General Mathematics: Business Mathematics and Sequences & Series

Flashcards covering salary deductions, arithmetic sequences and series, geometric sequences and series, and profit and loss calculations.

What are the four statutory and mandatory deductions from earnings in the Philippines?

1. SSS (Social Security System), 2. PhilHealth (Philippine Health Insurance Corporation), 3. Pag-IBIG (Home Development Mutual Fund / HDMF), and 4. Income tax under the TRAIN Law (Tax Reform for Acceleration and Inclusion Law).

Which Philippine law governs the Social Security System (SSS) contribution?

Republic Act 11199.

Under the Universal Health Care Act, what is the PhilHealth premium rate and its salary floor and ceiling as of 2025?

The premium rate is $5\%$ of an employee's monthly basic salary, with a salary floor of $\text{₱}10{,}000$ and a ceiling of $\text{₱}100{,}000$.

How is the PhilHealth contribution determined for monthly basic salaries less than $\text{₱}10{,}000$ and for salaries $\text{₱}100{,}000$ or more?

If the monthly basic salary is less than $\text{₱}10{,}000$, the contribution is fixed at $\text{₱}500$. If it is $\text{₱}100{,}000$ or more, it is capped at $\text{₱}5{,}000$. The computed amount is divided equally between the employee and employer.

Under the HDMF Law of 2009, how is the Pag-IBIG fund contribution calculated for 2025 for an employee earning over $\text{₱}1{,}500$ monthly?

It is the lower of $2\%$ of actual monthly salary or $2\%$ of the $\text{₱}10{,}000$ Monthly Fund Salary (MFS) cap, resulting in a maximum of $\text{₱}200$ per month for the employee and $\text{₱}200$ from the employer (a total of $\text{₱}400$ monthly).

What are the employee's and employer's shares for Pag-IBIG fund contributions if the monthly basic salary is $\text{₱}1{,}500$ or below?

The employee's share is $1\%$ (capped at $\text{₱}200$) and the employer's share is $2\%$ (capped at $\text{₱}200$), totaling $3\%$.

According to the 2025 BIR Tax Table under the TRAIN Law, what is the income tax rate for net taxable income of $\text{₱}250{,}000$ and below?

$0\%$ (tax-exempt).

According to the 2025 BIR Tax Table under the TRAIN Law, what is the tax rate for net taxable income between $\text{₱}250{,}000$ and $\text{₱}400{,}000$?

$15\%$ of the excess over $\text{₱}250{,}000$.

What is gross pay, and what is the formula to compute net pay (take-home pay)?

Gross pay is the total amount of money earned before any deductions. The formula for net pay is: $\text{Net Pay} = \text{Gross Pay} - \text{Total Deductions}$.

In the worked example for Mr. Dan Delo with a gross monthly salary of $\text{₱}34{,}500$, what was his total deduction amount?

$\text{₱}4{,}419.38$, which includes SSS Employee Share of $\text{₱}1{,}725.00$, PhilHealth Employee Share of $\text{₱}862.50$, Pag-IBIG Employee Share of $\text{₱}200.00$, and Monthly Income Tax of $\text{₱}1{,}631.88$.

In the worked example for Mr. Dan Delo, what was his final net take-home pay?

$\text{₱}30{,}080.57$.

What is the general mathematical definition of a sequence?

An ordered list of numbers that follow a certain pattern.

What is a series, and how is it symbolically represented?

A series is the sum of the terms of a sequence, represented as: $S_n = a_1 + a_2 + a_3 + \dots + a_n$.

For the sequence $3, 7, 11, 15, \dots$, what are $a_1$ and $d$, and what is the 10th term ($a_{10}$)?

$a_1 = 3$, $d = 4$, and $a_{10} = 3 + (10 - 1)(4) = 39$.

For the arithmetic sequence $3, 7, 11, 15, \dots$, what is the sum of the first 10 terms ($S_{10}$)?

$S_{10} = \frac{10}{2}[2(3) + (10 - 1)(4)] = 5[6 + 36] = 5(42) = 210$.

What defines an arithmetic sequence, and what is its constant called?

An arithmetic sequence is a sequence in which a constant, called the common difference ($d$), can be added to each term to get the next term.

How is the common difference ($d$) determined from consecutive terms in an arithmetic sequence?

By subtracting any term from one that follows it ($d = a_n - a_{n-1}$).

What is the formula to determine the $n\text{th}$ term of an arithmetic sequence?

$a_n = a_1 + (n - 1)d$.

What formula calculates the common difference ($d$) directly from any two terms $a_n$ and $a_m$ when $n > m$?

$d = \frac{a_n - a_m}{n - m}$.

What is the 28th term of the arithmetic sequence $5, 15, 25, 35, \dots$?

$a_{28} = 5 + (28 - 1)(10) = 5 + 270 = 275$.

What is the 16th term of an arithmetic sequence if the first term is $-10$ and the common difference is $2$?

$a_{16} = -10 + (16 - 1)(2) = -10 + 30 = 20$.

What is the 18th term of an arithmetic sequence if the first term is $-6$ and the common difference is $-4$?

$a_{18} = -6 + (18 - 1)(-4) = -6 + (-68) = -74$.

If the 2nd term of an arithmetic sequence is $8$ and the 12th term is $48$, what is the common difference ($d$)?

$d = \frac{48 - 8}{12 - 2} = \frac{40}{10} = 4$.

If the 4th term of an arithmetic sequence is $25$ and the 15th term is $80$, what is the common difference ($d$)?

$d = \frac{80 - 25}{15 - 4} = \frac{55}{11} = 5$.

If the 2nd term of an arithmetic sequence is $8$ and the 12th term is $48$, what is the first term ($a_1$) and the 18th term ($a_{18}$)?

