Jack is four years older than Mark and eight years older than Dave. The product from Mark's and Paul's age is greater by 16 than the product from Jack's and Dave's ages. In this group of 4 children, two of them are twins. Who are the twins? What is the age of each of the four boys?
Mark's age = X years.
Jack's age = X+4 years.
Dave's age = (X+4)-8 = X-4 years.
J*D = (x+4)(x-4) = x^2-16.
a. M*P = (x+4)(x-4)+16 = x^2-16+16=X^2
= x*x. Therefore, Mark and Paul are each X years old and twins.
j = m+4 = 8+d
mp = 16+jd
clearly, Jack is not a twin with either Mark or David.
So, Paul is one of the twins. Either
p=j or p=m or p=d
See what you can do with that. You now will have three equations in three variables, after getting rid of p.
To solve this problem, we'll start by assigning variables to the ages of the four boys. Let's say Mark's age is M, Jack's age is J, Dave's age is D, and Paul's age is P.
According to the given information:
1) Jack is four years older than Mark, so J = M + 4.
2) Jack is also eight years older than Dave, so J = D + 8.
Now let's consider the second part of the information:
3) The product of Mark's and Paul's age is greater by 16 than the product of Jack's and Dave's ages. This can be represented as (M * P) = (J * D) + 16.
We have three equations:
J = M + 4
J = D + 8
(M * P) = (J * D) + 16
Now let's substitute the value of J from equation 1 into equations 2 and 3:
(M + 4) = D + 8 --> M - D = 4 (Equation 4)
(M * P) = ((M + 4) * D) + 16 --> (M * P) = (M * D) + (4 * D) + 16 --> M * (P - D) = 4D + 16 --> M = (4D + 16) / (P - D) (Equation 5)
Now we need to find possible values for D and P that satisfy equations 4 and 5.
Let's start with different values for D, and we can then calculate M using equation 5. We will consider positive integer values for D.
For example, let D = 1. Substituting this into equation 4, we get M - 1 = 4, which gives us M = 5. Now we can find P using equation 5: P = (4*1 + 16) / (P - 1) = 20 / (P - 1).
Let's list out a few possible values for D, M, and P:
D = 1, M = 5, P = 20/0 (not possible)
D = 2, M = 6, P = 20/1 = 20 (possible)
D = 3, M = 7, P = 20/2 = 10 (possible)
D = 4, M = 8, P = 20/3 (not possible)
From here, we can see that the possible ages for the four boys are:
Mark is 6 years old (M = 6)
Jack is 10 years old (J = M + 4 = 6 + 4 = 10)
Dave is 2 years old (D = 2)
Paul is 20 years old (P = 20)
Therefore, the twins in this group are Mark and Dave, with ages 6 and 2 respectively, and Jack is 10 years old while Paul is 20 years old.