Anuitas | Matematika kelas XI SMA/SMK Kurikulum Merdeka

Summary

This video script delves into the concept of annuities in mathematics for high school students, particularly under the independent curriculum. It explains annuities as a series of payments, including principal and interest, using the formula for calculating annuity payments. The script provides a detailed example of how to determine monthly payments for a loan, including the initial principal and interest over time, and illustrates the changing proportion of principal and interest in each payment. It also includes a second example to demonstrate the application of the annuity formula in different scenarios, offering a comprehensive understanding of annuity calculations in finance.

Takeaways

• 📚 The video discusses the concept of annuities in mathematics for high school level, specifically for the independent curriculum.
• 💡 Annuities are compared to installment credit, which includes bank loans and leasing, like for motorcycles, and are paid in regular installments.
• 🔢 The formula for annuity includes principal payments (A) and interest payments (B), with the total payment varying each month due to changing interest and principal components.
• 📈 The initial payments have higher interest and lower principal, while the later payments have higher principal and lower interest.
• 🧮 The annuity formula is derived from borrowing a sum 'M' with an interest rate 'I' over 'n' periods, using the formula M * i * (1 + I)^n / (1 + I)^N - 1.
• 📝 To find the principal payment of the first period (A1), the formula A1 = M * i / (1 + I)^n is used, which requires knowing the total borrowed amount, interest rate, and number of periods.
• 📉 The interest payment for any period can be found by subtracting the principal payment from the total annuity payment for that period.
• 🏦 The script provides an example of Mr. Bagas taking a loan from Bank ABC for 40 million with a 12-month repayment period and a 6% annual interest rate, which is divided by 12 for a monthly rate.
• 📊 The script also explains how to calculate the total annuity payment, principal payment for the first and last periods, and the interest for the last payment using the provided formulas.
• 📋 The video concludes with a table summarizing the annuity payments, showing how the principal and interest payments change over the repayment period.
• 🤔 The video encourages viewers to understand these calculations as they are used by banks and other institutions when calculating credits and loans.

Q & A

What is the topic of the video?

The video discusses the concept of annuities in the context of high school mathematics, specifically for the Indonesian curriculum.

What is an annuity and how is it related to installment payments?

An annuity is a financial product similar to installment payments, which can be found in banking, leasing for motorcycles, and other services. It consists of regular payments that include both principal and interest.

What are the components of an annuity payment?

An annuity payment is composed of the principal payment and the interest payment. The principal is the amount borrowed, and the interest is the cost of borrowing money over time.

What is the formula used to calculate the annuity payment?

The formula to calculate the annuity payment is the borrowed amount (M) multiplied by the interest rate (i), then multiplied by (1 + I) to the power of n, and finally divided by (1 + I) to the power of N - 1.

How does the interest rate affect the annuity payment over time?

The interest rate affects the annuity payment by making the interest component higher in the beginning and lower towards the end of the payment period, while the principal payment is smaller in the beginning and larger at the end.

What is the first step in calculating the annuity payment for the first period?

The first step is to calculate A1, which is the principal payment for the first period. This is done using the formula for the annuity payment, but with adjustments specific to the first period.

What is the example given in the video about Mr. Bagas' loan from Bank ABC?

Mr. Bagas takes a loan of 40 million with a repayment period of 12 months, with an annual interest rate of 6%, which is then divided by 12 to get a monthly interest rate of 0.5%.

How is the total annuity payment calculated for Mr. Bagas' loan?

The total annuity payment is calculated by using the formula with the known values of the loan amount, the monthly interest rate, and the number of periods, and then using a calculator to find the result.

What is the difference between the principal payment and the interest payment in the first and last periods of the loan?

In the first period, the principal payment is smaller, and the interest payment is larger. In the last period, the principal payment is larger, and the interest payment is smaller.

What is the second example provided in the video about Bu Desi's loan?

Bu Desi takes a loan with a repayment period of 3 years, which is equivalent to 36 months. The annual interest rate is 12%, which is divided by 12 to get a monthly interest rate of 1%.

How is the loan amount determined in Bu Desi's example?

The loan amount is determined by using the annuity formula with the known values of the annuity payment, the monthly interest rate, and the number of periods, and then solving for the unknown loan amount (M).

Outlines

00:00

📚 Introduction to Annuities in Mathematics

This paragraph introduces the concept of annuities in the context of Indonesian high school mathematics curriculum. It explains annuities as a series of payments, similar to installment credit, which can be used in various financial contexts such as banking and leasing. The formula for calculating annuities is presented, involving principal payments and interest (referred to as 'KN' and 'BN' in the script). The paragraph emphasizes the importance of understanding the formula for annuity calculations, which involves borrowing an amount 'M' at an interest rate 'I' over 'N' periods, and provides a step-by-step guide to finding the monthly payment 'A', the principal payment 'A1', and the interest 'BN' at the end of the period.

