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The Jones family has 3 children. Amy's age plus Betty's age is 29. Amy's age plus Carl's age is 27. Betty's age and Carl's age is 22.
What is the difference between the oldest and the youngest child?
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Solution 1 (Easy and fast solution to the system of 3 linear equations)
a Amy's age;
b Betty's age;
c Carl's age.
a + b = 29, (1)
a + c = 27, (2)
b + c = 22. (3)
You may struggle on solving this system, as "josgarithmeic" does.
But this system is special. It has a special structure and admits admirable solution. You can solve it even mentally !
Add equations (1), (2) and (3). You will get
2a + 2b + 2c = 29 + 27 + 22 = 78.
Divide it by 2 (both sides). You will get
a + b + c = 39. (4)
Now subtract equation (3) from the equation (4). You will get
a = 39 - 22 = 17. Amy's age is 17.
Next subtract equation (2) from the equation (4). You will get
b = 39 - 27 = 12. Betty's age is 12.
Finally, subtract equation (1) from the equation (4). You will get
c = 39 - 29 = 10. Carl's age is 10.
Answer. Amy is 17, Betty is 12 and Carl is 10 years old. The maximal difference of ages is 7 years.
Solved.
This problem is SPECIALLY designed for you to astonish you and to teach you on how to solve such systems of equations.
The method can be directly generalized and extended to much larger systems of equations.
There are MANY lessons in this site where I demonstrate the methods and its applications. See the lessons
- The tricks to solve some word problems with three and more unknowns using mental math
- Joint-work problems for 3 participants
in this site.
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Solution 2 (by applying a brain without solving equations)
Notice that the condition does not require to find individual ages !
It asks about the maximal difference of the ages.
So, we can solve the problem much easier.
It is clear that maximal sum of ages 29 is the sum of the ages of the oldest person and the middle-aged person.
It is also clear that minimal sum of ages 22 is the sum of the ages of the youngest person and the middle-aged person.
Then the difference 29 - 22 = 7 will be (and is) the difference of ages between the oldest and the youngest persons.
We got again 7 years as the difference of ages between the oldest and the youngest persons.
The problem is solved (twice !!)
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Happy learning !!
The Jones family has 3 children. Amy's age plus Betty's age is 29. Amy's age plus Carl's age is 27. Betty's age and Carl's age is 22. What is the difference between the oldest and the youngest child?
Let Amy’s, Betty’s, and Carl’s ages, be A, B, and C, respectively
Then:
A + B = 29 ------- eq (i)
A + C = 27 ------- eq (ii)
B + C = 22 ------- eq (iii)
B – C = 2 -------- Subtract eq (ii) from eq (i) ------- eq (iv)
2B = 24 ------- Adding eqs (iv) & (iii)
B, or Betty is
12 + C = 22 ------- Substituting 12 for B in eq (iii)
C, or Carl is: 22 – 12, or 10 years-old
A + 12 = 29 ------- Substituting 12 for B in eq (i)
A, or Amy’s age = 29 – 12, or 17
Difference between the oldest (Amy), at 17, and the youngest (Carl), at 10 =