew that the children received 20,000 and that the total given was 60,000.
if you were asked how much each grandchild got, then the answer would be 8,000.
Answer by MathTherapy(10551)  (Show Source):
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