in direct proportion.

Direct Proportion:
When two variables are in direct proportion, it means that as one variable increases, the other variable also increases in the same ratio, and vice versa. Mathematically, this can be represented as:

x ∝ y

where ∝ denotes the proportionality symbol.

Inverse Proportion:
Now, let's consider the reciprocals of x and y, which are 1/x and 1/y, respectively. We want to determine the relationship between 1/x and 1/y.

Reciprocals of Variables:
To find the relationship between the reciprocals, we can rewrite the direct proportion equation as:

y = kx

where k is the constant of proportionality.

Taking the reciprocal of both sides of the equation gives:

1/y = 1/(kx)

Simplifying the equation:

1/y = 1/x * 1/k

Since 1/k is a constant, let's represent it as a new constant, m:

1/y = m * 1/x

Conclusion:
From the above equation, we can see that 1/y is in inverse proportion to 1/x. This means that as 1/x increases, 1/y decreases in the same ratio, and vice versa.

To summarize, if x and y are in direct proportion, then 1/x and 1/y are in inverse proportion. This relationship can be represented by the equation 1/y = m * 1/x, where m is a constant.

1/x : : 1/y as x : : y