the age of the older child is x.
the age of the younger child is x-5.
the age of the parent is 2 * (x+x-5) = 2 * (2x-5) = 4x-10.
20 years from now:
the age of the older child is x+20.
the age of the younger child is x+15.
the age of the parent is 4x+10.
you are given that, 20 years from now, 15 * the age of the parent will be the product of the age of both children.
this means that 15 * (4x+10) = (x+20) * (x+15).
simplify this equation to get 60x + 150 = x^2 + 35x + 300
subtract (60x + 150) from both sides of the equation to get:
(60x + 150) - (60x + 150) = x^2 + 35x + 300 - (60x + 150).
simplify this to get 0 = x^2 - 25x + 150.
factor the right side of this equation to get 0 = (x-10) * (x-15).
solve for x to get x = 10 or x = 15.
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when x = 10, x-5 = 5 and 4x-10 = 30
this means the age of the older child is 10 and the age of the younger child is 5 and the age of the parent is 30.
20 years from now, the age of the older child is 30 and the age of the younger child is 25 and the age of the parent is 50.
15*50 = 750 = 15 * the age of the parent 20 years from now.
30*25 = 750 = the product of the age of both of the children 20 years from now.
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when x = 15, x-5 = 10 and 4x-10 = 50
this means the age of the older child is 15 and the age of the younger child is 10 and the age of the parent is 50.
20 years from now the age of the older child is 35 and the age of the younger child is 30 and the age of the parent is 70.
15*70 = 1050 = 15 * the age of the parent 20 years from now.
35*30 = 1050 = 15 * the age of the parent 20 years from now.
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all formulas check out and so the possible ages of the children are (15 and 10) or (10 and 5).
your questions follow with their answers.
A
a) Write an expression for the current age of the young child.
x-5
b) Write an expression for the current age of the parent.
4x-10
B
In twenty years time, the product of the children's age will be 15 times the age of their parent.
a) Form an equation in x and hence determine the present possible ages of the elder child
(x+20) * (x+15) = 15 * (4x+10) 20 years from now.
solve for x to get x = 10 or 15.
b) Find the present possible ages of the parent
present possible ages for the parent are 30 if the oldest child is 10, and 50 if the oldest child is 15.
c)Determine the possible ages of the younger child in 20 years time.
younger child could be 30 or 25 in 20 years time.
graphical solutions based on the ages of the children and parent in 20 years time and the given relationship between them.
formula is (x+20) * (x+15) = 15 * (4x+10).
you can graph each equation separately and then look for the intersection, or you can take the second graph and subtract it from the first graph to get one single graph that then shows the solution.
coordinates in both graphs are shown in (x,y) format.
the x part of the coordinate point tells you the age of the oldest child.
equations for the first graph are:
y = (x+20) * (x+15)
y = 15 * (4x+10)
the first equation is the product of the children's ages 20 years from now.
the second graph is the age of the father 20 years from now, multiplied by 15.
the intersection of the graph of these two equations shows the solution.
equation for the second graph is:
y = (x+20) * (x+15) - 15 * (4x+10)
this is the result of taking (x+20) * (x+15) = 15 * (4x+10) and then subtracting ((15 * (4x+10)) from both sides of the equation.
you get (x+20) * (x+15) - 15 * (4x+10) = 0.
to graph, just replaced the 0 with y.
your solution is when y = 0.
this equation is simplified to y = x^2 + - 25x + 150.
both equations create the exact same graph shown below.
