Simple equations (involving non-integral coefficients or constants)

Q: What is a variable in mathematics?

A variable is a letter (like $x$ or $y$) representing an unknown number in an expression or equation.

Q: What is the definition of an equation in mathematics?

An equation is a mathematical sentence stating that two expressions are equal, connected by an equals sign ($=$).

Mathematics · Primary School · 5 min read · Updated: 18 Aug 2026

Welcome to the World of Simple Equations!

Hello, super mathematicians! Get ready for an exciting adventure into the world of Simple Equations with Decimals and Fractions (non-integral numbers). Think of it like being a detective. You get a puzzle with a missing piece, and you have to use your math skills to find it!

In this chapter, you'll learn how to solve equations where the numbers might be decimals or fractions. It's a very important skill because real life often involves parts of numbers, like $\$1.50$ for a snack or $\frac{1}{2}$ a kilogram of flour. Let's get started!

The Building Blocks: Meet the Variable and Non-Integral Numbers

What is a Variable? The Mystery Letter!

A letter like x or y in a math problem is called a variable. It's like a mystery box holding a number we don't know yet. In Primary 6, the numbers inside the box or multiplying it can be whole numbers, decimals, or fractions!

Example: If a recipe needs 'x' kilograms of sugar and we have 0.4 kg, 'x' represents an unknown amount.

Reading Algebraic Expressions

When we combine numbers and variables, we get an algebraic expression.

- If you see 1.5x, it means "1.5 times x".
- If you see $\frac{2}{3}y$, it means "$\frac{2}{3}$ times y" or "2 times y divided by 3".
- If you see m + 0.8, it means "an unknown number plus 0.8".

What is an Equation?

The Balancing Act!

An equation is like a perfectly balanced scale. The equals sign (=) is the balance point. It tells us that the left side has the exact same value as the right side!

$x + 1.2 = 3.5$

Key Term: An equation is a math sentence stating that two expressions are equal.

The Golden Rule of Equations

To keep the scale balanced, whatever you do to one side of the equation, you MUST do the exact same thing to the other side. If you subtract 1.2 from the left, you must subtract 1.2 from the right!

Solving One-Step Equations with Decimals and Fractions

We isolate the variable using inverse operations (opposite actions):

- The opposite of adding (+) is subtracting (-).
- The opposite of subtracting (-) is adding (+).
- The opposite of multiplying (×) is dividing (÷).
- The opposite of dividing (÷) is multiplying (×).

1. Decimal Addition and Subtraction Equations

Example: $x + 1.8 = 4.2$

Step 1: To get $x$ alone, subtract 1.8 from both sides.
$x + 1.8 - 1.8 = 4.2 - 1.8$
Step 2: Calculate the difference:
$x = 2.4$

2. Decimal and Fraction Multiplication Equations

Example with a decimal: $0.5x = 4$

Step: Divide both sides by 0.5:
$\frac{0.5x}{0.5} = \frac{4}{0.5}$
$x = 8$

Example with a fraction: $\frac{3}{4}y = 9$

Step: Multiply both sides by the reciprocal $\frac{4}{3}$ (or multiply by 4 then divide by 3):
$y = 9 \times \frac{4}{3}$
$y = 12$

Level Up! Two-Step Equations (Up to Two Steps)

Two-step equations involve two operations. Remember the Golden Rule: Undo addition or subtraction first, then undo multiplication or division.

Example 1 (Type: ax + b = c with Decimals)

Equation: $1.2x + 3.4 = 7$

Step 1 (Undo Addition): Subtract 3.4 from both sides.
$1.2x = 7 - 3.4$
$1.2x = 3.6$

Step 2 (Undo Multiplication): Divide both sides by 1.2.
$x = \frac{3.6}{1.2}$
$x = 3$

Check: $1.2(3) + 3.4 = 3.6 + 3.4 = 7$. Correct!

