Introduction

In mathematics, proportion describes the relationship between two quantities. When one value changes, how does the other react?

Understanding whether a relationship is direct or inverse is essential for everything from cooking recipes to physics and economics.

In this book, we will break down:

• How to identify the type of proportion.
• How to calculate the constant of proportionality ($k$).
• Real-world applications for both types.

🍕 Case Study: The Ultimate Pizza Party

Before we look at formulas, let’s look at a party. Imagine you are the lead organizer for the End of Year Pizza Bash.

Two very different mathematical "laws" are happening at the same time. Let's break them down.

🟢 Scenario 1: The Grocery Bill

The Rule: Every pizza costs exactly $10.

The Discovery: As the number of pizzas increases, the cost increases. If you buy twice as many pizzas, you pay twice as much money.

This is Direct Proportion. They move in the same direction.

🔴 Scenario 2: Sharing the Slices

The Rule: You have one giant box of 24 slices.

The Discovery: As the number of friends increases, the amount of pizza each person gets decreases. If you double the number of friends, everyone gets half as much food.

This is Inverse Proportion. They move in opposite directions.

💡 Summary Table

🧠 Brain Teaser

Think about your journey to the pizza shop:

1. If you walk at a constant speed, does the distance you cover relate to time directly or inversely?
2. If the shop is a fixed distance away, does your speed relate to the time it takes to get there directly or inversely?

(Check the next chapters to see if you're right!)

Direct Proportion

Direct proportion occurs when two quantities increase or decrease at the same rate. If you double one, the other doubles as well.

The Mathematical Formula

The relationship is expressed as: $$y = kx$$

Where:

• $y$ and $x$ are the variables.
• $k$ is the constant of proportionality.

Key Characteristics

1. The ratio $\frac{y}{x}$ is always constant ($k$).
2. The graph is always a straight line passing through the origin (0,0).

Example

If 5 apples cost $10, how much do 8 apples cost?

1. Find $k$: $\frac{10}{5} = 2$. (Each apple costs $2).
2. Apply to new value: $y = 2 \times 8 = 16$. Answer: $16.

Inverse Proportion

Inverse proportion occurs when one quantity increases while the other decreases. If you double one, the other is halved.

The Mathematical Formula

The relationship is expressed as: $$y = \frac{k}{x} \quad \text{or} \quad xy = k$$

Key Characteristics

1. The product $x \times y$ is always constant ($k$).
2. The graph is a curve (hyperbola) that never touches the axes.

Example: Speed and Time

If it takes 4 hours to travel a distance at 60 mph, how long will it take at 80 mph?

1. Find $k$: $60 \times 4 = 240$ (The total distance).
2. Apply to new value: $t = \frac{240}{80} = 3$. Answer: 3 hours.

Practice Problems

Test your understanding of direct and inverse proportions with these word problems.

Problem 1: The Bakery (Direct)

A bakery uses 12 cups of flour to make 30 large cookies. How many cups of flour are needed to make 75 cookies?

Problem 2: The Construction Crew (Inverse)

It takes 6 workers 10 days to build a small shed. If the owner wants the shed finished in only 4 days, how many workers are needed in total?

Problem 3: Fuel Consumption (Direct)

A car travels 150 miles on 5 gallons of gas. How far can it travel on a full tank of 14 gallons?

Problem 4: Pumping a Tank (Inverse)

Three identical pumps can empty a water tank in 8 hours. How long would it take if 2 more pumps were added?

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🏛️ Archimedes: The Father of Mathematics

Archimedes of Syracuse (c. 287 – 212 BC) was a Greek mathematician, physicist, and engineer. He is most famous for his "Eureka!" moment and his ability to see the world through the lens of ratios and proportions.

👑 The Case of the Golden Crown

The most famous story involving Archimedes is the Case of the Golden Crown. King Hiero II of Syracuse suspected a goldsmith had cheated him by mixing silver into a "pure gold" crown. He asked Archimedes to prove it without damaging the crown.

The Discovery: Displacement

While stepping into a bathtub, Archimedes noticed the water level rose. He realized that the volume of water displaced was directly proportional to the volume of the object submerged.

