Inverse proportion

The more hands, the fewer hours

Two painters paint a fence in 6 hours. How much time do four painters need? Not 12 hours — 3 hours. More hands, less time.

This is inverse proportion: when one quantity grows several times, the other falls the same number of times. Twice as many painters, half the time.

The sign is the opposite of last time: with direct proportion the quotient was steady, with inverse the product is steady. Painters times hours: 2 × 6 = 12. And also 4 × 3 = 12. Even 1 × 12 = 12. That steady number is the total work — twelve “painter-hours” that the fence simply needs.

You meet inverse proportion with speed and time (you go faster, you are there sooner), with sharing supplies (more people, smaller portions) and with rectangles of the same area. Today you will learn to spot it and calculate via a constant product.

Ema and Adam prepare 60 sandwiches for a school event. “Alone I would do it in 3 hours,” Adam sighs. Ema: “In two, an hour and a half… no, wait.” She stops and counts: one person 3 hours, that is a product 1 × 180 minutes. Two people: 180 : 2 = 90 minutes. “An hour and a half, I guessed well.” Then Tereza comes: “In three?” — 180 : 3 = 60 minutes. Adam laughs: “We invite the whole class and we are done in five minutes.” Ema: “Except that only about five people fit around one bowl.”

A constant product and calculating across

The inverse proportion test: I double one — does the other fall to a half? Speed and travel time: yes. Number of workers and time of work: yes (if they do not get in each other’s way). Number of notebooks and price: no, that is direct.

The method via a constant product:

Step 1: multiply the given pair → you get the constant product. Step 2: divide the product by the new value → you get the other quantity.

Example: 3 pumps empty a pool in 8 hours. How much time do 6 pumps need? Product: 3 × 8 = 24. New value: 24 : 6 = 4 hours.

An inverse proportion table: pumps 1, 2, 3, 4, 6 — hours 24, 12, 8, 6, 4. The product in every column is 24. When one quantity grows, the other falls, but the product stands.

A word check always comes first: more pumps must give fewer hours. If more comes out, you used direct proportion by mistake.

The limits of the model: maths assumes everyone works the same and they do not get in each other’s way. A hundred painters will not paint one fence a hundred times faster — they will not fit around it. The question holds in reasonable bounds.

A car goes at 60 km/h on average and the journey takes 2 hours. How long will it take at 80 km/h?

Test: I go faster → I am there sooner. More means less → inverse proportion.

Step 1: constant product = speed × time = 60 × 2 = 120. (That is the distance: 120 km.)

Step 2: new time: 120 : 80 = 1.5 hours.

Result: 1.5 hours, so 90 minutes. Check: a higher speed gave a shorter time. That checks out. The constant product here has a clear meaning — the length of the journey does not change, however you drive.

Supplies for 12 campers last 10 days. How long will they last for 8?

Test: fewer eaters → the food lasts longer. Inverse proportion.

Step 1: product: 12 × 10 = 120 (person-days — food for one person for 120 days).

Step 2: for 8 campers: 120 : 8 = 15 days.

Result: 15 days. Check: fewer people, longer time. That checks out. Notice the direction — if campers grew to 15, the supplies last 120 : 15 = 8 days. The product 120 stays, it just gets shared out.

A rectangular bed should have an area of 36 m². Fill in the sizes.

The area of a rectangle = length × width. Constant product 36 — that is inverse proportion between the sides.

Options: width 2 m → length 36 : 2 = 18 m. Width 3 m → 12 m. Width 4 m → 9 m. Width 6 m → 6 m (a square).

All the beds have the same area, but a different shape. The wider, the shorter — a constant product in action. Geometry and proportions are one family; we will come back to that with areas in a few lessons.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. Start with ten. When it goes well, add more.

Calculate via a constant product: multiply the given pair, then divide. Write only the number.

Practise the questions

On paper: more means less

For each question first write the sentence “more … means less …” and the constant product.

1. Five machines finish a job in 12 days. In how many days will 4 machines finish it? And 10 machines?

2. A cyclist goes on a trip of 24 km. Make a table of time for speeds 8, 12, 16 and 24 km/h.

3. An uncle shares 120 sweets equally. How many does each get if 6, 8, 10 or 15 children come?

4. Decide the kind of proportion and solve: 3 kg of potatoes cost 66 Kč, how much do 5 kg cost? (Watch — is this inverse at all?)

Now you 💪

1. I decide the kind of proportion with a sentence: more means more (direct), or more means less (inverse)?
2. With inverse proportion I calculate via a constant product.
3. I check the result by direction: more machines must give fewer days.

What to take from this lesson

• Inverse proportion: how many times one grows, that many times the other falls.
• The sign: a constant product of both quantities.
• The method: multiply the given pair, divide by the new value.
• The sentence “more means less” decides before the numbers.
• The model holds in reasonable bounds — a hundred painters will not fit around a fence.

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