05:03

🔢 Calculation of Annuity Payments

This paragraph delves into the practical calculation of annuity payments using a specific example. It outlines the process of determining the total monthly payment, which consists of both the principal and the interest component. The example involves calculating the first and last monthly payments for a loan taken out by Pak Bagas from Bank ABC. The calculations are performed using a calculator, and the results are presented, showing how the principal and interest change over the course of the loan term. The paragraph also explains how to find the principal payment for the first period (A1) and the last period (A12), as well as the interest for the last payment (B12).

10:05

📈 Loan Calculations with Different Interest Rates

The final paragraph presents a different scenario involving Bu Desi taking out a loan with a repayment period of 3 years, which equates to 36 months. The paragraph discusses the process of determining the amount borrowed using the annuity formula with an interest rate of 12% per annum, which is then divided by 12 to get the monthly interest rate. The calculations lead to the determination of the total amount borrowed, which is rounded to the nearest million. The paragraph concludes with a general statement about the applicability of annuity calculations in various financial contexts and encourages further learning and exploration of the topic.

Keywords

💡Anuity

An annuity is a type of financial product that provides a series of payments to an individual over a specified period. In the video, an annuity is discussed in the context of a loan repayment plan where the borrower makes regular payments consisting of principal and interest. The script uses the term to explain the mathematical formulas and calculations involved in determining the monthly payments for a loan.

💡Principal

The principal refers to the original amount of a loan or investment. In the video, the principal is the amount of money borrowed, which is used in the formula to calculate the annuity payment. It's the base amount on which interest is calculated and is repaid over time through installments.

💡Interest

Interest is the cost of borrowing money, expressed as a percentage of the principal. In the context of the video, interest is a crucial component of the annuity payments. It's calculated monthly and added to the principal portion of each payment, making the total monthly payment.

💡Loan

A loan is a sum of money that is borrowed and expected to be paid back with interest. The video discusses loans in the context of annuity repayments, where the borrower takes a loan from a bank and repays it over a fixed period in the form of annuity payments.

💡Formula

In mathematics and finance, a formula is a concise way of expressing information in a logical manner. The script introduces several formulas used to calculate annuity payments, interest, and principal repayments, which are essential for understanding the mechanics of loan repayment.

💡Bank

A bank is a financial institution that offers services such as accepting deposits, providing loans, and offering various financial products. In the video, banks are mentioned as the entities that provide loans and calculate annuity payments for their customers.

💡Repayment Period

The repayment period is the duration over which a loan is to be repaid. The script discusses different repayment periods, such as 12 months or 3 years, which affect the calculation of monthly annuity payments.

💡Credit

Credit refers to the ability to obtain goods or services before payment, based on the trust that payment will be made in the future. In the video, credit is discussed in the context of loans and annuities, where credit is extended by banks to borrowers.

💡Percentage

A percentage is a way of expressing a number as a fraction of 100. In the script, percentages are used to denote interest rates, which are then applied to the principal to calculate the interest component of each annuity payment.

💡Calculation

Calculation refers to the process of computing or determining something by mathematical methods. The video script includes several examples of calculations needed to determine the amount of each annuity payment, including the breakdown into principal and interest.

💡Example

An example is a representative illustration of a concept or process. The script provides examples of annuity calculations, such as Mr. Bagas taking a loan from Bank ABC, to demonstrate how the formulas are applied in real-world scenarios.

Highlights

Introduction to the concept of annuities in mathematics for high school students.

Explanation of annuities as installment credit in banking and leasing contexts.

The formula for calculating annuities, including principal and interest components.

The difference in interest and principal payment amounts at the beginning and end of the annuity period.

The mathematical formula to find the annuity amount (A), involving the principal (M), interest rate (I), and number of periods (N).

How to derive the formula for annuities, though not explained in detail in the video.

Calculation of the principal payment for the first period (A1) using a specific formula.

The relationship between the annuity payment and the principal payment for any period (An).

Method to determine the total interest payment for the entire annuity period.

Example problem involving Mr. Bagas taking a loan from Bank ABC, with calculations for monthly payments, interest, and principal.

Detailed step-by-step calculation of the annuity payment for Mr. Bagas's loan.

How to calculate the principal and interest for the last payment period of the annuity.

The significance of the changing ratio of principal to interest payments over the annuity period.

A second example involving Bu Desi taking a loan with a 3-year repayment period and calculations for the loan amount.

