Five years ago Katie was ten times as old as her daughter Melanie was. In 15 years
Katie will be three times as old as Melanie will be in 11 years. Use simultaneous
equations to determine Katie and Melanie’s current ages.
let Katie's present age be k
let Melanie's present age be m
Five years ago:
Katie ---- k - 5
Melanie -- m-5
k-5 = 10(m-5)
k - 10m = -45 **
2nd condition:
Katie's age in 15 years = 3(Melanies age in 11 years)
k + 15 = 3(m + 11)
k+15 = 3m + 33
k - 3m = 18 ***
subtract ** from ***
7m = 63
m = 9
in ** , k - 90 = -45
k = 45
Katie is now 45 years old and Melanie is now 9.
check:
5 years ago, Melanie was 4 and Katie was 40
10 times as old? YES, check!
in 15 years, Katie will be 60
in 11 years Melanie will be 20.
3 times as old? YES, Check!
Thank you so much Reiny. I have spent 2 hours looking at this and it has driven me crazy. I really need some practice in unravelling the questions!
To solve this problem using simultaneous equations, let's first assign variables to Katie and Melanie's current ages.
Let's say:
- Katie's current age is K.
- Melanie's current age is M.
Now, let's translate the given information into equations:
1. Five years ago, Katie was ten times as old as her daughter Melanie was.
This can be represented as:
K - 5 = 10(M - 5)
2. In 15 years, Katie will be three times as old as Melanie will be in 11 years.
This can be represented as:
K + 15 = 3(M + 11)
Now we have a system of two equations with two variables. We can solve this system using the method of substitution or elimination:
Method 1: Substitution
- Solve one equation for one variable (e.g., equation 1 for K).
- Substitute the expression for that variable into the other equation.
- Solve the resulting equation for the other variable.
- Substitute the value of that variable back into the first equation to find the value of the first variable.
Method 2: Elimination
- Multiply one or both equations by appropriate constants to make the coefficients of one variable the same or multiples of each other.
- Add or subtract the resulting equations to eliminate that variable.
- Solve the resulting equation for the remaining variable.
- Substitute the value of that variable back into one of the original equations to find the value of the other variable.
Both methods will lead to the same solution: Katie's current age (K) and Melanie's current age (M).