Algebra age riddle word problem

[late347]

Homework Statement

son and father and granpa have same value for their birthday (except the year ostensibly)

father is nine times the son's age now. calculate the granpa's age now.

clues
- five years ago, grandpa was 32 years older than what the father was at five years ago.
-in five years time, granpa will be 56 years older than what the son would be in five years time.

Homework Equations

The Attempt at a Solution

son's age = x

father's age = 9x

granpa = y

five years ago, everybody was younger and limber and more active at these values

x-5
9x-5
y-5 (however granpa was at this age already somewhat older than the father.)

it was said that five years ago granpa was much older than the father at that same time. we must balance the equation between granpa and father

y-5 = 9x-5 +56

y= 9x +56I got stuck there with nowhere to go about solving the problem.

This was prob the second most difficult problem in our math class...
Clues would be welcome as to what would be prudent tactic to solve this dreadful age riddle .

 

[ehild]

Homework Statement

son and father and granpa have same value for their birthday (except the year ostensibly)

father is nine times the son's age now. calculate the granpa's age now.

clues
- five years ago, grandpa was 32 years older than what the father was at five years ago.
-in five years time, granpa will be 56 years older than what the son would be in five years time.

Homework Equations

The Attempt at a Solution

son's age = x

father's age = 9x

granpa = y

five years ago, everybody was younger and limber and more active at these values

x-5
9x-5
y-5 (however granpa was at this age already somewhat older than the father.)

it was said that five years ago granpa was much older than the father at that same time. we must balance the equation between granpa and father

y-5 = 9x-5 +56

y= 9x +56
.

Five years ago grandpa was 32 years older than what the father was at five years ago.
And there is the other condition, five years later.

 

[late347]

Five years ago grandpa was 32 years older than what the father was at five years ago.
And there is the other condition, five years later.

Y-5= 9x -5 +32
Y= 9x +32

The other equation was that the granpa and son

X+5 +56 = y+ 5 ((now granpa and son are equal age as prescribed))I was wondering about the mathematical theory behind the solving methods for simultaneous equations

What exactly speaking is allowed in terms of operations and manipulation with regard to these groups or pairs of equations. (Without disturbing the solution for the group or pair)

In any normal equation like... only one of them.

You are allowed to put things or take things from both sides in equal measure. BUT don't multiply or divide by zero (I think)

This lack of knowledge really affected my confidence at tackling the problem at first.

It seems to me though that the sons age and fathers age cannot really be tied into the simultaneous equations as the third equation? There were no other clues than the father is 9 times son's age currently.

 

[ehild]

There were 3 statements about the ages of son, father and grandpa.

father is nine times the son's age now. calculate the granpa's age now.

clues
- five years ago, grandpa was 32 years older than what the father was at five years ago.
-in five years time, granpa will be 56 years older than what the son would be in five years time.

This resulted in the equations you found at the end:
Y-5= 9x -5 +32
X+5 +56 = y+ 5
or eliminating the "5" :
Y=9x+32
x+56=Y

You can make the Y-s equal :9x+32=x+56 and solve for x. Then add 56 to get Y.
Watch this video about methods of solution of linear systems of equations.

 

[late347]

I made one equation pair such that

Y= 9x + 32
Y=× +56

I assume that the truth value of simulatenous equa stays the same when one flips the equation from one side simply by rearranging it.

Y= k × a
K×a = y
a×k =y
Y= a×k

Multiply by (-1) and add together
-y = -x -56

Y= 9x +32

0 = 8x -24
24 = 8x
X= 3
Son's age currently seems to be 3

Plug x=3 into for example the
Y= 9x +32
Y= 27 +32
=59

Granpa is 59 years currently.

I thought about making a table at the end to see whether or not all the clues are fulfilled or not

In future the son is 3+5=8
In future the granpa is 59 + 5= 64
64-8 =56
Age difference is good.

Currently son is 3 and father is 9x3= 27

Seems plausible. The father is old enough to be an adult and have children biologically.

In the past father was 27-5=22
In the past granpa was 32+27-5=54

54 -22= 32 age difference is good

 

[ehild]

Well done!

I made one equation pair such that

Y= 9x + 32
Y=× +56

You can simply write that 9x+32=x+56.

 

[late347]

Well done!You can simply write that 9x+32=x+56.

Can you answer my question about whether it is allowed to rearrenge individual equation inside the simultaneous equations bracket?

Rearrenge one equation witj respect to the others such that

A×B +c= D
D= c + B×A

Etc
...

 

[late347]

Does the truth value remain the same even with the above type of rearranging?

 

[late347]

My own guess is that the ordering of the terms anf the equations in simultanous equations only matters when...
One uses computer solver
One uses matrix technique with pen and paper (I once knew how to do the matrix method but I have forgotten long since everytging about it)

 

[ehild]

You can apply all identities of addition and multiplication inside every equations. And an equations is valid either you write A=B or B=A. You spend the same amount of money if you buy two pens, each for 1$ first, then a book for 3$; or a book for 3$ first, and then two pens, each for 1$ .