Advanced Word Problems • Topic 1 of 5

Mixture Problems

Mixture problems combine two quantities at different prices, concentrations or rates and ask for the blend. The fastest tool is alligation: the cheaper and dearer items are mixed in the ratio (dear − mean) : (mean − cheap). To find a blend price directly, use the weighted average — total cost divided by total amount. For two amounts x and y at prices p₁ and p₂, the mean price is (x·p₁ + y·p₂)/(x + y). Decide whether the question wants a ratio or an average value, set up the matching formula, and the rest is arithmetic. The same idea covers salt solutions, alloys and investment blends.

Alligation cross for mixing two prices to a mean priceAlligation crosscheap adear bmean mb − mm − aratio (cheap : dear) = (b − m) : (m − a)

✅ Solved examples

1. Mix tea at $20/kg and $30/kg to get $24/kg. In what ratio (cheap:dear)?
Ratio = (30 − 24) : (24 − 20) = 6 : 4 = 3 : 2.
2. 2 kg at $10 mixed with 3 kg at $20. Average price per kg?
(2·10 + 3·20)/(2 + 3) = 80/5 = $16.
3. Mix $40 and $60 coffee to get $45. Ratio (cheap:dear)?
(60 − 45) : (45 − 40) = 15 : 5 = 3 : 1.
4. 4 L at $5 with 1 L at $10. Mean price per litre?
(4·5 + 1·10)/5 = 30/5 = $6.

✏️ Practice — try these, take hints as needed

1. Mix rice at $30 and $50 to get $35. Ratio (cheap:dear)?
(dear − mean):(mean − cheap).
(50 − 35):(35 − 30).
15:5.
3 : 1.
2. 3 kg at $20 with 2 kg at $30. Average price per kg?
Weighted average.
(60 + 60)/5.
—
$24.
3. Mix $10 and $16 syrup to get $12. Ratio (cheap:dear)?
(16 − 12):(12 − 10).
4:2.
—
2 : 1.
4. 1 L at $8 with 3 L at $12. Mean price per litre?
(8 + 36)/4.
—
—
$11.
5. Mix $25 and $40 to get $30. Ratio (cheap:dear)?
(40 − 30):(30 − 25).
10:5.
—
2 : 1.

📝 Topic test — 8 questions

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