 ## WELCOME!

Hello to all SPM 2019 candidates, we know that you are clicking in just for the full solutions of Add Maths Project, you are at the right place! We provide you with the complete answer for you below (for education reference only)

***We have amended answers for Part 2.

#### Question Update

This year's SPM 2019 Add Maths Project mostly revolves around real life applications and observations such as compound interest, index number and Golden ratio. The upside of this is that it is more in touch with our lives.

However, there was a mistake in the question paper given. Please download the latest question paper from here.

## PART 1

##### a. Explain the meaning of Compound Interest.

Sample 1

Compound interest is the summation of interest to the principal sum of a deposit or loan. In other words, it is interest upon interest. This is the result of reinvesting interest rather than paying it out, resulting the interest in the next period earned is based on the principal sum plus the previously accrued interest. Compound interest is standard in economics and finance.

This concept of compounding interest is thought to have originated from the 17th Century Italy as “interest on interest” that also implies a sum will grow at a more rapid rate than of simple interest-that is calculated only based on the principal amount.

The rate at which compound interest accumulates is dependent on the frequency of compounding, such that the increase in the number of compounding periods will result in a greater compound interest. Hence, the amount of compound interest accrued on RM 10,000 compounded at 5% annually will be lower than that on RM 10,000 compounded at 2.5% semi-annually over the same period. Professionals across the finance and economics industries coined the phrase “miracle of compound interest” since the interest-on-interest effect can generate increasingly positive returns based on the initial principal amount.

Sample 2

The definition of compound interest is the summation of interest to the principal sum of a deposit or loan. In other words, it is interest upon interest. This is the result of reinvesting interest rather than paying it out, resulting the interest in the next period earned is based on the principal sum plus the previously accrued interest. This concept of compounding interest is thought to have originated from the 17th Century Italy as “interest on interest” that also implies a sum will grow at a more rapid rate than of simple interest- that is calculated only based on the principal amount. Compound interest is standard in economics and finance.

The rate at which compound interest accumulates is dependent on the frequency of compounding, such that the increase in the number of compounding periods will result in a greater compound interest. Hence, the amount of compound interest accrued on RM 20,000 compounded at 4% annually will be lower than that on RM 20,000 compounded at 2.4% semi-annually over the same period. Professionals across the finance and economics industries coined the phrase “miracle of compound interest” since the interest-on-interest effect can generate increasingly positive returns based on the initial principal amount.

##### b. How can Mathematics be used to calculate the Compound Interest?

Compound interest is calculated based on the principal sum, interest rate-annual percentage rate (APR), and the time involved using these formulas.

This formula can be applied when the compounding of interest in a year is more than once. For instance, if the interest is compounded yearly, then n = 1; if semi-annually, then n = 2; quarterly, then n = 4; monthly, then n = 12; weekly, then n = 52; daily, then n = 365; and so forth, regardless of the number of years involved.

## PART 2

##### a. Based on the situation given, if you did not withdraw any savings for 25 years, construct a table to show the total amount of savings for 25 years. Give your answer to the nearest RM. *Amended*

Using Microsoft excel, we can generate the table as below. Note that the amount saved in the current year is 1.1 times the amount saved in the preceding year. The cumulative amount is the sum of the amount saved in the current year and the cumulative amount saved in the preceding year.

##### b. Show that the total sum, that you should receive can be calculated using the formula. Give your answer to the nearest RM. *Amended*

Since the calculated values using the formula is tallied with the values tabulated at the table above using Microsoft Excel. Therefore, the formula of J is applicable.

##### c. Will your dream of saving at least RM 1 million after 25 years be realised? Give a reason for your answer. *Amended*

Therefore, we can conclude that the dream of saving at least RM1 million after 25 years can not be realised as RM 983,471 is less than RM 1,000,000.

## PART 4

##### a. Based on Diagram 2, copy Stage 3 and then construct Stage 4 of the outer squares of the Pythagoras’ tree.

Stage 3 is copied using the initial length of square as 4 inches.