With $d = 4$, $a_1 = 8 - (1)(4) = 4$, and $a_{18} = 4 + (18 - 1)(4) = 4 + 68 = 72$.

If the 3rd term of an arithmetic sequence is $15$ and the 10th term is $50$, what is the 26th term ($a_{26}$)?

With $d = \frac{50 - 15}{10 - 3} = 5$ and $a_1 = 5$, $a_{26} = 5 + (26 - 1)(5) = 5 + 125 = 130$.

A train has 125 passengers in the first carriage, 150 in the second, and 175 in the third. How many passengers are in the 10th carriage?

$a_{10} = 125 + (10 - 1)(25) = 125 + 225 = 350$ passengers.

What is the formula for the sum of an arithmetic series ($S_n$) when the last term ($a_n$) is given?

$S_n = \frac{n(a_1 + a_n)}{2}$.

What is the sum of the first 50 even numbers ($2, 4, 6, 8, 10, \dots, 100$)?

$S_{50} = \frac{50(2 + 100)}{2} = 25(102) = 2{,}550$.

A concert hall has 20 rows with 25 seats in row 1, 27 in row 2, and 29 in row 3. At a ticket price of ₱2,300, what are the total sales if all seats are taken?

Total seats: $S_{20} = \frac{20}{2}[2(25) + (20 - 1)(2)] = 10[50 + 38] = 880$. Total sales: $880 \times \text{₱}2{,}300 = \text{₱}2{,}024{,}000.00$.

What is the definition of a geometric sequence?

A sequence in which the same number is multiplied by each term to get the next term in the sequence.

What is the common ratio ($r$) in a geometric sequence and how is it found?

The common ratio is found by dividing any given term by the preceding term ($r = \frac{a_n}{a_{n-1}}$).

When treating a geometric sequence as a function, what are the input and the output?

The term number (or position in the sequence, $n$) is the input, and the term itself ($a_n$) is the output.

What is the formula to find the $n\text{th}$ term of a geometric sequence?

$a_n = a_1 r^{n-1}$.

Why must parentheses enclose a negative common ratio when writing its exponential term?

Because without parentheses, the negative sign is not raised to the exponent (e.g., $-2^{12} \neq (-2)^{12}$).

What is the 6th term of the geometric sequence $2, 6, 18, \dots$?

$a_6 = 2(3)^{6-1} = 2(3)^5 = 2(243) = 486$.

In a geometric sequence where the first term is $\frac{1}{8}$ and the seventh term is $8$, what is the common ratio ($r$)?

$8 = \frac{1}{8}(r)^{7-1} \implies 64 = r^6 \implies r = 2$.

A ball dropped from a tower bounces to heights of $300\,\text{cm}$, $150\,\text{cm}$, and $75\,\text{cm}$. What is the height of the 6th bounce?

With $a_1 = 300\,\text{cm}$ and $r = 0.5$, $a_6 = 300(0.5)^{6-1} = 300(0.5)^5 = 9.375\,\text{cm}$.

In 2024, 2,200 learners enrolled in UST AC. If the enrollment doubles every year, how many learners should enroll in 2030 ($n = 7$)?

$a_7 = 2{,}200(2)^{7-1} = 2{,}200(2)^6 = 2{,}200(64) = 140{,}800$ learners.

A DepEd Teacher 1 with an initial salary of ₱25,125.00 receives a 7% annual raise. What is the salary at the start of year six ($n = 6$)?

$a_6 = 25{,}125(1.07)^{6-1} = 25{,}125(1.07)^5 \approx \text{₱}35{,}239.11$.

What are geometric means?

The terms between any two given terms in a geometric sequence.

What is the geometric mean between $4$ and $16$?

$8$ or $-8$ (since $r = \pm 2$).

What are the four geometric means inserted between $3$ and $96$?

$6, 12, 24, 48$ (derived from $a_1 = 3$, $a_6 = 96$, yielding $r = 2$).

What formula is used for the sum of the first $n$ terms of a geometric sequence ($S_n$) when $r > 1$?

$S_n = \frac{a(r^n - 1)}{r - 1}$.

What is the sum of the first 20 terms of the geometric sequence $1, 2, 4, 8, \dots$?

$S_{20} = \frac{1(2^{20} - 1)}{2 - 1} = 1{,}048{,}576 - 1 = 1{,}048{,}575$.

Under what condition does a business experience a profit, and under what condition does it experience a loss?

Profit occurs when the selling price (or total revenue) is greater than the cost price (or total expenses). Loss occurs when the selling price is less than the cost price.

What are the formulas for calculating profit and loss in absolute terms?

$\text{Profit} = \text{Selling Price} - \text{Cost Price}$ and $\text{Loss} = \text{Cost Price} - \text{Selling Price}$.

What are the formulas for calculating Profit Percentage and Loss Percentage?

$\text{Profit Percentage} = \left(\frac{\text{Profit}}{\text{Cost Price}}\right) \times 100$ and $\text{Loss Percentage} = \left(\frac{\text{Loss}}{\text{Cost Price}}\right) \times 100$.

In Example 14, an electronic gadget was bought for ₱2,000 and sold for ₱2,350. What were the profit and profit percentage?

Profit: $\text{₱}2{,}350 - \text{₱}2{,}000 = \text{₱}350$. Profit Percentage: $\left(\frac{350}{2{,}000}\right) \times 100 = 17.5\%$.

In Example 15, Alex purchased an e-bike for ₱49,000 and sold it for ₱46,500. What were the loss and loss percentage?

Loss: $\text{₱}49{,}000 - \text{₱}46{,}500 = \text{₱}2{,}500$. Loss Percentage: $\left(\frac{2{,}500}{49{,}000}\right) \times 100 \approx 5.1\%$.