John is three times as old as his son, Michael, and in x year time he will be twice as old as him
(i) write down expressions for Johns and Michaels age in x years time
(ii) write down and solve an equation in x
I know the answers as they are in the back of my book although I don't understand WHY that is, so please can someone explain it to me? Thank you so much in advance!
call Michael's age m
then John's age is 3 m
in x years, Michael will be (m + x) years old
in x years, John's age will be 3 m + x
The problem statement says that
3 m + x = 2 (m + x)
3 m + x = 2 m + 2 x
x = m
for example if today Michael is 10 and John is 30
then x = m = 10
and in 10 years
Michael is 20
and John is 40
and sure enough 40 is twice 20
You are supposed to write down two equations and solve them to obtain rhe age of John (J) and of Michael (M).
Let John's age now be J. Let Michael's age now be M. In x years, John's age will be
J' = J + x
and Michael's age will be
M' = M + x
If J'/M' = 2, then
(J+x)/M+x) = 2
J +x = 2M + 2x
You also know that J = 3M, so
3M + x = 2M + 2x
M = x
No matter what John's age is now, there is an x value that satisfies the two requirements. That is because you have only two equations in three unknowns.
1--J = 3S
2-- J + X = 2(S + X)
3--3S + x = 2S + 2x
4--S = x
5--J = 3x
6--Select an x and solve for J and S
7--x........10...15...20...25...etc.
...S........10...15...20...25...etc.
...J........30...45...60...75...etc.
.S+x........20...30...40...50...etc.
.J+x........40...60...80..100...etc.
(J+x)/(S+x)..2....2....2....2
Sure! I'd be happy to explain it to you.
Let's start by breaking down the information given in the problem. Here are the key points:
1. John is currently three times as old as his son, Michael.
2. In x years, John's age will be twice as old as Michael's.
Now let's find the expressions for John's and Michael's ages in x years.
(i) Expressions for John's and Michael's ages in x years:
Let's say Michael's current age is M. According to the first point, John's current age would be three times Michael's age, so John's current age is 3M.
In x years, John's age will be 3M + x, since he is currently 3M years old and x years have passed.
Michael's age in x years can be expressed as M + x, as he is currently M years old and x years have passed.
Therefore, the expressions for John's and Michael's ages in x years are:
John's age in x years: 3M + x
Michael's age in x years: M + x
(ii) Equation to solve in x:
Based on the second point, John's age in x years will be twice as old as Michael's age in x years. So we can equate the expressions we found:
3M + x = 2(M + x)
Now let's simplify and solve this equation to find the value of x:
3M + x = 2M + 2x
Subtract 2M from both sides:
M + x = 2x
Subtract x from both sides:
M = x
So the value of x is M.
To summarize, the expressions for John's and Michael's ages in x years are:
John's age in x years: 3M + x
Michael's age in x years: M + x
And the equation to solve in x is M = x.