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In this chapter, we will focus on Percentage increase and decrease, Profit and Loss, Hire Purchase, Commission, Bills, and Simple and Compound Interest.

Key Formulae:

Example 1

#### 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

#### Solution:

% Decrease = 80%;

% Remaining = 100% - 80% = 20%

Therefore:

(20 / 100) x 360

= 72 bottles

Example 1

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

Increase = Sh. 24,000 - Sh. 18.000 = Sh. 6,000

Therefore:

= (6,000 ÷ 18,000) x 100%

= 33.33% (2 decimal places)

Mary had 360 bottles of soda. During a wedding she sold 80% of soda. How many bottles was she left with?

% Remaining = 100% - 80% = 20%

Therefore:

(20 / 100) x 360

= 72 bottles

Key Formulae:

Profit = Selling Price (SP) - Buying Price (BP)

Loss = Buying Price (BP) - Selling Price (SP)

Example 1

#### Solution:

% Buying Price = 100%

% 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

#### 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

#### 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%

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?

% 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

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?

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

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?

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%

Key Formulae:

Discount (D) = Marked Price (MP) - Buying Price (BP)

Example 1

#### 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

#### 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

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?

% 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

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?

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

Key Formulae:

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

Cash Price (CP) is ALWAYS 100%

Example 1

#### 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

#### 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

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.

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

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?

Hire purchase = 100% + 20% = 120% of the cash price

Hence:

HP = (120 / 100) x 18,000 = sh. 21,600

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

Key Formula:

Example 1

#### 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)

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?

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)

Key Formulae:

Amount = Principal + Interest

Example 1

#### 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

#### 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

#### 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

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?

Principal = sh. 4,000

Time = 8 Months or

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

= 80000 ÷(32000 / 12)

= 80000 x 12 / 32000

=30%

Please note: You can simplify in the early stages of the solution

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?

Principal = sh. 54,000

Time = 8 Months or

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

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?

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)

=

Key Formula:

Example 1

METHOD 1

#### Solution:

First, we find the total interest

I = (Principal x Rate x 1) / 100

1^{st} Year Principal = sh. 10,000

I = (10,000 x 15 x 1) / 100

= sh. 1,500

2^{nd} 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)

Example 1

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

METHOD 1

I = (Principal x Rate x 1) / 100

1

I = (10,000 x 15 x 1) / 100

= sh. 1,500

2

I = (11,500 x 15 x 1) / 100

= sh. 1,725

I = (13,225 x 15 x

= 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

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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