Question 498846: the sum of the ages of a man and his wife is 6 times the sum of the ages of their children.two years ago the sum of their ages was 10 times the sum of the ages of their children at that time.after six years the sum of their ages will be 3 times the sum of the ages of their children . How many children do they have ?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! we let a = the sum of the ages of the adults.
we let c = the sum of the ages of the children.
we know that the number of adults is equal to 2.
we set the number of children to be equal to x.
our formulas becomes:
first formula:
a = 6c
this means that sum of the ages of the adults today is equal to 6 times the sum of the ages of the children today.
second formula:
a - 2*2 = 10 * (c - 2*x)
this means that the sum of the ages of the adults 2 years ago minus 2 * 2 was equal to the sum of the ages of the children 2 years ago minus 2 * x.
since there were 2 adults, the sum of their ages was decreased by 2 * 2 = 4
since there were x children the sum of their ages was decreased by 2 * x.
simplification of the formula gets:
a - 4 = 10 * (c - 2*x)
third formula:
a + 6*2 = 3 * (c + 6*x)
this means that the sum of the ages of the adults 6 years from now + 6*2 will be equal to the sum of the ages of the children + 6*x.
we simplify this equation to get:
a + 12 = 3 * (c + 6*x)
we have 3 formulas to work with:
they are:
a = 6*c (equation 1)
a - 4 = 10 * (c - 2*x) (equation 2)
a + 12 = 3 * (c + 6*x) (equation 3)
since a = 6*c from the first equation, we can substitute for a in equation 2 and equation 3 to get:
6*c - 4 = 10 * (c - 2*x) (equation 4)
6*c + 12 = 3 * (c + 6*x) (equation 5)
we simplify both equations to get:
6c - 4 = 10c - 20x (equation 6)
6c + 12 = 3c + 18x (equation 7)
i omitted the * which means multiply since it is implied and so does not need to be shown here. if there is confusion on whether multiplication is indicated or not, i will show the * to indicate multiplication.
we subtract 6c from both sides of each of these equations to get:
-4 = 4c - 20x (equation 8)
12 = -3c + 18x (equation 9)
equations 8 and 9 need to be solved simultaneously so we can find the values of x and c.
multiply both sides of equation 8 by 3 and multiply both sides of equation 9 by 4 to get:
-12 = 12c - 60x (equation 10)
48 = -12c + 72x (equation 11)
add these 2 equations together to get:
36 = 12x
divide both sides of this equation by 12 to get:
x = 3
that should be your answer.
we can now go back and solve for c.
we'll use equation 8.
equation 8 says:
-4 = 4c - 20x
we replace x with 3 to get:
-4 = 4c - 60
we add 60 to both sides of this equation to get:
4c = 56
we divide both sides of this equation by 4 to get:
c = 14
since a = 6c (from our first formula way up top), we get:
a = 6 * 14 = 84
we now have:
a = 84
c = 14
x = 3
the number of children is equal to 3.
if this is correct, then all the other formulas should work out.
our first formula says that a = 6c.
84 = 6 * 14 becomes 84 = 84.
that is correct so we're good here.
our second formula says that:
a - 4 = 10 * (c - 2*x)
substituting 84 for a and 14 for c and 3 for x, we get:
84 - 4 = 10 * (14 - 2*3) which becomes 80 = 10 * (14-6) which becomes 80 = 10*8 which becomes 80 = 80.
that is correct so we're good here.
our third formula says that:
a + 12 = 3 * (c + 6*x)
substituting 84 for a and 14 for c and 3 for x, we get:
84 + 12 = 3 * (14 + 6*3) which becomes 96 = 3*(14 + 18) which becomes 96 = 3 * (32) which becomes 96 = 96.
that is correct so we're good here.
looks like we're good all around.
the answer to the question is:
the number of children is equal to 3.
the bonus answers are:
the sum of the ages of the adults is 84.
the sum of the ages of the children is 14.
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