# Percentage Increase and Decrease, Profit and Loss, Simple and Compound Interest

## Percentage Increase and Decrease

Key Formulae:

Example 1
Okumu's salary in 2013 was sh. 18,000. In 2018, it was increased to 24,000. What was the percentage increase?

#### Solution:

Increase = Sh. 24,000 - Sh. 18.000 = Sh. 6,000
Therefore:
= (6,000 ÷ 18,000) x 100%
= 33.33% (2 decimal places)
Example 2
Mary had 360 bottles of soda. During a wedding she sold 80% of soda. How many bottles was she left with?

#### Solution:

% Decrease = 80%;
% Remaining = 100% - 80% = 20%
Therefore:
(20 / 100) x 360
= 72 bottles

### Profit and Loss

Key Formulae:
Profit = Selling Price (SP) - Buying Price (BP)
Loss = Buying Price (BP) - Selling Price (SP)

Example 1
A retailer bought 30 pens. He later sold each of the pens at Sh. 6 making a 20% profit. How much had he bought the pens?

#### Solution:

% Profit = 20%
% Selling Price = 120%
Selling Price (SP) = Sh. 30 x 6 = Sh. 180
Therefore:
Sh. 180 = 120%
Sh. ?     = 100%

Buying Price (BP) = Sh. 150
Example 2
A shopkeeper bought 3 trays of eggs at sh 150 per tray. On the way to the shop, he realized 20% of the eggs were broken. He sold the rest at sh 72 per dozen. How much loss did he make?

#### Solution:

B.P for 3 trays = 3 x 150 = sh. 450
Number of eggs = 3 x 30 = 90 eggs
20% eggs broke = (20/100) x 90 = 18 eggs broken
Therefore remained = (90 - 18) eggs = 72 eggs
1 dozen = 12 eggs
Hence 72 eggs = (72 / 12) = 6 dozen
If:
1 dozen = Sh. 72
6 dozen = Sh. ?

Loss = BP - SP
450 - 432
= Sh. 18

Example 3
A chips vendor bought 3 bags of potatoes at sh. 800 per bag. She then paid sh. 50 per bag for transport to her business location. She has 3 workers, each being paid sh. 200 per day. If she sold 150 plates of chips each at sh. 50, how much profit or loss did she make from the sale in one day?

#### Solution:

B.P for 3 bags = 3 x 800 = sh. 2,400
Transport = 3 bags x 50 = 150
Workers' Pay = 5 x sh. 200 sh. 1000
Total BP = sh. 2,400 + sh. 150 + sh. 1,000 = sh. 3, 550
To get Selling Price:
SP = 150 plates x sh. 50 = sh. 7,500
Since the SP is higher, we get a profit:
Profit = SP - BP
sh. 7,500 - sh. 3,550 = sh. 3,950 (Profit)
Additional Exercise: Can you calculate the Percentage Profit / Loss?
Remember: % Profit = (Profit / B.P) x 100%

### Discount

Key Formulae:
Discount (D) = Marked Price (MP) - Buying Price (BP)

Example 1
After being given a 20% discount, John bought a radio for sh. 5,600. What was the marked price?

#### Solution:

% Marked Price (MP) = 100%
% Discount = 20%
Hence, Buying Price (BP) = 100% - 20% = 80%
If:
80% = sh. 5,600
100% = sh. ? (MP)
MP = (100% / 80%) x sh. 5,600
MP = sh. 7,000
Example 2
Peter paid sh. 4,400 for a bicycle after he was given a 12% discount. Julius bought the same item from a different shop and was given a 15%. How much more than Julius did Peter pay for the bicycle?

#### Solution:

Peter's BP = sh. 4,400
Peter % BP = 100% - 12% = 88%
Therefore (4400 / 88) x 100 = sh. 5000 (Marked Price)
Julius BP = 100% - 15% = 85%
Therefore: (5000 x 85) / 100 = sh. 4,250
Difference: sh. 4400 - sh. 4250 = sh. 150

### Hire Purchase

Key Formulae:
Hire Purchase (H.P) = Deposit + Total Monthly Instalments

Cash Price (CP) is ALWAYS 100%

Example 1
In a certain shop, the hire purchase terms of a new bicycle is a deposit of sh. 12,500 and 15 monthly instalments of sh. 7,500 each. This is 25% more than the cash price. What is the cash price of the bicycle.

