Bel is 5 times older than Tina.
6 years ago,
Bel was 7 times older than Tina.
what are their ages now.
a crucial phrase is "5 times older" and "7 times older".
this is not the same thing as "5 times as old" and "7 times as old".
if i am 2 times as old as you, then, if you are 10 years old, i am 20 years old.
if i am 2 times older than you, then, if you are 10 years old, i am 20 years older than you which makes my age 10 + 20 = 30.
i'll solve this problem with that interpretation to see where that leads us.
let y = bel's age and let x = tina's age.
the first equation becomes y = x + 5 * x which simplifies to y = 6 * x.
the second equation becomes y - 6 = (x - 6) + 7 * (x - 6) which becomes y - 6 = 8 * (x - 6).
solve for y in the second equation to get y = 8 * (x - 6) + 6
your 2 equations that need to be solved simultaneously are:
y = 6 * x
y = 8 * (x - 6) + 6
simplify the second equation to get y = 8 * x - 48 + 6.
this simplifies further to get y = 8 * x - 42
your 2 equations that need to be solved simultaneously become:
y = 6 * x
y = 8 * x - 42
subtract the first equation from the second to get 0 = 2 * x - 42
add 42 to both sides of that equati0on to get 42 = 2 * x
solve for x to get x = 21
remember, y is bel's age and x is tina'a age, so this is saying that tina is 21 years old.
since y = 6 * x, then y = 126.
since y is bel's age, then this is saying that bel is 126 years old.
that's very old, but that's the way the math came out.
we now know that tina is 21 years old and bel is 26 years old.
the first problem statement says that bel is 5 times older than tina.
tina is 21.
5 * 21 = 105
21 + 105 = 126.
the first problem statement is confirmed as true.
the second problem statement says that, 6 years ago, bel was 7 times older than tina.
6 years ago, bel was 126 - 6 = 120.
6 years ago, tina was 21 - 6 = 15.
this made bel 120 - 15 = 105 years older than tina.
105 is equal to 7 * 15.
the second problem statement is confirmed to be true.
you can solve these two equations graphically as shown below.
the first equation is y = 6 * x
the second equation is (y - 6) = 8 * (x - 6).
the intersection of the 2 lines generate is at x = 21 and y = 126.
NOTE !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
if the problem statement was:
Bel is five times as old as Tina.
six years ago Bel was seven times as old as Tina.
what are their ages now?
then you would get a different answer as shown below:
your 2 equations that need to be solved simultaneously become:
y = 5 * x
y - 6 = 7 * (x - 6)
in the second equation, replace y with 5 * x to get:
5 * x - 6 = 7 * (x - 6)
simplify to get 5 * x - 6 = 7 * x - 42
subtract 5 * x from both sides of this equation and add 42 to both sides of this equation to get 36 = 2 * x
solve for x to get x = 18
since y = 5 * x, then y = 90
you have x = 18 and y = 90
tina is 18 and bel is 90.
90 / 18 = 5.
bel is 5 times as old as tina, confirming the first statement is correct.
6 years ago, bel is 84 and tina is 12.
7 * 12 = 84, confirming the second statement is correct.
you have 2 answers.
if the problem statement states "older", then tina is 21 and bel is 126.
if the problem statement states "as old as", then tina is 18 and bel is 90.
bel being 90 is easier to accept than bel is 126.
please check to make sure whether you meant "older" or "as old as".