The Math: $$V_{\text{object}} \propto V_{\text{water displaced}}$$

By comparing the amount of water displaced by the crown versus a lump of pure gold of the exact same weight, he proved the crown was less dense—meaning it was mixed with silver!

🏗️ Archimedes and the Lever

Archimedes was obsessed with the Inverse Proportion found in simple machines. He famously said:

"Give me a place to stand, and I shall move the Earth."

The Law of the Lever

He discovered that for a lever to balance, the weight ($w$) is inversely proportional to its distance ($d$) from the pivot point (fulcrum).

The Formula: $$w_1 \cdot d_1 = w_2 \cdot d_2$$

To lift a very heavy weight (increase $w$), you must increase your distance from the pivot (increase $d$) so the effort required remains small.

📜 His Mathematical Legacy

Archimedes didn't just solve kingly puzzles; he laid the groundwork for modern geometry:

• Calculating Pi ($\pi$): He used polygons to find a very accurate ratio of a circle's circumference to its diameter.

• The Archimedes Screw: A device for raising water that is still used in irrigation today.

• Sphere and Cylinder: He proved that the volume of a sphere is exactly two-thirds the volume of a cylinder that encloses it.

💡 Why this matters today

Archimedes showed us that nature follows mathematical rules. Whether it's the density of a metal (Direct Proportion) or the physics of a see-saw (Inverse Proportion), his work proves that math is the "language" of the physical world.

📐 Geometry Formula Reference

This page serves as a quick-start guide for the most common geometry formulas used in middle school.

🟦 2D Shapes: Area and Perimeter

🧊 3D Shapes: Volume and Surface Area

1. Rectangular Prism

• Volume: $V = lwh$
• Surface Area: $SA = 2(lw + lh + wh)$

2. Cylinder

• Volume: $V = \pi r^2 h$
• Surface Area: $SA = 2\pi rh + 2\pi r^2$

3. Sphere

• Volume: $V = \frac{4}{3}\pi r^3$
• Surface Area: $SA = 4\pi r^2$

📐 Angles and Triangles

Pythagorean Theorem

For any right-angled triangle, the square of the hypotenuse ($c$) is equal to the sum of the squares of the other two sides ($a$ and $b$). $$a^2 + b^2 = c^2$$

Angle Sums

• Triangle: The interior angles always sum to $180^\circ$.
• Quadrilateral: The interior angles always sum to $360^\circ$.

💡 Geometry and Proportion Connection

Remember:

1. Direct Proportion: If you double the radius of a circle, the circumference doubles (Linear).
2. Square-Cube Law: If you double the side of a cube, the surface area increases by 4x ($2^2$), but the volume increases by 8x ($2^3$)!

📐 High School Geometry & Trigonometry

At the high school level, we focus on the relationships between angles and sides, as well as the properties of circles and coordinate planes.

1. Right Triangle Trigonometry

When dealing with right triangles, we use the ratios of the sides relative to an angle $\theta$.

SOH CAH TOA:

• Sine: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
• Cosine: $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
• Tangent: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$

2. Laws for Non-Right Triangles

For any triangle with sides $a, b, c$ and opposite angles $A, B, C$:

Law of Sines

The ratio of the length of a side to the sine of its opposite angle is constant. $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$

Law of Cosines

Useful for finding a missing side when you have two sides and the included angle (SAS). $$c^2 = a^2 + b^2 - 2ab \cos(C)$$

3. Coordinate Geometry

Formulas used for points $(x_1, y_1)$ and $(x_2, y_2)$ on a Cartesian plane.

• Distance Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
• Midpoint Formula: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$
• Slope ($m$): $m = \frac{y_2 - y_1}{x_2 - x_1}$

4. Circles & Conic Sections

Beyond area, we look at the equation of a circle and arc properties.

• Equation of a Circle: $(x - h)^2 + (y - k)^2 = r^2$ (where $(h, k)$ is the center).
• Arc Length ($s$): $s = r\theta$ (where $\theta$ is in radians).
• Sector Area ($A$): $A = \frac{1}{2}r^2\theta$

5. Advanced Volume (Pyramids, Cones, Spheres)

💡 The Calculus Connection

As you move toward Calculus, remember that Volume is the integral of Area, and Area is the integral of Perimeter (or circumference).