The use of the annuity formula to determine the initial loan amount based on known annuity payments and interest rate.

The importance of understanding annuity calculations in the context of bank loans and other credit transactions.

Closing remarks encouraging further learning and application of annuity concepts.

Transcripts

[Musik]

Assalamualaikum warahmatullahi

wabarakatuh jumpa lagi dengan dokter

online

video kali ini akan membahas tentang

anuitas materi kelas 11 matematika SMA

ya untuk kurikulum merdeka

secara umum anuitas itu secara mudah

seperti yang angsuran ya angsuran kredit

baik kredit di perbankan maupun leasing

sepeda motor dan lain-lain

jadi rumusnya seperti ini

ini adalah anuitas ya atau angsuran

dengan anuitas

ini terdiri dari angsuran pokok an ini

yang angsuran pokok ya pada pembayaran

KN dan BN adalah bunga pada pembayaran

jadi setiap kali angsuran itu terdiri

dari angsuran pokok ditambah dengan

bunga

ini nanti tiap bulannya berbeda

di awal bunganya besar angsuran pokoknya

kecil nanti di akhir kebalikannya

angsuran pokoknya besar dan bunganya

kecil

rumus anuitas ya untuk mencari a ini

diperoleh dari

misalkan kita pinjam uang sebesar m ya

jadi jumlah pinjamannya adalah m ini

kemudian bunga yang berlaku adalah suku

bunganya I sekian persen maka rumusnya

seperti ini m * i

dikalikan dengan 1 + I pangkat n dibagi

1 + I ^ N - 1 ini perlu dihafalkan ya

Ini cara memperoleh rumus ini cukup

panjang ya di video ini tidak saya

jelaskan bagaimana cara memperoleh

rumusnya

kemudian

untuk mengetahui besar angsuran pokok

pada periode pertama atau A1

jadi misalkan kita pengen tahu nih pada

angsuran pertama itu angsuran pokoknya

berapa ya

Kemudian untuk mencari besar angsuran

pokok pada periode ke-n atau an ini ya

maka rumusnya ternyata seperti ini jadi

kita wajib mencari A1 dulu untuk bisa

mencari an karena rumusnya sama dengan

A1 dikali 1 + I pangkat n min 1

untuk mengetahui banyaknya bunga maka

dari rumus ini ya dari rumus ini kita

akan memperoleh rumus yang ini dengan

memindahkan luas ya diperoleh BN = a -

seperti itu ini rumus-rumus yang wajib

kita ketahui untuk bisa menyelesaikan

masalah tentang anuitas ini

contoh ya

Pak Bagas mengajukan pinjaman kur ke

bank ABC sebesar 40 juta

ini berarti m nya ya ini m nya

dengan periode pengembalian selama 12

bulan ini berarti n-nya ini n

jika besar suku bunga 6% per tahun

perhatikan ini yang berlaku per tahun

Padahal kita setorannya dalam perbulan

maka 6% ini perlu dibagi dengan 12 jadi

nanti ketemunya

0.5% per bulan

pertanyaannya adalah Tentukan yang

pertama besar angsuran yang harus

dibayar Pak Bagas setiap bulan berarti a

ya anuitasnya ini pertanyaannya berarti

anuitas kemudian besar bunga dan

angsuran pokok yang harus dibayarkan

pada bulan terakhir ini berarti bunga

berarti B ya B12 karena ada 12 bulan

terakhir berarti 12 ditambah a 12 Oke

untuk mencari ini semua kita catat dulu

yang diketahui apa ini dia yang

diketahui m nya 40 juta n-nya 12i ya

adalah 6% dibagi 12 bulan = 0,5% per

bulan

Kemudian untuk mencari a kita masukkan

rumusnya masing-masing

m * i

* 1 + I pangkat n dibagi 1 + x -1

kemudian Apa yang diketahui di sini kita

masukkan ya m nya diganti dengan 40 juta

seperti ini diganti 0.5% ini juga sama

kemudian n-nya 12 bagian bawah juga sama

i-nya 0,5% kawatnya n12 kurangi 1 ini

dihitung silakan pakai kalkulator Ya

wajib pakai kalkulator

ketemunya ternyata seperti ini

ini 40.000 40 juta kali 0,5% ketemunya

200.000 kemudian 1 tambahan 0,5%

ketemunya 1,005 pangkatkan 12 per 1,005

pangkat 12 kurangi 1

hasil akhirnya adalah ini ini adalah

anuitasnya atau angsuran totalnya ya per

bulan ini nanti terdiri dari

