Question

The numbers 1/2,x,y,3/4 are in increasing order of size. The differences between successive numbers in this list are all the same. What is the value of y?

Answer 1

If the differences between successive numbers in the list are all the same, then we can use these differences to set up equations involving the given numbers.

Let d be the common difference between the numbers. Then we have:

x - 1/2 = d
y - x = d
3/4 - y = d

Adding the first and third equations, we get:

(3/4 - 1/2) + (x - y) = 2d

Simplifying, we have:

1/4 + (x - y) = 2d

Substituting the second equation into this expression, we get:

1/4 + d = 2d

Solving for d, we get:

d = 1/4

Substituting this value of d into the first equation, we get:

x - 1/2 = 1/4

x = 3/4

Substituting this value of x into the second equation, we get:

y - 3/4 = 1/4

y = 1

Therefore, the value of y is 1.

Answer 2

Let's start by finding the common difference between successive numbers. Since the differences between successive numbers are the same, the difference between 1/2 and x is the same as the difference between x and y, which is the same as the difference between y and 3/4. Therefore:

x - 1/2 = y - x = 3/4 - y

Simplifying each of these equations, we get:

2x - 1 = 2y - 1/2
2y - 1/2 = 3/4 - y

Solving for x in the first equation, we get:

x = y + 1/4

Substituting this into the second equation, we get:

2y - 1/2 = 3/4 - (y + 1/4)

Simplifying this equation, we get:

3y = 1/2

Therefore, y = 1/6.