How to solve the linear equation



Unit 9A Percentage

Learning Objectives

The students should be able to:

• Solve simple percentage problems

• Calculate percentage change

• Solve problems about loss and profit

Percentages

Percentage is a fraction with the denominator equal to 100

e.g.:

10% =[pic]

“%” is the symbol of percentage and often is called “Per cent”. Per cent means “per one hundred”.

A fraction can be converted to percentage.

e.g. :

[pic]

As you know, a fraction can be converted to decimal, too. Therefore, percentage can also be converted to decimal and decimal can be converted to percentage as well.

e.g. 43% = 0.43

e.g. 0.54 = 54%

Activity I:

Example 1:

In a hall, there were 8000 female persons, which was 20% of the total persons. Given that there were 80000 seats, find the percentage of seats occupied.

Solution:

Let n be the total number of persons in the hall.

n × 20% = 8000

n = 40000

∴ Percentage of seats occupied :

[pic]

Example 2:

Peter participates in a 10km race. He has finished 5000m already. What percentage of the whole race does Peter not finish?

Solution:

The remaining distance that Peter need to run :

10000 –5000 = 5000m

Percentage of race not cmopleted

[pic]

Some other general equations

Percentage Increase and Decrease

Percentage increase or decrease is commonly used in our daily life.

In general,

|New value = Original value (1 + Percentage Increase) |

|New value = Original value (1 – Percentage Decrease) |

|Percentage Increase = [pic] |

|× |

|Percentage Decrease =[pic] |

Activity II:

Example 3:

Last year, David was 200cm tall. His height is increased by 10% this year. What is his height now?

Solution:

His height now:

200 ∗ (1 + 10%)

= 200 ∗ 1.1

= 220cm

Example 4:

The price of a P4 computer was $10000 last year. What is the price of the computer for this year if the price is decreased by 20%?

Solution:

The price of the computer this year

$10000 × (1 –20%)

= $10000 × 0.80

= $8000

Example 5:

The price of a house decreases from $2,000,000 to $1,800,000. How to find the percentage decrease from the data provided?

Solution:

The decrease of the price of the house= $2,000,000 – $1,800,000 = $200,000

The percentage decrease of the price of the house:

[pic]

Example 6:

The Heng Seng Index increases from 15000 to 15500. How to find the percentage increase from the data provided?

Solution:

The increase in Heng Seng Index = 15500 –15000= 500

The percentage increase in Heng Seng Index:

[pic] (correct to 3 sig. fig.)

However, we can summarize the two formulae into one and call percentage change. If the result of the percentage change is positive, it is percentage increase. In other words, if the result of the percentage change is negative, it is percentage decrease.

|Percentage change = [pic] |

Activity III:

Example 7:

The temperature was 32°C in the afternoon and the temperature dropped to 28°C in the evening. Find the percentage change in the temperature.

Solution:

Percentage change in the temperature:

[pic]

∴The percentage of temperature drop is 12.5%

Example 8:

The sales of a company increases from $100,000 last month to $120,000 this month. What is the percentage change in profit for the month?

Solution:

Percentage change in profit:

[pic]

∴The percentage increase in profit is 20%

Profit and Loss

In our daily life, selling and buying goods are commonly activities. We also often use percentages in these activities.

Before learning how to calculate the profit and loss, we should know some special words.

I. Cost / Cost price is the value of the good that the merchant paid for.

II. Selling price is the value of the good that the merchant got in selling the good.

III. Marked price is the value of the good displayed in the shop.

IV. Profit percent(%) is used to compare the profits. Profit percent is the profit expressed as a percentage of the cost price.

V. Loss percent(%), is the loss expressed as a percentage of the cost price, is used to compare the loss.

|Profit % = [pic] |

|Loss % = [pic] |

Activity IV:

Example 9:

A shop bought 1000 pieces of mother boards at $1200 each and 1000 pieces of rams at $150 each. The shop sold mother boards at $1800 and the rams at a special price of $100 each if people bought mother board at the same time. Find the profit % after selling all mother board and the rams at the special price.

Solution:

Total cost:

1000*1200+1000*150

= $1350000

Total selling price:

1000*1800+1000*100

= $1900000

∴ The profit % :

[pic] (correct to 2 sig. fig.)

Example 10:

A boat shop sold a ship at the profit % of 20%.

(a) If the profit was $40000, what was the cost of the ship?

(b) What was the selling price?

Solution (a):

[pic]

(b) The selling price :

$200000 + $40000

= $240000

Solutions of the problems

1. The price of an alarm clock is $200. If the price decreases by $60, what is the percentage decrease in price? (3 marks)

|Percentage decrease = [pic] |

2. If Mr.Chan's salary is increased by 10%, it would become $11000. What is his salary? (3 marks)

|Salary *1.1=$11000, Salary =$10000 |

1. The dimensions of a rectangle are 10 cm(width) by 20 cm(length). If the length is increased by 10% while the width is increased by 150%, find

a) the increase in area of the rectangle, and (4 marks)

b) the percentage increase in area. (4 marks)

|New length=20*1.10=22 |

|New width=10*(1+1.5)=25 |

|New area = 22 cm × 25 cm = 550 sq cm Increase in area = 350 sq cm |

|(b) Percentage increase in area = (350/200)*100=175% |

4. Carmen is 15% lighter than Mary and Tomas is 25% heavier than Carmen. Between Mary and Tomas, who is heavier? By what percentage? (4 marks)

|Let Mary weight=1, Carmen weight=0.85 Tomas=0.85*1.25=1.0625 |

|Tomas is heavier. |

|He is heavier than May by 6.25%. ((1.0625-1)*100%=6.25%) |

5. The cost of making a table is calculated as in the following table. Now, the cost of wood increases by 30%, that of metal increases by 20%, that of paint decreases by 15% and that of labour increases by 10%.