Example: $\frac{d}{dr}(\text{Volume of Sphere}) = \frac{d}{dr}(\frac{4}{3}\pi r^3) = 4\pi r^2$ (Surface Area!)

🎭 Introduction to Polynomials

In our earlier chapters, we looked at simple relationships like $y = kx$. A Polynomial is simply a more complex version of these algebraic expressions.

The word comes from:

• Poly- (meaning many)
• -nomial (meaning terms)

1. What is a Polynomial?

A polynomial is an expression consisting of variables (like $x$), coefficients (numbers in front of $x$), and exponents (powers).

The Golden Rules:

To be a polynomial, an expression CANNOT have:

1. Negative or fractional exponents (No $x^{-2}$ or $x^{1/2}$).
2. Variables in the denominator (No $1/x$).
3. Variables under a square root (No $\sqrt{x}$).

2. Anatomy of a Polynomial

Let's look at $5x^3 - 2x + 7$:

• Terms: There are three terms ($5x^3$, $-2x$, and $7$).
• Degree: The highest exponent. Here, the degree is 3.
• Leading Coefficient: The number attached to the highest power ($5$).
• Constant: The number without a variable ($7$).

3. Classifying Polynomials

We name polynomials based on how many terms they have and their degree.

By Number of Terms:

• Monomial: 1 term (e.g., $4x^2$)
• Binomial: 2 terms (e.g., $3x + 5$)
• Trinomial: 3 terms (e.g., $x^2 - 4x + 4$)

By Degree:

4. Polynomials in Standard Form

Standard form means writing the terms in order from the highest exponent to the lowest.

• Messy: $5 + 2x^3 - x$
• Standard Form: $2x^3 - x + 5$

5. Why do we care?

In the real world, polynomials model everything from the trajectory of a basketball (Quadratic) to the growth of a savings account with compound interest.

Connection to Proportion: A linear polynomial with no constant ($y = mx$) is exactly what we studied in Direct Proportion! As the degree increases, the relationships just become more "curvy."

🧮 Polynomial Operations & Theorems

Before we can use advanced theorems, we must understand how to combine polynomial terms.

1. Adding and Subtracting: "Like Terms"

You can only add or subtract terms that have the same variable and the same exponent.

Example: $(3x^2 + 5x - 7) + (x^2 - 2x + 1)$

1. Group the $x^2$ terms: $3x^2 + 1x^2 = 4x^2$
2. Group the $x$ terms: $5x - 2x = 3x$
3. Group the constants: $-7 + 1 = -6$ Result: $4x^2 + 3x - 6$

2. The Remainder Theorem

This theorem is a massive time-saver. Instead of performing long division, you can find the remainder of a polynomial division instantly.

The Theorem: If you divide a polynomial $P(x)$ by $(x - c)$, the remainder is simply $P(c)$.

Example: Find the remainder when $P(x) = x^3 - 2x^2 + 3$ is divided by $(x - 2)$.

• Instead of dividing, just plug in $2$ for $x$:
• $P(2) = (2)^3 - 2(2)^2 + 3$
• $P(2) = 8 - 8 + 3 = 3$
• The Remainder is 3.

3. The Factor Theorem

The Factor Theorem is a special case of the Remainder Theorem. It helps us find the "roots" or "zeros" of a polynomial.

The Theorem: $(x - c)$ is a factor of $P(x)$ if and only if $P(c) = 0$.

Why is this useful?

If $P(c) = 0$, it means the division has no remainder, so the polynomial passes exactly through the x-axis at that point.

Example: Is $(x - 1)$ a factor of $P(x) = x^2 - 3x + 2$?

1. Plug in $1$: $P(1) = (1)^2 - 3(1) + 2$
2. Calculate: $1 - 3 + 2 = 0$
3. Yes! Because the result is $0$, $(x - 1)$ is a factor.

4. Synthetic Division: The Shortcut

To apply these theorems quickly, we often use Synthetic Division, a shorthand method of polynomial division.

💡 Pro-Tip for your Book

When graphing these, remember:

• If $(x - c)$ is a factor, the graph crosses the x-axis at $c$.
• If the remainder is positive, the graph is above the axis at that point.

