Direct and inverse proportion: understanding recipes, map scales and speed

Your mum is going to prepare a recipe for four people for six. On a map two cities are four centimetres apart; how many kilometres is that in reality? If you travel the route twice as fast, how much earlier do you arrive? These three questions look different but rest on the same two ideas: direct proportion and inverse proportion. In this article we explain both with everyday examples and show which to use when, and the most common mistakes.

Direct proportion: they grow and shrink together

If one of two quantities increases by a certain factor and the other increases by the same factor, the two quantities are directly proportional. In maths: their ratio is constant.

Recipe example. A recipe for four people uses 200 g of flour. How much flour is needed for six? The number of people increased by $6/4 = 1.5$ times, so the flour must increase 1.5 times: $200 \cdot 1.5 = 300$ g. This scaling of a recipe is the most familiar form of direct proportion.

A common mistake is “it went from 4 to 6, two more people, so I will add 2 to every ingredient”. That is scaling by addition. In direct proportion the change is by multiplication: every ingredient grows by the same factor. You can try this hands-on in the Scaling a recipe lesson.

A map scale is direct proportion too

The scale on a map gives the constant ratio of length on the map to length in reality. If the scale is $1:2\,500\,000$, 1 cm on the map means 2 500 000 cm in reality. For a distance of four centimetres:

If the length on the map doubles, the real distance doubles too: direct proportion. If you are curious about old maps and how maps are drawn, have a look at our article on the Piri Reis map.

Inverse proportion: when one grows, the other shrinks by the same factor

If one of two quantities increases by a certain factor and the other decreases by the same factor, the quantities are inversely proportional. This time what is constant is the product:

Speed and time. Covering the same 120 km, at 60 km/h the time is 2 hours. Double the speed to 120 km/h and the time halves to 1 hour. The product ($\text{speed} \cdot \text{time}$) is the distance, so it is constant: $60 \cdot 2 = 120 \cdot 1 = 120$. So when the distance is fixed, speed and time are inversely proportional. You can find the relationship between distance, speed and time in detail in the Speed, distance and time lesson.

Workers. If 6 workers finish a job in 12 days (assuming everyone works at the same rate), the total work is $6 \cdot 12 = 72$ “worker-days”. 9 workers finish the same job in $72 / 9 = 8$ days. Workers increased 1.5 times, and time decreased by the same factor.

In the Inverse proportion lesson you can try this relationship with a simulation; direct proportion is in the Direct proportion lesson.

How do you tell which one it is?

Ask yourself one question: “If one doubles, what happens to the other?”

• If the other doubles too: direct proportion.
• If the other halves: inverse proportion.
• If it neither doubles nor halves: there is no proportion.

A short checklist:

1. Is the constant thing between the two quantities the ratio (y/x) or the product (x·y)?
2. Do both become zero together at the start (at 0)? In direct proportion, when $x = 0$ then $y = 0$.
3. Write the values in a table and look at the ratios or the products.

Common mistakes

• “One increases while the other decreases, so it is inverse.” Not every decreasing relationship is inverse proportion. For example, if you read 20 pages a day from a book, the pages left decrease, but the product of pages left and number of days is not constant. Inverse proportion needs the same-factor rule.
• Scaling by adding. Adding the same amount to every ingredient in a recipe. The correct way is multiplication.
• Mixing up units. Skipping the conversion from cm to km in a map scale. Remember that 100 000 cm = 1 km.
• Setting up a proportion where there is none. Percentages and discounts are about ratios too, but need care; our article on discount traps helps here. To connect the sense of ratio with fractions, see Understanding fractions.

In mathematics education research, proportional reasoning is regarded as an important bridge in middle-school mathematics, because it connects fractions, ratios and later algebra topics. That is why seeing the multiplicative relationship first matters more than memorising a formula.

Summary

• Direct proportion: $y/x$ is constant; whatever factor one increases by, the other increases by the same factor (recipe, map scale).
• Inverse proportion: $x \cdot y$ is constant; whatever factor one increases by, the other decreases by that factor (speed-time, workers-days).
• To tell them apart, ask “if one doubles, what happens to the other?”.
• Use multiplication, not addition, to scale; check the units.
• Not every decrease is inverse proportion; the product must stay constant.