Using initial length of square of 4 inches, the Stage 4 Pythagoras’ Tree is constructed.

## FURTHER EXPLORATION

##### ​

The ratios obtained from 6 different people are tabulated in the Table 3. The values of the distance from tip of nose to bottom of chin (a) and the distance from centre of lips to bottom of chin (b) that are taken above are independent and random. Both a and b are related by taking the ratio a:b which will give us values that are approximated to the golden ratio that has an approximate value of 1.618 with an uncertainty of (plus minus) 0.1.

The difference between the ratio of the gross readings taken and the divine ratio is 1.680 - 1.618 = 0.062 which is negligible. Therefore, the ratio of the gross readings taken is nearing the divine ratio (Golden ratio).

##### What is the Golden Ratio?

The golden ratio phi is a special irrational number that is approximately equal to 1.618. It appears many times in geometry, art, architecture and other fields.The Golden ratio is a special number found by dividing a line into two parts so that the longer part divided by the smaller part is also equal to the whole length divided by the longer part. In an equation form, it looks like this:

Historically, the number can be seen in the architecture of many ancient creations, like the Great Pyramids and the Parthenon. In the Great Pyramid of Giza, the length of each side of the base is 756 feet with a height of 481 feet. The ratio of the base to the height is roughly 1.5717, which is close to the Golden ratio. Around 1200, the mathematician Leonardo Fibonacci discovered the distinctive properties of the Fibonacci sequence. This sequence links directly into the Golden ratio because any ratio of two consecutive Fibonacci numbers is very close to the Golden ratio. As the numbers increase, the ratio becomes even closer to 1.618. For instance, the ratio of 3 to 5 is 1.666; the ratio of 13 to 21 is 1.625; the ratio of 144 to 233 is 1.618, which eventually converges to 1.618.

The Golden Rectangle is also closely related to the Golden ratio as its proportion of its dimension is approximately to the value 1.618.This is known as one of the most visually satisfying of all geometric forms– hence, the appearance of the Golden ratio in art. The Golden rectangle is also related to the Golden spiral, which is created by making adjacent squares of Fibonacci dimensions. In the beauty industry, studies have shown that when test subjects view random faces, the ones they deem most attractive are those with solid parallels to the Golden ratio. Faces judged as the most attractive depict Golden ratio proportions between the width of the face and the width of the eyes, nose, and eyebrows. In toto, the Golden ratio is more than an obscure term found in Mathematics and Physics. It also revolves around us in our daily lives.

## REFLECTION

Sample 1

While I was conducting this project, I have learnt how to appreciate and enjoy the beauty of Mathematics. I have learnt how to apply my Index Number skills, calculating compound interest skills, identifying progression skills and tabulating data skills more efficaciously. I also learnt that the skills mentioned are very crucial in our life; it can help us solve our daily-life problems that relate to finance.

The moral values that I have put into practice is compassion and dedication in conducting this project. I have poured my heart, mind and soul into finishing this project with the best result. I also put in a lot of hard work and commitment in order to complete this project. I also did not give up in doing my project; I have completed it to the end. In the making of this project, I have spent countless nights researching and completing this project. I realise this subject is compulsory for me. Without it, I could not fulfill my abjured hopes and dreams. These are the benefits of the project.

Sample 2

In the making of this project, I have spent countless nights researching and completing this project. I realise this subject is compulsory for me. Without it, I could not fulfill my abjured hopes and dreams. These are the benefits of the project.

While I was conducting this project, I have learnt how to appreciate and enjoy the beauty of Mathematics. I have learnt how to apply my Index Number skills, calculating compound interest skills, identifying progression skills and tabulating data skills more efficaciously. These skills are very crucial in our life; it can help us solve our daily-life problems that relate to finance.

The moral values that I have put into practice is compassion and dedication in conducting this project. I have poured my heart, mind and soul into finishing this project with the best result. I also put in a lot of hard work and commitment in order to complete this project. I also did not give up in doing my project; I have completed it to the end. 