CBSE Class 10 Mathematics Commercial Mathematics Notes Set 06

Percentage

x percent means x/100. To increase a quantity by x% multiply by (1 + x/100); to decrease multiply by (1 − x/100). Successive changes multiply: an increase of a% followed by a decrease of b% gives a net multiplier (1 + a/100)(1 − b/100).
If a price changes by p%, the consumption must change by 100p/(100 − p)% (for a fall) to keep the expenditure constant. Percentage change always uses the original value as the base.

Profit, Loss and Discount

Profit = SP − CP and Loss = CP − SP. Profit% = (Profit/CP) × 100 and Loss% = (Loss/CP) × 100. SP = CP × (100 + gain%)/100 or CP × (100 − loss%)/100.
Discount = Marked price − Selling price, calculated on the marked price: SP = MP × (100 − discount%)/100. Successive discounts of a% and b% multiply the price by (1 − a/100)(1 − b/100). Overheads such as transport are added to the cost price.

Ratio and Proportion

The ratio a : b is the fraction a/b. If a : b = c : d then ad = bc. To share a total in the ratio m : n, the parts are m/(m + n) and n/(m + n) of the total. When quantities change by percentages, multiply each term of the ratio by its multiplier.
Mixtures: the average cost of a mixture is the weighted average of the costs by quantity. A fixed part (such as solid matter or water) is used to solve dilution and drying problems.

Solved Examples

Example 1. A price rises by 20% and then falls by 20%. What is the net change?
Answer:
1.2 × 0.8 = 0.96.
A 4% decrease.

Example 2. A shirt is sold for Rs. 540 at a 10% loss. Find its cost price.
Answer:
CP = 540/0.9.
Rs. 600

Example 3. The marked price of a bag is Rs. 800 and it is sold at 15% discount. Find the selling price.
Answer:
800 × 0.85.
Rs. 680

Example 4. Find a single discount equal to successive discounts of 20% and 10%.
Answer:
Net multiplier 0.8 × 0.9 = 0.72.
28%

Example 5. Divide Rs. 1200 in the ratio 3 : 5.
Answer:
Parts = 1200/8 = 150 each.
Rs. 450 and Rs. 750

Example 6. If the price of sugar rises by 25%, by what percent must consumption fall to keep spending unchanged?
Answer:
Fall = 25/125 = 20%.
20%

Example 7. Two numbers are in the ratio 4 : 5 and their sum is 72. Find the numbers.
Answer:
Parts = 72/9 = 8.
32 and 40

Example 8. A trader buys 20 kg of rice at Rs. 30 per kg and 30 kg at Rs. 40 per kg and mixes them. Find the cost per kg of the mixture.
Answer:
Total cost = 600 + 1200 = 1800 for 50 kg.
Rs. 36 per kg

Example 9. A man sells an article at a gain of 10%. If he had bought it for 10% less and sold it for Rs. 22 less, he would have gained 20%. Find the cost price.
Answer:
Let CP = x. 1.1x − 22 = 1.2(0.9x) = 1.08x, so 0.02x = 22.
x = Rs. 1100

Example 10. The ratio of boys to girls in a class is 3 : 2. If 6 more girls join, the ratio becomes 1 : 1. Find the number of boys.
Answer:
3k = 2k + 6, so k = 6.
18 boys