CBSE Class 10 Mathematics Partnership, Mixtures and Alligation Notes Set 08

Partnership

In a partnership, profit or loss is shared in the ratio of the partners' investments when the time is the same. If the time differs, the share is in the ratio of (capital × time), so convert every period to months before comparing.
If a partner's capital changes during the year, add (capital × months) for each period. A working partner may first receive a fixed salary or commission, and the rest is shared by the capital-time ratio.
For two partners investing C1 and C2 for T1 and T2 months, profit ratio = C1T1 : C2T2.

Mixtures

A simple mixture combines two ingredients, for example milk and water. If the ratio is a : b, the fraction of the first ingredient is a/(a + b).
If a vessel holds V litres of a pure liquid and x litres are drawn off and replaced by water n times, the pure liquid left is \( V(1 - x/V)^n \) litres.
When water of no value is added to milk, the price of the mixture per litre = price of milk × milk fraction.

Alligation Rule

When two ingredients of costs c1 (cheaper) and c2 (dearer) are mixed to get a mean cost m, the quantities are in the ratio (c2 − m) : (m − c1) = quantity of the cheaper : quantity of the dearer.
The rule works for prices, percentages of a solution, speeds and ages of groups, wherever a weighted average is involved. Write the three values in a cross and subtract along the arms.

Worked Examples

Illustration 1. A invests Rs. 6000 for 12 months and B invests Rs. 8000 for 6 months. Divide a profit of Rs. 12000.
Answer:
A : B = 6000 × 12 : 8000 × 6 = 72 : 48 = 3 : 2.
A gets Rs. 7200 and B gets Rs. 4800.

Illustration 2. In what ratio must rice at Rs. 40 per kg be mixed with rice at Rs. 55 per kg so that the mixture costs Rs. 48 per kg?
Answer:
Cheaper : dearer = (55 − 48) : (48 − 40) = 7 : 8.

Illustration 3. A vessel contains 40 litres of milk. 4 litres are drawn off and replaced by water, and this is done once more. Find the milk left.
Answer:
Milk left = 40 × (1 − 4/40)² = 40 × 0.81 = 32.4 litres.