Ten years ago Jane was four times as old as Bianca. Now she is only twice as old as Bianca. Find their present ages.
J - 10 = 4 (B - 10)
J = 2 B
substituting ... 2 B - 10 = 4 B - 40 ... 30 = 2 B
Let's assume Jane's age 10 years ago was x, and Bianca's age 10 years ago was y.
According to the given information, Jane was four times as old as Bianca 10 years ago, so we can write the equation: x = 4y.
Now, let's determine their present ages:
After 10 years, Jane's age will be x + 10, and Bianca's age will be y + 10.
According to the second part of the given information, Jane is now only twice as old as Bianca, so we can write the equation: x + 10 = 2(y + 10).
Simplifying this equation, we get: x + 10 = 2y + 20.
By substituting x = 4y from the first equation into the second equation, we can solve for y:
4y + 10 = 2y + 20,
2y = 10,
y = 5.
Now, substitute the value of y into the first equation to find x:
x = 4 * 5 = 20.
Therefore, Jane's present age is 20 years, and Bianca's present age is 5 years.
To find their present ages, we can use algebraic equations. Let's assume Jane's present age is J and Bianca's present age is B.
According to the problem, ten years ago, Jane was four times as old as Bianca. So we can write the first equation:
J - 10 = 4(B - 10)
Now, the second equation states that Jane is currently only twice as old as Bianca. So we can write the second equation:
J = 2B
Now we have a system of two equations. We can solve this system to find the values of J and B.
First, let's simplify the first equation:
J - 10 = 4B - 40
Simplifying further:
J = 4B - 30
Now we substitute the value of J from the second equation into the first equation:
2B = 4B - 30
Subtracting 2B from both sides:
0 = 2B - 30
Adding 30 to both sides:
30 = 2B
Dividing both sides by 2:
15 = B
Now we substitute this value of B into the second equation to find J:
J = 2B
J = 2(15)
J = 30
So Jane's present age is 30, and Bianca's present age is 15.