A string is cut into two parts.The ratio of lengths of the string and the smaller part is six times the ratio of lengths of the smaller and the larger parts .Find the ratio of the lenghths of the larger and the smaller parts of the string.

Help me solve this question.

let the string be 1 unit of length

shorter piece --- x
longer piece ---- 1-x

ratio of string to smaller part = 1/x
ratio of smaller part to larger part = x/(1-x)

1/x = 6(x/(1-x)
1/x = 6x/(1-x)
6x^1 = 1 - x
6x^2 + x - 1 = 0
(3x - 1)(2x + 1) = 0
x = 1/3 or x = -1/2, the last answer makes no sense

so smaller is 1/3, larger is 2/3
ratio of larger to smaller
= (2/3) / (1/3)
= 2 : 1

Ah, the mysterious case of the cut string! Well, my mathematical friend, let's crack this puzzle together. So we have a string that's cut into two parts, and we need to find the ratio of the lengths of the larger and smaller parts.

Let's call the length of the string "S" (because it's easier than saying "length of the string" every time), and let's call the length of the smaller part "x" and the length of the larger part "y". According to our problem, we know that:

S : x = 6x : y

Now let's simplify this equation a bit. We can cross-multiply to make it easier:

S * y = 6x * x

Simplifying even further:

Sy = 6x^2

Now, we just need to solve for the ratio of y to x. We can divide both sides of the equation by x:

Sy / x = 6x^2 / x

Simplifying once again:

y = 6x

And voila! We've found that the ratio of the lengths of the larger and smaller parts is 6:1. So for every 6 units of length in the larger part, there is 1 unit of length in the smaller part.

I hope that helps, and remember, if you ever need more assistance, I'm here to clown around with your questions!

Let's assume the length of the string is 'L' and the length of the smaller part is 'S'.

According to the problem statement, the ratio of the lengths of the string and the smaller part is six times the ratio of the lengths of the smaller and the larger parts.

Mathematically, we can represent this as:

L / S = 6 * (S / (L - S))

To simplify the equation, we can cross multiply:

L * (L - S) = 6 * S * S

Expanding and rearranging the terms, we get:

L^2 - LS = 6S^2

Now, let's solve this quadratic equation.

L^2 - LS - 6S^2 = 0

We can now factorize this equation:

(L - 3S)(L + 2S) = 0

So, we have two possible solutions:

1. L - 3S = 0
L = 3S

2. L + 2S = 0
L = -2S (which does not make sense for lengths)

Therefore, the ratio of the lengths of the larger and smaller parts can be written as:

L / S = 3S / S = 3:1

So, the ratio of the lengths of the larger and smaller parts is 3:1.

To solve this question, let's assume the length of the string is represented by the variable "x", and let's also assume the length of the smaller part is represented by the variable "a".

We are given that the ratio of the lengths of the string and the smaller part is six times the ratio of the lengths of the smaller and larger parts. Mathematically, this can be represented as:

x/a = 6(a/(x-a))

To find the ratio of the lengths of the larger and smaller parts, we need to find the value of x/a.

Let's solve the equation step by step:

x/a = 6(a/(x-a))

Multiply both sides of the equation by a(x-a) to eliminate the denominators:

x(x-a) = 6a^2

Expanding the equation:

x^2 - xa = 6a^2

Rearranging the terms to form a quadratic equation:

x^2 - 6a^2 - xa = 0

Now, since we are looking for the ratio of the lengths, we can assume a constant value for either "a" or "x". Let's assume a = 1.

Substituting a = 1 into the equation, we get:

x^2 - 6 - x = 0

x^2 - x - 6 = 0

Factoring the quadratic equation:

(x - 3)(x + 2) = 0

This gives us two possible values for x: x = 3 or x = -2. Since length cannot be negative, we can discard the x = -2 solution.

Therefore, the length of the string (x) is 3 times the length of the smaller part (a).

The ratio of the lengths of the larger and smaller parts is x/a, which is 3/1 or simply 3:1.

So, the ratio of the lengths of the larger and smaller parts of the string is 3:1.