the points A,B,C and D lie on a straight line. The ratio of the length of AB to the length of BC is 3:4 and the ratio of the length of AC to the length of CD is 2:1. Find the ratio of the length of BC to the length of AD ?

AC = AB+BC = 3+4 = 7

AC/CD = 2/1, so CD = 7/2

AD = AC+CD, so
BC/AD = 4/(7 + 7/2) = 8/21

To find the ratio of the length of BC to the length of AD, we need to find the individual lengths of BC and AD.

Let's assume that the length of AB is 3x and the length of BC is 4x (according to the given ratio). Similarly, assume the length of AC is 2y and CD is y.

Now, we have the following lengths:
AB = 3x,
BC = 4x,
AC = 2y,
CD = y.

Since points A, B, C, and D are on a straight line, we can add their lengths to get the total length:

AB + BC + CD + AD = AC.

Substituting the lengths we have:

3x + 4x + y + AD = 2y.

Combining like terms:

7x + y + AD = 2y.

Moving all terms involving AD to one side:

AD = 2y - y - 7x.

Simplifying:

AD = y - 7x.

Now that we have the individual lengths of BC and AD, we can find the ratio:

Ratio of BC to AD = BC / AD.

Substituting the lengths we found:

Ratio of BC to AD = 4x / (y - 7x).

Therefore, the ratio of the length of BC to the length of AD is 4x / (y - 7x).

AB:BC=3:4, so AC=AB+BC=3+4=7;

AC:CD=2:1, so AC=2;

because we get 2 different length for AC, we change the given ratio AC:CD into an equivalent one where AC=7 as follows: AC:CD = 2 : 1 = 7 : 3.5 (multiply both sides by 3.5)

Now we have AC:CD = 7 : 3.5 and AD = AC+CD= 7+3.5 = 10.5

and the answer is: the ratio of BC:AD = 4: 10.5

AB:BC=3:4, so AC=AB+BC=3+4=7;

AC:CD=2:1, so AC=2;

because we get 2 different length for AC, we change the given ratio AC:CD into an equivalent one where AC=7 as follows: AC:CD = 2 : 1 = 7 : 3.5 (multiply both sides by 3.5)

Now we have AC:CD = 7 : 3.5 and AD = AC+CD= 7+3.5 = 10.5

and the answer is: the ratio of BC:AD = 4: 10.5