CBSE Class 10 Mathematics Commercial Mathematics Notes Set 21

Partnership

Profit is shared in the ratio of (capital × time) of the partners. If a partner joins later or withdraws part of the capital, add the capital-months of each period.

Time and Work

If a job takes n days, one day's work is 1/n. Rates of workers add when they work together. A pipe filling a tank has a positive rate and a leak a negative rate. Efficiency is inversely proportional to the time taken.

Time, Speed and Distance

Speed = distance/time. To convert km/hr to m/s multiply by 5/18. A train crossing a pole covers its own length; crossing a platform or another train covers the sum of the lengths. Relative speed is the sum of the speeds for opposite directions and the difference for the same direction.

Simple and Compound Interest

SI = PRT/100. Compound amount A = P(1 + R/100)ⁿ; for half-yearly compounding use R/2 and 2n periods. For two years, CI − SI = P(R/100)². For a fractional period, use compound interest for the whole years and simple interest for the fraction. Depreciation uses a multiplier (1 − R/100)ⁿ.

Solved Examples

Example 1. A and B invest Rs. 4000 and Rs. 6000 for a year. Divide a profit of Rs. 2500 between them.
Answer:
Ratio 4 : 6 = 2 : 3, so shares are 2/5 and 3/5 of 2500.
Rs. 1000 and Rs. 1500

Example 2. P invests Rs. 5000 for 12 months and Q invests Rs. 6000 for 8 months. Find the ratio of their profits.
Answer:
5000 × 12 : 6000 × 8 = 60 : 48.
5 : 4

Example 3. A can do a job in 10 days and B in 15 days. How long will they take together?
Answer:
Rate = 1/10 + 1/15 = 1/6.
6 days

Example 4. A pipe fills a tank in 6 hours and a leak empties it in 10 hours. In how long is the tank filled with both active?
Answer:
Net rate = 1/6 − 1/10 = 1/15.
15 hours

Example 5. A 150 m long train passes a pole in 15 seconds. Find its speed in km/hr.
Answer:
Speed = 150/15 = 10 m/s.
10 × 18/5 = 36 km/hr

Example 6. Two trains of lengths 100 m and 120 m run in opposite directions at 40 km/hr and 50 km/hr. In how many seconds do they cross?
Answer:
Relative speed = 90 km/hr = 25 m/s; distance = 220 m.
220/25 = 8.8 seconds

Example 7. Find the simple interest on Rs. 2000 at 6% per annum for 3 years.
Answer:
SI = 2000 × 6 × 3/100.
Rs. 360

Example 8. Find the compound interest on Rs. 5000 at 10% per annum for 2 years.
Answer:
Amount = 5000 × 1.1² = 6050.
CI = Rs. 1050

Example 9. Find the compound interest on Rs. 8000 at 10% per annum for 1 year, compounded half-yearly.
Answer:
Rate 5% per half-year, 2 periods: 8000 × 1.05² = 8820.
CI = Rs. 820

Example 10. In what ratio must rice at Rs. 40 per kg be mixed with rice at Rs. 55 per kg to get a mixture at Rs. 50 per kg?
Answer:
Alligation: (55 − 50) : (50 − 40) = 5 : 10.
1 : 2