Question 6

Mbox laptop store offers a $20\%$ discount for purchasing series $A$ laptops. Purchases at the store can be paid in monthly installments (with equal amounts, starting from the second month) with a simple interest of $6\%$ per year. The down payment required is at least $10\%$ of the initial price, and every subsequent minimum multiple will get an interest reduction of $1\%$ per year, up to a maximum of $2\%$ per year.

At the annual special event, the store offers an installment program without a down payment, with the first, second, third, fourth, and subsequent monthly installments being $\text{Rp}2,000,000.00$ , $\text{Rp}1,800,000.00$ , $\text{Rp}1,600,000.00$ , $\text{Rp}1,400,000.00$ , and so on until zero rupiah for purchasing a series $B$ laptop priced at $\text{Rp}10,000,000.00$ . With this purchase method, the price of the series $B$ laptop is ... percent more expensive compared to the initial price.

Explanation

We will calculate the total cost to be paid through this special installment scheme and compare it with the initial price of the laptop.

Installment Pattern

The installments paid form a decreasing arithmetic sequence:

First installment ( $a$ ) = $2,000,000$
Second installment = $1,800,000$
Difference ( $b$ ) = $1,800,000 - 2,000,000 = -200,000$

The installments continue until the value approaches zero (the last valid payment cannot be zero rupiah as it adds no cost, so we calculate until the last positive payment).

We determine the number of installments ( $n$ ) assuming the last installment ( $U_n$ ) is $200,000$ (one step before zero):

U_n = a + (n-1)b

200,000 = 2,000,000 + (n-1)(-200,000)

We divide both sides by $200,000$ to simplify:

1 = 10 + (n-1)(-1)

1 = 10 - n + 1

n = 10

So, there are 10 installments.

Calculating Total Payment

The total payment is the sum of the arithmetic series ( $S_{10}$ ):

S_n = \frac{n}{2}(a + U_n)

S_{10} = \frac{10}{2}(2,000,000 + 200,000)

S_{10} = 5(2,200,000)

S_{10} = 11,000,000

The total paid is $\text{Rp}11,000,000.00$ .

Calculating Price Increase Percentage

The initial price of the series $B$ laptop is $\text{Rp}10,000,000.00$ . Price difference = $11,000,000 - 10,000,000 = 1,000,000$ .

Percentage price increase:

\text{Percentage} = \frac{\text{Difference}}{\text{Initial Price}} \times 100\%

\text{Percentage} = \frac{1,000,000}{10,000,000} \times 100\%

\text{Percentage} = 10\%

So, the price of the series $B$ laptop is $10\%$ more expensive compared to the initial price.

Command Palette

Number 6

Explanation

Installment Pattern

Calculating Total Payment

Calculating Price Increase Percentage