📝 Worked Example: The Horner Method

Let's solve a problem step-by-step to see how the numbers move through the "Bagan Horner."

Problem: Find the quotient and remainder when: $$P(x) = x^4 - 3x^2 + 2x - 5$$ is divided by $(x - 2)$.

1. Identify the Coefficients

Before starting, look for "missing" powers of $x$. In this polynomial, there is no $x^3$ term. You must use 0 as a placeholder.

• Coefficients: 1 ($x^4$), 0 ($x^3$), -3 ($x^2$), 2 ($x$), -5 (constant)
• Divisor ($k$): Since we divide by $(x - 2)$, our $k = 2$.

2. The Horner Table Setup

Step-by-Step Calculation:

1. Drop: Bring the first coefficient (1) straight down.
2. Multiply & Add: $1 \times 2 = 2$. Add to $0 \rightarrow$ 2.
3. Multiply & Add: $2 \times 2 = 4$. Add to $-3 \rightarrow$ 1.
4. Multiply & Add: $1 \times 2 = 2$. Add to $2 \rightarrow$ 4.
5. Multiply & Add: $4 \times 2 = 8$. Add to $-5 \rightarrow$ 3.

3. Interpreting the Result

The final number in the row is your Remainder, and the preceding numbers are the coefficients of the Quotient.

Important: The degree of the quotient is always one less than the original. Since we started with $x^4$, our result starts with $x^3$.

• Quotient ($Q(x)$): $1x^3 + 2x^2 + 1x + 4$
• Remainder ($R$): $3$

4. Verification (Remainder Theorem)

To ensure the Horner Method worked correctly, we can plug $x=2$ into the original equation: $$P(2) = (2)^4 - 3(2)^2 + 2(2) - 5$$ $$P(2) = 16 - 12 + 4 - 5$$ $$P(2) = 3$$

The results match!

⚠️ Common Pitfall: The "Zero" Placeholder

If you forget to include the 0 for the $x^3$ term, the entire calculation shifts:

• Wrong: $[1, -3, 2, -5]$ ❌
• Correct: $[1, 0, -3, 2, -5]$ ✅

Always double-check that your coefficients represent every power from the highest degree down to the constant.

✍️ Practice Set: Polynomial Challenges

Test your skills using the Horner Method, Remainder Theorem, and Factor Theorem. Try to solve these on paper first, then click the "Solution" toggle to check your work.

🟢 Level 1: The Basics

Problem 1: Use the Horner Method to find the quotient and remainder of $(x^3 - 4x^2 + 2x + 5) \div (x - 3)$.

🟡 Level 2: The Missing Term

Problem 2: Find the remainder of $(2x^4 - 5x^2 + 6) \div (x + 2)$. Hint: Don't forget the placeholders for $x^3$ and $x$!

🟠 Level 3: The Factor Theorem

Problem 3: Is $(x - 1)$ a factor of $P(x) = x^3 + 2x^2 - x - 2$? Prove it using the Factor Theorem.

🔴 Level 4: Solving for $k$

Problem 4: The polynomial $P(x) = x^3 + kx^2 - 4x + 1$ has a remainder of $5$ when divided by $(x - 2)$. Find the value of $k$.

🟣 Level 5: Horner-Kino Challenge

Problem 5: Divide $P(x) = x^4 - 3x^3 + 5x^2 - 2x + 1$ by $(x^2 - x - 2)$.

📊 Latihan Rumus Excel Interaktif

Gunakan tabel di bawah ini untuk menjawab soal-soal latihan. Masukkan rumus Excel yang tepat (gunakan huruf kapital) lalu klik tombol Cek Jawaban.

Tabel Data Referensi:

Soal 1: Penjumlahan Dasar

Hitung total seluruh produk yang terjual (Kolom C)! Input jawaban (Contoh: =SUM(C1:C3)):

Soal 2: Perkalian & Total Harga

Hitung total pendapatan untuk Kopi saja (Harga Satuan $\times$ Jumlah Terjual)! Input jawaban:

Soal 3: Fungsi Logika (IF)

Jika jumlah terjual (C3) lebih dari 15, maka tampilkan "Laris", jika tidak tampilkan "Biasa". Input jawaban: