Sally is thinking of a ratio.the simplest form of the ratio is 2:3.if she subtracts 1 from the first number in her ratio and adds 2 to the second number of her ratio,the simples form of new ratio is 1:2.Find the ratio that Sally is thinking of
x/y = 2/3
(x-1)/(y+2) = 1/2
The original ratio was 8/12
10/20
To find the ratio that Sally is thinking of, we can start by representing it as a fraction. Let's assume that Sally is thinking of the ratio as "x:y".
From the given information, we know that the simplest form of the original ratio, "x:y", is 2:3. This can be written as the fraction (x/y) = 2/3.
Next, we are told that Sally subtracts 1 from the first number and adds 2 to the second number of her ratio. This results in a new ratio, "x-1:y+2", with the simplest form being 1:2. This can be written as the fraction (x-1)/(y+2) = 1/2.
Now we have two equations:
(x/y) = 2/3 ---(1)
(x-1)/(y+2) = 1/2 ---(2)
To solve the system of equations, we can use cross multiplication or substitution. Let's use substitution here.
From equation (1), we can write x = (2/3)y.
Substituting this value of x into equation (2):
((2/3)y - 1)/ (y+2) = 1/2
Next, we can cross multiply:
2[ (2/3)y - 1 ] = (y+2)
Expanding and simplifying:
(4/3)y - 2 = y + 2
Let's isolate the variable y:
(4/3)y - y = 2 + 2
(4/3 - 1)y = 4
(-1/3)y = 4
y = -4 * (-3/1) (multiply both sides by -3)
y = 12
Now that we've found the value of y, we can substitute it back into equation (1) to find x:
x/y = 2/3
x/12 = 2/3
3x = 24
x = 8
Therefore, the ratio that Sally is thinking of is 8:12, which simplifies to the simplest form of 2:3.