Example 2 (Type: ax - b = c with Fractions)

Equation: $\frac{2}{3}x - \frac{1}{4} = \frac{5}{12}$

Step 1 (Undo Subtraction): Add $\frac{1}{4}$ to both sides.
$\frac{2}{3}x = \frac{5}{12} + \frac{1}{4} = \frac{5}{12} + \frac{3}{12} = \frac{8}{12} = \frac{2}{3}$

Step 2 (Undo Multiplication): Divide both sides by $\frac{2}{3}$.
$x = \frac{2}{3} \div \frac{2}{3}$
$x = 1$

Special Types of Equations

1. Equations with Brackets: $a(x + b) = c$ and $a(x - b) = c$

Equation: $2.5(x - 1.2) = 10$

Method 1 (Divide First):
Divide both sides by 2.5:
$x - 1.2 = \frac{10}{2.5} = 4$
Now add 1.2 to both sides:
$x = 4 + 1.2 = 5.2$

Method 2 (Expand First):
Multiply 2.5 into the brackets:
$2.5x - 3 = 10$
Add 3 to both sides: $2.5x = 13$
Divide by 2.5: $x = \frac{13}{2.5} = 5.2$

2. Combining Terms: $dx + ex = c$ and $dx - ex = c$

When the same variable appears more than once on the same side, combine their coefficients first.

Addition Example: $1.5x + 2.5x = 16$

Step 1: Combine terms: $(1.5 + 2.5)x = 4x = 16$
Step 2: Divide by 4: $x = 4$

Subtraction Example: $\frac{5}{6}y - \frac{1}{6}y = 8$

Step 1: Combine terms: $\frac{4}{6}y = \frac{2}{3}y = 8$
Step 2: Solve for y: $y = 8 \times \frac{3}{2} = 12$

Using Non-Integral Equations to Solve Word Problems

Word Problem Example

Problem: Mary bought 3 identical notebooks and a pen that cost $\$4.50$. She spent a total of $\$22.50$. How much did each notebook cost?

Step 1: Let $n$ be the cost of one notebook in dollars.
Step 2: Form the equation: $3n + 4.5 = 22.5$
Step 3: Subtract 4.5 from both sides: $3n = 18$
Step 4: Divide by 3: $n = 6$
Step 5: Check: $3(6) + 4.5 = 18 + 4.5 = 22.5$.
Answer: Each notebook cost $\$6$.

Chapter Summary

- An equation is a balanced scale with an = sign in the middle.
- Equations can contain fractions and decimals as coefficients or constants.
- The Golden Rule: Do the exact same operation to both sides.
- Solutions in Primary 6 take at most two calculation steps.
- Always check your answer by substituting it back into the original equation!

References

Primary 6 Assessment: Scope of Assessment and Basic Competencies (Mathematics) — Hong Kong Examinations and Assessment Authority (HKEAA)

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is a variable in mathematics?

A variable is a letter (like $x$ or $y$) representing an unknown number in an expression or equation.

What does an expression like $1.5x$ or $\frac{2}{3}y$ represent?

It represents the coefficient multiplied by the variable (e.g., $1.5 \times x$ or $\frac{2}{3} \times y$).

What is the definition of an equation in mathematics?

An equation is a mathematical sentence stating that two expressions are equal, connected by an equals sign ($=$).

What is the Golden Rule for solving equations?

Whatever operation you perform on one side of the equation, you must perform the exact same operation on the other side.

How do inverse operations help solve equations?

Inverse (opposite) operations undo calculations to isolate the variable on one side.

How do you solve a multiplication equation involving a fraction coefficient like $\frac{3}{4}y = 9$?

Multiply both sides by the reciprocal $\frac{4}{3}$ (or multiply by the denominator $4$ and divide by the numerator $3$).

What order of operations should be followed when solving two-step equations of the form $ax + b = c$?

First undo addition or subtraction, then undo multiplication or division.

What are two valid methods for solving an equation with brackets like $a(x - b) = c$?

You can either divide both sides by $a$ first, or expand the brackets by distributing $a$ and then solve.

How do you solve equations where like variable terms are added or subtracted, such as $dx + ex = c$?

Combine the coefficients of the like terms first into $(d + e)x$, then solve for $x$.

How do you set up and solve a word problem using an algebraic equation?

Define a variable for the unknown quantity, write an equation matching the problem, solve for the variable, and verify the answer.

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