Question

Point A,B,C, and D lie on a single line in that order. The length of BC is p percent of AB. The length of BC is 25% of the length of CD. The length of AD is 20. What is p?

A) 1
B) 20
C) 30
D) 40
E) cannot be determined

Answer 1

To find the value of p, we express the segments AB, BC, CD, and AD in terms of x and p. Using the given information, we establish an equation and solve for p to find that it is 20. Hence, the correct option is B.

Step-by-step explanation:

To solve the problem, let's denote the length of AB as x. If the length of BC is p percent of AB, then BC is (p/100) × x. We are also told that the length of BC is 25% of the length of CD, which means that CD is four times BC, so CD is 4 × (p/100) × x. Given that the length of AD is 20, we can express AD as the sum of AB, BC, and CD:

AD = AB + BC + CD

20 = x + (p/100) × x + 4 × (p/100) × x

Using the distributive property, we get:

20 = x (1 + (p/100) + 4(p/100))

20 = x (1 + 5(p/100))

Since x is part of AB and we know that AB is a segment of AD, which is 20 units long, we can infer that x cannot be greater than 20. Therefore, we can divide both sides of the equation by x to get:

20/x = 1 + 5(p/100)

By isolating p, we get:

p = 20(1 - 1/x) × 20

Since x must be a positive fraction of 20 (because AB is a part of AD), p is a fraction of 20 squared. Only one of the options provided is a fraction of 20 squared:

p = 20 (option B)