Two plus lenses have focal lengths in the ratio 7:1. When placed together in close contact the focal length of the combination is +12.5cm. What are the focal lengths of the two lenses?

Let's denote the focal length of the first lens as x, and the focal length of the second lens as y.

According to the given information, the ratio of the focal lengths is 7:1. This can be written as:

x:y = 7:1

Now, when the lenses are placed together in close contact, their focal lengths add up. Therefore, we can write the equation:

x + y = 12.5 cm

We have two equations:

x:y = 7:1 --(Equation 1)
x + y = 12.5 --(Equation 2)

To solve these equations, we can use the fact that the ratio of 7:1 implies that the focal lengths are in the form 7k and k, where k is a constant.

Substituting the values of x and y in equation 1, we get:

(7k):(k) = 7:1

This implies that 7k = 7 and k = 1.

Substituting k = 1 in equation 2, we get:

x + y = 12.5
7(1) + 1 = 12.5
7 + 1 = 12.5
8 = 12.5

This is not a valid solution, which means that our assumption that the lenses have focal lengths in the ratio 7:1 was incorrect.

Therefore, there is no valid solution to this problem with the given information. Please double-check the values and ratios provided.

To find the focal lengths of the two lenses, we can set up a proportion using the given information.

Let's assume the focal lengths of the two lenses are f1 and f2, respectively.

According to the problem, the ratio of the focal lengths is 7:1. Therefore, we can set up the proportion as follows:

f1/f2 = 7/1

Now, we know that when the lenses are placed together in close contact, the focal length of the combination is +12.5 cm. This means that the combined focal length is equal to the sum of the individual focal lengths.

So, we can set up another equation:

1/f1 + 1/f2 = 1/12.5

To solve this system of equations, we can rearrange the first equation to solve for one variable in terms of the other:

f1 = 7f2

Substituting this expression into the second equation, we get:

1/(7f2) + 1/f2 = 1/12.5

Simplifying this equation, we get:

1/f2 + 7/f2 = 1/12.5

Combining the fractions on the left side, we get:

8/f2 = 1/12.5

Cross-multiplying, we have:

8 * 12.5 = f2

f2 = 100 cm

Now, substituting the value of f2 into the first equation, we get:

f1 = 7 * 100 = 700 cm

Therefore, the focal lengths of the two lenses are 700 cm and 100 cm, respectively.