Proportion Portal

Proportion Portal

Two quantities can be linked in two opposite ways. In direct proportion they grow together; in inverse proportion, as one grows the other shrinks. Both follow the same reliable script, and both come with a quick test you can run on a table of numbers to decide which you are looking at. Recognising the type is worth more marks than the arithmetic that follows it.

Direct and inverse

Everything about the two types is a mirror image, including the test you run on a table of values:

Which one is inverse?

Which of these describes INVERSE proportion?

• The number of workers on a job and the time it takes to finish it
• The number of litres of petrol bought and the amount paid
• A person's age and their height
• The day of the week and the outdoor temperature

The four-step script

Every proportion question, direct or inverse, is the same four moves. Writing them out is what earns the method marks even when the arithmetic goes wrong. 1. Write the relationship with a constant: y = kx, or y = k ÷ x. 2. Substitute a known pair of values. 3. Solve for k, and write out the full equation. 4. Evaluate for the value you were asked about. Worked on a direct case: if y = 18 when x = 6, then 18 = k × 6, so k = 3 and the equation is y = 3x. Every later question in the module runs this same script.

Find k

y is directly proportional to x, and y = 15 when x = 3. What is the constant k?

Now use it

Still with y directly proportional to x, and y = 15 when x = 3. What is y when x = 7?

Plot the next point

The points (0, 0), (1, 2) and (2, 4) lie on the direct-proportion line y = 2x. Plot the next point on the line, where x = 3.

The job that does not change

Four people paint a fence in 6 hours. How long would three people take? The tempting move is to scale down, and it gives the wrong answer. What is actually fixed here is the size of the job, and that is the product: 4 workers × 6 hours = 24 worker-hours. That 24 is the constant k. So with three workers, 3 × time = 24, giving 8 hours. Fewer people, longer job, and the product unchanged. This is why x × y = k is worth knowing as well as y = k ÷ x. Given any inverse-proportion pair, multiplying them gives you k in one step, and you can sanity-check every later answer by multiplying it back.

Inverse in two steps

y is inversely proportional to x, and y = 6 when x = 2. What is y when x = 4?

Backwards through the portal

Two quantities are linked by y = 12 ÷ x. For what value of x is y equal to 8?

• x = 1.5, because 8 = 12 ÷ x rearranges to x = 12 ÷ 8
• x = 96, because 12 × 8 = 96
• x = 4, because y falls as x rises and 8 is two-thirds of 12
• x = 20, because 12 + 8 = 20

Reading it off a table

Exam questions often give a table of values and ask which kind of proportion it shows, or whether it shows either. Four things to have ready:

What does this table show?

A table gives three pairs of values: x = 2 with y = 30, x = 4 with y = 15, and x = 5 with y = 12. What kind of relationship is this?

• Inverse proportion, because x × y comes to 60 for every pair
• Direct proportion, because both columns change steadily
• Neither, because the numbers do not double or halve neatly
• It cannot be decided from only three pairs of values

Which are true?

Select ALL THREE statements that are TRUE.

• A direct-proportion graph must pass through the origin, because zero of one quantity means zero of the other
• For inverse proportion the product x × y stays constant, which makes a table quick to test
• Two quantities can both increase together without being in direct proportion, because proportion needs a constant multiplier
• Any straight-line graph shows direct proportion
• In inverse proportion, adding a fixed amount to x subtracts the same amount from y
• The constant k changes as x and y change

The two portals

When y is directly proportional to x the equation is y = _____, its graph is a straight line through the _____, and dividing y by x gives the same answer for every pair. When y is inversely proportional to x the equation is y = k ÷ x, and it is the _____ of x and y that stays constant. In both cases the method is the same: write the relationship with a constant, substitute a known pair to find _____, then evaluate.