angsuran pokok dan bunga seperti itu

untuk mengerjakan soal yang B lanjut

kita akan mencari besar bunga dan

angsuran pokok pada periode ke-12 ya

berarti pertama kita harus mencari A1

dulu

jadi kita harus mencari besar angsuran

pokok pada periode pertama atau A1 ini

rumusnya seperti ini

ini mirip dengan ini Ya hanya

dihilangkan 1 + n nya untuk mempermudah

kita menghafal

kita masukkan m nya 40 juta i-nya 0,5%

seperti ini ya kemudian kita hitung

pakai kalkulator hasilnya seperti ini

hasil akhirnya adalah

3.242.657,19 jadi ini adalah angsuran

pokoknya

berarti kalau cari bunga di periode

pertama tinggal mengurangkan a ini

dengan

A1 nanti ketemu B1

lanjut kita akan mencari a 12

jadi besar angsuran pokok pada periode

12 ini ya A 12

rumusnya seperti ini

a 12 = A1 * 1 + I ^ 12 - 1

tinggal kita masukkan saja A1 nya yang

sudah diperoleh ini ya

diperoleh ini dia kita hitung hasilnya

adalah 3.425.000

529.54 ini adalah a 12

Bagaimana cara mencari B12

rumusnya adalah seperti ini

B12 diperoleh dari a kurangi A12 jadi

nilai anuitas itu untuk setiap bulannya

selalu sama ya selalu sama banyak

besaran bunga sama angsuran pokoknya

yang berbeda

hanya adalah ini

3.442

657,19 dikurangi A12 yaitu 3 juta

425.529,54

seperti ini ya hasilnya adalah

17127,65 jadi ini bunganya ya bunga pada

angsuran ke-12 ini angsuran pokoknya

selengkapnya biar lebih paham akan saya

Tampilkan dalam bentuk tabel jadi

seperti ini

jadi angsuran anuitasnya itu yang

sebelah kanan ini ya selalu tetap ini

dari bulan ke bulan selalu tetap

coba perhatikan angsuran pokok dan bunga

pada bulan ke-1 angsuran pokoknya

ternyata adalah ini 3 juta 242

657,19 sedangkan bunganya adalah 200

ribu ya

sedangkan pada angsuran terakhir

bunganya tinggal 17.000 saja sedangkan

angsuran pokoknya yang besar

semakin lama periode

peminjamannya misalkan sampai 10 tahun

15 tahun maka nanti akan kelihatan

sekali ya bahwa kalau angsuran pokok

pada periode pertama itu sangat kecil

yang besar adalah bunganya bahkan lebih

besar dari angsuran pokoknya nanti pada

bulan terakhir surat pokoknya besar

bunganya semakin kecil ya

perhitungan inilah yang dipakai ketika

bank menghitung kredit ya kemudian

mungkin dealer menghitung kredit motor

dan lain-lain seperti itu

contoh yang kedua

Bu Desi mengambil pinjaman dengan jangka

waktu pengembalian 3 tahun perhatikan

3 tahun Berarti masing-masing tahun 12

bulan ya berarti nanti ini n-nya adalah

36

dan angsuran yang harus dibayarkan

setiap bulan adalah ini ini berarti

diketahui anuitasnya ya

jika suku bunga sebesar 12% pertahun

ini Iya Ini berarti harus dibagi dengan

12 berarti ketemunya satu persen per

bulan

Tentukan berapa uang yang dipinjam Bu

Desi

kita akan gunakan rumus

ini yang diketahui

n-nya 36 ini sama ya i-nya 1% kemudian

ini rumusnya kita tinggal masukkan saja

hanya adalah satu juta 660.000 715,49

m nya belum tahu ya Karena yang

ditanyakan adalah m tentukan berapa uang

berarti ini adalah m yang ditanyakan

i-nya sudah ada tinggal dimasukkan saja

n-nya sudah ada yaitu 36

jadi seperti ini ya Ini a Ini m belum

diketahui kita cari Q nya 1% n-nya

adalah 36 selanjutnya tinggal kita

hitung saja pakai kalkulator ya

tentunya seperti ini sederhananya 1 + 1%

berarti 1,01

kemudian kita hitung lagi

ini dia ketemunya

lanjut lagi

ini harus pakai kalkulator ya

Oke sampai ketemu 50 juta koma 107 ini

ya ini dibulatkan saja berarti ini

pinjamnya adalah

50 juta

seperti itu

ya demikian pembahasan anuitas soalnya

nanti bisa berkembang bisa ditanyakan

periodenya ya dan lain-lain bisa juga

memilih antara pinjaman di Bank a sama

bank b ya dan seterusnya Selamat belajar

semoga bermanfaat saya akhirnya

Assalamualaikum warahmatullahi

wabarakatuh