Mix, share, calculate
Two classics of today’s lesson: mixtures and shared work. They look different, but both solve the same thing — how parts add up into a whole.
Mixture: grandma mixes cheap nuts at 150 Kč/kg with more expensive ones at 250 Kč/kg and wants 2 kg of mix at 190 Kč/kg. How much of which to pour?
Shared work: one painter paints a fence in 6 hours, the other in 3. How long will they take together?
With mixtures the golden rule holds: the value of the whole mix = the sum of the values of the parts. Kilograms add and crowns add too — only the average price must not be calculated as a simple average (190 is not the average of 150 and 250, because there is not the same amount of each).
With work you add rates: how much work each person manages in an hour. A painter with a fence in 6 hours paints a sixth of the fence in an hour.
Both problems end with an equation. And both have a check against the question that will hold you up.
Sofie and mum are making homemade lemonade for a party. They have syrup that is diluted, and they want 3 litres of drink that is exactly 20% syrup. “How much syrup do I pour?” Sofie asks. Mum: “Twenty percent of three litres.” Sofie calculates: 0.2 · 3 = 0.6 litres of syrup, the rest 2.4 litres of water. At the tasting the lemonade is just right. “And if we had a weaker lemonade already made and wanted to flavour it?” — “Then it would be an equation. The parts of syrup add, the whole adds too.”
With mixtures always write a table: the rows are the ingredients and the mix, the columns are amount, price per kilo (or percents) and total value. The equation then falls out of the table by itself.
Two patterns: mixture and shared work
Pattern 1: mixture. Nuts at 150 Kč/kg and at 250 Kč/kg, I want 2 kg of mix at 190 Kč/kg.
Mark: x = kilograms of cheap nuts. The more expensive ones are then (2 − x).
Crowns add: value of cheap + value of expensive = value of the mix:
150x + 250(2 − x) = 190 · 2
150x + 500 − 250x = 380 → −100x = −120 → x = 1.2.
Cheap 1.2 kg, expensive 0.8 kg. Check: 180 + 200 = 380 Kč and 380 : 2 = 190 Kč/kg. It fits.
Pattern 2: shared work. The first paints the fence in 6 h → in an hour 1/6 of the fence. The second in 3 h → in an hour 1/3.
Together in an hour: 1/6 + 1/3 = 1/6 + 2/6 = 3/6 = 1/2 of the fence.
If in an hour they manage a half, they paint the whole fence in 2 hours.
By equation the same: t · (1/6 + 1/3) = 1 (one whole fence) → t/2 = 1 → t = 2.
Remember: with mixtures you add crowns (or grams of pure stuff), with work rates per hour. Never add “prices per kilo” or “hours for the whole” directly.
Example 1: tea from two kinds
A shopkeeper mixes tea at 300 Kč/kg with tea at 500 Kč/kg. They want 10 kg of mix at 360 Kč/kg. How much of which kind?
x = kg of the cheaper, (10 − x) = kg of the more expensive.
300x + 500(10 − x) = 360 · 10
300x + 5000 − 500x = 3600 → −200x = −1400 → x = 7.
Cheaper 7 kg, more expensive 3 kg. Check: 2100 + 1500 = 3600 Kč for 10 kg = 360 Kč/kg. It fits. There is less of the more expensive — logical, the mix is closer in price to the cheap one.
Example 2: tidying a room as a pair
Ema tidies the room in 30 minutes, Adam in 60. How long will they take together?
Rates per minute: Ema 1/30 of the room, Adam 1/60.
Together: 1/30 + 1/60 = 2/60 + 1/60 = 3/60 = 1/20 of the room per minute.
The whole room: 20 minutes.
Sense check: together it must be faster than the fastest of them (30 min), but not zero. 20 minutes fits. If 45 minutes came out, there is a mistake somewhere — a helper does not slow the work down.
Example 3: diluting juice
You have 2 litres of juice with 30% sugar. How much water must you add so the mix has only 20% sugar?
The sugar does not change when you pour water: there is 0.3 · 2 = 0.6 litres of it.
x = litres of water added. New volume: (2 + x). Sugar should make 20%:
0.6 = 0.2 · (2 + x) → 0.6 = 0.4 + 0.2x → 0.2x = 0.2 → x = 1 litre.
Check: 3 litres of mix, sugar 0.6 litres, share 0.6 : 3 = 0.2 = 20%. It fits. The trick of dilution problems: follow the stuff that does NOT change.
The price of a mix is not the average of the ingredient prices — unless there is exactly the same amount of each. And with shared work never add the times: two painters of six hours each do not paint the fence in 12 hours, but in 3. You add rates, not hours.
Practice: now you do it 🎲
Now try it for real. You will see all the questions at once, like on paper. Mulai with ten. When it goes well, add more.
Practise a percent of a number — mixtures stand on it: 20% of 3 litres you must have off by heart.
On paper
A table of ingredients, an equation, a check against the question. With work, rates per hour.
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Muesli at 80 Kč/kg is mixed with nuts at 200 Kč/kg. You want 3 kg of mix at 120 Kč/kg. Build a table and an equation, solve.
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Jonáš washes the dishes in 20 minutes, Filip in 30. Calculate through rates how long they take together.
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You have a litre of juice with 40% fruit. How much water do you add so it has 25%? (Follow what does not change.)
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Check all three results against the question: do the total price, the time and the percents fit?
Now you 💪
- With mixtures I add crowns (or the pure stuff), not prices per kilo.
- With shared work I add rates per hour, not times.
- With diluting I follow the stuff that does not change.
Done when: You have a mixture, shared work and a dilution solved, all with checks, and a run of five correct in practice.
What to take from this lesson
- The value of a mix = the sum of the values of the ingredients. A table helps.
- Rate = work per hour. Together = the sum of the rates.
- When diluting, the pure stuff does not change — build the equation on it.
- Sense check: a helper speeds things up, a mix sits in price between the ingredients.
- Both classics are just equations in another costume.
© 2026 Ing. Martin Polak / AlgoRhino · Ketentuan penggunaan konten