(a) Using the above data, complete the following table. (8 marks)

| |Original cost |Final cost |

|Wood |$1200 |$1200 × (1 + 30%) = $1560 |

|Metal |$ 800 |$ 800 × (1 + 20%) = $ 960 |

|Paint |$ 200 |$ 200 × (1 – 15%) = $ 170 |

|Labour |$ 950 |$ 950 × (1 + 10%) = $1045 |

|Total |$3150 |$3735 |

(b) What is the percentage increase in the cost of making a table? (4 marks)

|[(3735 – 3150)/3150] × 100 = 18.57 %(to 2 sig) |

6. The volume of water in a pool was increased by 20% in July, then decreased by 5% in August and increased by 25% in September. What was the overall percentage change in the volume of water between the beginning of July and the end of September? (4 marks)

|Let the volume of water in the beginning of July be V m cube. |

|Volume of water at the end of September=V (1 + 20%)(1 – 5%)(1 + 25%) m cube =1.425V m cube Percentage increase = 42.5% |

7. The population of a city is originally 6 800 000. If the population increases steadily at a rate of 2% each year, what will be the population next year? (4 marks)

|Population next year = 6 800 000 * (1 + 5%)= 7140 000 |

8. The gross profit of a company is $1000000 and the operating expenses are $500000. Find the profits tax payable by the company if tax rate for the year is 15%.(4 marks)

|(1000000-500000)*15%=$75000 |

9. The value of a ship decreases at a rate of 25%. If the value of the ship now is $200 000, find its value

c) last year, (4 marks)

d) one year later. (4 marks)

|(a) Let be the value last year of the ship n(1-0.25)=200 000 n=$266 666.67 (to 2 sig.) |

|(b) $200000*(1-0.25)=150000 |

10. In a hall, there were 10000 female persons, which was 25% of the total persons. Given that there were 60000 seats, find the percentage of seats occupied. (4 marks)

Solution:

Let n be the total number of persons in the hall.

n × 25% = 10000

n = 40000

∴ Percentage of seats occupied :

[pic]

11. Peter participates in a 10km race. He has finished 8000m already. What percentage of the whole race does Peter not finish? (4 marks)

Solution:

The remaining distance that Peter need to run :

10000 –8000 = 2000m

Percentage of race not finished

[pic]

12. Last year, David was 150cm tall. His height is increased by 15% this year. What is his height now? (4 marks)

Solution:

His height now:

150∗ *(1 + 15%)

= 150 ∗ 1.15

= 172.5cm

13. The price of a P4 computer was $12000 last year. What is the price of the computer for this year if the price is decreased by 10%?(4 marks)

Solution:

The price of the computer this year

$12000 × (1 –10%)

= $12000 × 0.90

= $10800

14. The price of a house decreases from $2,500,000 to $2,000,000. How to find the percentage decrease from the data provided? (4 marks)

Solution:

The decrease of the price of the house= $2,500,000 – $2,000,000 = $500,000

The percentage decrease of the price of the house:

[pic]

15. The Heng Seng Index increases from 14500 to 14900. How to find the percentage increase from the data provided? (4 marks)

Solution:

The increase in Heng Seng Index = 14900 –14500= 400

The percentage increase in Heng Seng Index:

[pic] (correct to 3 sig. fig.)

16. The temperature was 30°C in the afternoon and the temperature dropped to 26°C in the evening. Find the percentage change in the temperature. (4 marks)

Solution:

Percentage change in the temperature:

[pic](correct to 2 sig. fig.)

∴The percentage of temperature drop is 13.33%

17. The sales of a company increases from $150,000 last month to $200,000 this month. What is the percentage change in profit for the month? (4 marks)

Solution:

Percentage change in profit:

[pic](correct to 2 sig. fig.)

∴The percentage increase in profit is 33.33%

18. A shop bought 500 pieces of mother boards at $1800 each and 500 pieces of rams at $350 each. The shop sold mother boards at $2500 and the rams at a special price of $180 each if people bought mother board at the same time. Find the profit % after selling all mother board and the rams at the special price. (8 marks)

Solution:

Total cost:

500*1800+500*350

= $1075000

Total selling price:

500*2500+500*180

= $1340000

∴ The profit % :

[pic] (correct to 2 sig. fig.)

19. A boat shop sold a ship at the profit % of 30%.

(a) If the profit was $50000, what was the cost of the ship? (5marks)

(b) What was the selling price? (5 marks)

Solution (a):

[pic]

(b) The selling price :

$166666.7 + $50000

= $21666.7(correct to 1 sig. fig.)

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