We sometimes need to find an equation for two quantities that are in proportion.

If *a* and *b* are in direct proportion then we can write this as *a* ∝ *b*.

∝ is the symbol for proportionality

We can use the example from the previous page.

It costs 72p for 12 pencils.

Work out an equation connecting cost and number of pencils, then find out how much 30 pencils will cost.

Let *a* be the cost in pence and *b* the number of pencils.

*a* ∝ *b*

Which means:

*a*= k x*b*- 72 = k x 12
- k = 72 ÷ 12
- k = 6

We can find the equation is *a* = 6*b*.

Now, we can calculate the price for 30 by substituting *b* with 30.

So:

*a*= 6 x 30*a*= 180p

If *a* and *b* are in indirect proportion then *a* ∝ ^{1}/_{b}.

We can work out the following questions in the same way.

**Example**

The time taken to dig a hole is indirectly proportional to the number of people doing the digging.

It takes 4 people 6 hours to dig the hole.

Find an equation connecting the time, *t*, to the number of people digging, *d*.

**Solution**

*t* and *d* are indirectly proportional, so:

*t*∝^{1}/_{d}

Which means:

*t*= k x^{1}/_{d}- 6 = k x
^{1}/_{4}

The equation is *t* = ^{24}/_{d}

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