Question 28702
1. rose is 6 years older than her sister cora. In four years, she will be nine years 
more than one-half as old as cora. how old are the girls now?

Let   Rose  be  a   years old   and  let   Cora be   b  years  old.
Rose is 6 years older than her sister cora.
a is 6 more than  b  which means
a= b+6  ----(1)
In four years, she will be nine years more than one-half as old as cora.
Rosa is now a yrs old and in 4 yrs she will be (a+4) yrs old
Cora is now b yrs old. and in 4 yrs she will be (b+4) yrs old
(a+4) will be 9 yrs more than (1/2)(b+4)
(a+4) - (1/2)(b+4) = 9
multiplying by 2 through out.
2(a+4)-(b+4)=(2X9)
2a+8-b-4=18
2a-b+8-4=18
2a-b=18-4
2a-b=14  ----(2)
using a= b+6  ----(1)  in this
2(b+6)-b = 14
2b+12-b=14
12+2b-b =14
12+b =14  gives b= 14-12 = 2
b= 2 in (1) we get  a= b+6 = 2+6 =8
a=8, b=2
Answer: Rosa is 8 yrs old and Cora is 2 yrs old.
Verification: Rosa is 8 yrs old and is of course 6 more than  2 which is   Cora’s age.
In  4 years Rosa wii be (8+4) yrs old and Cora will be (2 +4)yrs old.
And of course 12 is 9 more  than half  of  6

2.) Rolly is 9 years younger than greg. 
In seven years, he will be two years more than one-half as old as greg. find their present ages.

Let   Rolly  be  a   years old  and  let  Greg  be  b  years  old.
Rolly is 9 years younger than Greg 
a is 9 less than b
a = b-9   ----(1)
in seven years.he will be two years more than one-half as old as greg.
In 7 yrs  Rolly will be (a+7) and Greg will be (b+7)
(a+7) is 2 more than (1/2)(b+7)
(a+7) = (1/2)(b+7) +2
Multiplying by 2 through out
2(a+7)=(b+7)+(2X2)
2a+14=b+7+4
2a+14=b+11
2a-b = 11-14
2a-b=-3   ----(2)
Using (1) in (2)
2(b-9)-b=-3
2b-18-b=-3
(2b-b)=-3+18
b=15
Putting b=15 in (1)  a= b-9 = 15-9 = 6
Answer:Rolly is 6 yrs old and Greg is 15 years old
Verification: 6 is 9 less than  15 which answers Rolly is 9yrs younger than Creg
In 7 yrs Rolly’s age will be (6+7) = 13 years and Greg’s age will be (15+7) = 22 years and  then Rolly  should be 2 more than half of Greg and  of  course13  is 2 more than (1/2) of  (22)             

3)A mother is 20 years older than her daughter. five years from now, she will be thrice as old as her daughter will then be. how old is the daughter?  
Let the mother be M yrs old and  let the daughter  be D yrs  old.
M is 20 more than  D    That  is M= D+20  ----(1)
Five years from now, mother   will be (M+5 ) and daughter  will  be (D+5) yrs old.
Then mother will be thrice as old  as  the daughter 
(M+5) =  3(D+5)   ----(2)
Putting (1) in (2)
[(D+20)+5]  = 3D+15
D+25 = 3D+15
25-15=3D-D
10=2D
D=10/2 = 5
Putting  D=5 in    (1)
M=D+20----(1)
M=5+20=25
Answer: Mother  is 25 yrs old and daughter  is  5 yrs  old.
Verification: Mother is 25 and is  20 more than  5 which is the daughter’s  age.
After 5 years, mother will be 30 and daughter will be 10yrs and 30 is of course thrice 10       

4. A mother is twice as old as her daughter. eight years from now, she will be twenty years older than her daughter will then be. how old is the mother now?
Let the mother be M yrs old and  let the daughter  be D yrs  old.
M is twice D    That  is M= 2D  ----(1)
Eight years from now, mother will be (M+8 ) and daughter  will  be (D+8) yrs old.
Then mother will be 20 more than  the daughter 
(M+8) =  (D+8) +20  ----(2)
Putting (1) in (2)
(2D+8)  = (D+8) +20
2D-D = 20
D=20
Putting  D=20 in    (1)
M=2D----(1)
M=2X20=40
Answer: Mother  is 40 yrs old and daughter  is  20 yrs  old.
Verification: Mother is 40 and is  twice  20 which is the daughter’s  age.
After 8 years, mother will be 48 and daughter will be 28 yrs and 48 is of course 20 more than 28 
 
The problem is wrong. Here is the corrected  one.
5. Edwin is thrice as old as Ruel. In three years, Edwin will be 13 years more than twice as old as Ruel. find their present ages.
The problem is wrong. Here is the corrected  one.

Let  Edwin  be  E   years old  and  let  Ruel  be  R  years  old.
Edwin is thrice as old as Ruel
E=3R----(1)
In three years, Edwin will be 13 years more than twice as old as Ruel.
In 3 yrs  Edwin will be (E+3) and Ruel will be (R+3)
(E+3) is 13 yrs more than twice (R+3)
(E+3) =2(R+3)+13   ----(2)
 Putting (1) in (2)
(3R+3)=2R+6+13
3R-2R=19-3
R=16
Putting   R=16  in (1) gives  E = 3R = 3X16 = 48
Answer: Edwin is 48 years old and  Ruel  is 16 yrs   old

Verification: Edwin is  48 yrs old  which  is    thrice  16= Ruel’s age.
In 3 yrs Edwin’s age will be (4 8 +3) = 51 years and Ruel’s age will be (16+3) = 19 years and  then Edwin   should be 13 more than twice Ruel and  of  course 51  is 13 more than twice(19)