Let's begin by tackling each equation one by one.
Let's denote the numbers as follows:
First number: a
Second number: b
Third number: c
1. The sum of the numbers, a + b + c, is 145.
2. We are also told 7 times the second number, or 7 * b, is equal to twice the first number, or 2 * a.
3. Lastly, twice the second number, or 2 * b, is equal to six times the third number, or 6 * c.
We can simplify the second and third equations to make them easier to handle:
2 * a = 7 * b -> a = (7/2) * b
2 * b = 6 * c -> b = 3 * c
Now we substitute the above expressions into the first equation:
a + b + c = 145 then becomes:
(7/2)*b + b + c = 145,
(9/2)*b + c = 145,
(18/9)*b + (2/2)*c = (290/2),
2*b + c = 290 (Let's call this Equation 1)
Again, substitute b = 3 * c into Equation 1:
2*3*c + c = 290,
6*c + c = 290,
7*c = 290,
After solving for c, we get: c = 290 / 7,
c = 41.43 (approximately, if we round off to two decimal places).
To find b, we substitute c back into b = 3 * c:
b = 3 * 41.43,
b = 124.29 (approximately).
Now, to find a, substitute b back into a = (7/2) * b:
a = (7/2) * 124.29,
a = 435.01 (approximately).
Now, the initial numbers were a = 435.01, b = 124.29, and c = 41.43. The original ratio is a : b : c or 435.01 : 124.29 : 41.43.
Next, we decrease the first number, a, by 15 and increase the third number, c, by 15.
Then, we have:
new_a = a - 15 = 435.01 - 15 = 420.01,
new_c = c + 15 = 41.43 + 15 = 56.43.
Finally, the new ratio is new_a : b : new_c, or 420.01 : 124.29 : 56.43.