#### Solution:

Total Monthly Instalments = sh. 7,500 x 15 = sh. 112,500
Hire Purchase Price (HPP) = Deposit + Total Monthly Instalments
= sh. 12,500 + sh. 112,5000
= sh. 125,000
CP = 100%
HPP = 100% + 25% = 125%
If:
125% = sh. 112,500
100% = sh. ?
= (100% x sh. 125, 000) / 125%
CP = sh. 100,000
Example 2
The cash price of a microwave is sh. 18,000. The hire purchase price of the microwave is 20% more than the cash price. Bernice bought it on hire purchase terms by paying 40% of the hire purchase price as the deposit and the balance equal monthly instalments of sh. 1,620. How many instalments did she pay?

#### Solution:

Remember that the Cash Price (CP) = 100%
Hire purchase = 100% + 20% = 120% of the cash price
Hence:
HP = (120 / 100) x 18,000 = sh. 21,600
Deposit: 40% of the HP
D = (40/100) x 21600 = sh. 8,640
Number of Months = (Hire Purchase (HP) - Deposit(D)) / Monthly instalments (MI)
= (21,600 - 8,640) / 1,620 = 8 Months

### Commission

Key Formula:

Example 1
A salesman is paid a salary of sh. 3,500 per month plus a commission of 2.5% on the sale of goods above sh. 10,000. In one month, she was paid a total of sh. 3,800. How much was the sale of goods?

#### Solution:

Basic salary = sh. 3,500
Commission = basic salary - total salary
= 3,800 - 3,500 = sh. 300
If:
2.5% = sh. 300
100% = sh. ?
cross-multiply: (100 x sh. 300) / 2.5
= sh. 12,000
Since, the 12,000 is the above the limit of 10,000;
sh. 12,000 + 10,000 = 22,000
= sh. 22,000 (Total sales)

### Simple Interest

Key Formulae:
Amount = Principal + Interest

Example 1
Miriam lent Mugechi sh. 4,000 to pay school fees. AFter 8 months, Mugechi gave Miriam sh. 4,800 as refund including simple interest. At what rate per annum was Miriam charging the interest?

#### Solution:

Amount = sh. 4,800
Principal = sh. 4,000
Time = 8 Months or 8/12
Rate = ?
Interest = Amount - Principal
= sh. 4,800 - sh. 4,000 = sh. 800
Rate = (Simple Interest x 100) / (Principal x Time)
Rate = (800 x 100) ÷ (4000 x 8/12)
= 80000 ÷(32000 / 12)
= 80000 x 12 / 32000
=30%
Please note: You can simplify in the early stages of the solution
Example 2
Olang borrowed sh. 54,000 from a bank which charged interest at the rate of 18% p.a. He repaid the whole loan after 8 months. How much did he pay back?

#### Solution:

Amount = ?
Principal = sh. 54,000
Time = 8 Months or 8/12
Rate = 18%
Simple Interest = (Principal x Rate x Time) / 100
= (54,000 x 18 x 8) / (100 x 12)
= sh. 6,480
Hence:
Amount = P + I
= 54,000 + 6,480
= sh. 60, 480
Example 3
Beth deposited sh. 38,200 in a bank which paid simple interest at the rate of 12% p.a. For how long had her money stayed in the bank if he withdrew a total of sh. 39,346?

#### Solution:

Amount = sh. 39,346
Principal = sh. 38,200
Time = ?
Rate = 12%
Interest = amount - Principal
= 39,346 - 38,200
= sh. 1,146
Time = (Interest x 100) / (Principal x Rate)
= (1,146 x 100) / (38,200 x 12)
= 1/4 years or 3 months

### Compound Interest

Key Formula:

Example 1
Henry borrowed sh. 10,000 for a period of 21/2 years in a bank which charged a compound interest of 15% p.a. How much did he pay altogether?

METHOD 1

#### Solution:

First, we find the total interest
I = (Principal x Rate x 1) / 100
1st Year Principal = sh. 10,000
I = (10,000 x 15 x 1) / 100
= sh. 1,500
2nd Year Principal = sh. 10,000 + sh. 1,500 = sh. 11,500
I = (11,500 x 15 x 1) / 100
= sh. 1,725
1/2 Year Principal = sh. 11,500 + sh. 1,725 = sh. 13,225
I = (13,225 x 15 x 1/2) / 100
= sh. 991.88 (2 decimal places)
Total Interest = sh. 1,500 + sh. 1,725 + sh. 991.88 = sh. 4,216.88

Amount = Principal + Interest
= 10,000 + 4,216.88
= sh. 14,216.88

METHOD 2

#### Solution:

Use the formula:

Total Amount = 10,000 x ((15+100)/100) + ((15+100)/100) + ((15/2+100)/100)
Remember to divide the last Rate by 2 as it is half an year
= 10000 x 1.15 x 1.15 x 1.075
= sh. 14,216.88 (2 decimal places)

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