Direct Variation

In direct variation (or direct proportion) an increase or decrease in one quantity results in a proportional increase or decrease in the other. 

For example, if one quantity is doubled the other will be doubled and if one quantity is halved, the other will also be halved. 

For example: 

If y varies directly as x, then \( \scriptsize y \propto x \)

The symbol \( \scriptsize \propto\) means ‘is proportional to’ or varies directly with. The symbol \( \scriptsize \propto\) can be changed to an ‘=’ sign by introducing a constant k. 

i.e. if \( \scriptsize y \propto x \)

Then y = kx 

Where k is a constant (or a constant of proportionality) 

Example 8.2.1:

If y varies directly as x and y = 5 when x = 2. Find the value of y when x = 4. 

Solution 

If \( \scriptsize y \propto x \)

Then y = kx

where k is a constant

When y = 5 and x = 2 

y = kx

5 = 2k

k = \( \frac{5}{2} \)

But y = kx

Substitute k into the equation

y = \( \frac{5}{2} \scriptsize x \)

Now when x = 4 

y = \( \frac{5}{2} \scriptsize \; \times \; 4 \)

y = 5 × 2

y = 10

Example 8.2.2:

If p is directly proportional to q: 

1. Find the constant of proportionality 
2. Find the relationship between p and q if p = 16 when q = 12 
3. Find the value of p when q = 24 

Solution

1.

\( \scriptsize p \propto q \)

p = kq where k is a constant 

when p = 16 and q = 12 

then p = kq 

16 = 12k 

k = \( \frac{16}{12} \)

k = \( \frac{4}{3} \)