Simple algebraic word problems

Mathematics · Pre-Secondary One Hong Kong Attainment Test · 4 min read

Welcome to the World of Algebra!

Hello there! Today, we are going to learn one of the most powerful tools in mathematics: Algebra. Think of algebra like being a math detective. Sometimes, a number is "missing" or "hidden," and our job is to use clues from a story to find out what that number is. Don't worry if this seems tricky at first—once you learn the "code" to translate words into math, you'll be solving problems like a pro!

Section 1: The Basics - What is a Variable?

In algebra, we use letters (like $x$, $y$, or $a$) to represent numbers we don't know yet. These letters are called variables.

Quick Review: You have actually been doing algebra for years! Remember when you saw questions like $ \Box + 5 = 10 $? In algebra, we just replace that box with a letter: $ x + 5 = 10 $. It’s the same thing!

Key Takeaway:

A variable is just a placeholder for a secret number.

Section 2: Translating Words into Math

To solve word problems, we need to translate English words into mathematical symbols. This is often the part students find hardest, but there is a simple "decoder key" you can use!

The Math Decoder Key:

Addition (+): Sum, plus, increased by, more than, total.
Subtraction (-): Difference, minus, decreased by, less than, left.
Multiplication ($\times$): Product, times, twice (x2), triple (x3), of.
Division ($\div$): Quotient, divided by, shared equally, half ($\div 2$).
Equals (=): Is, results in, becomes, is the same as.

Did you know? In algebra, we usually don't use the $\times$ symbol because it looks too much like the letter $x$. Instead, we write $3 \times x$ as $3x$. If you see a number and a letter touching, it means they are being multiplied!

Key Takeaway:

Look for "keyword" clues in the story to decide which math operation ($+$, $-$, $\times$, $\div$) to use.

Section 3: The 4-Step Method to Solve Word Problems

Whenever you see a word problem, follow these four steps to stay organized. Let’s look at an example together.

Example Problem: "Peter has some candies. After his mother gives him 5 more candies, he has 12 candies in total. How many candies did Peter have at the start?"

Step 1: Identify the Unknown

What are we trying to find? Let's give it a name. We usually write a "Let" statement.
Let $x$ be the number of candies Peter had at the start.

Step 2: Build the Equation

Read the story again and turn it into math.
"Peter has some candies" $\rightarrow x$
"Mother gives him 5 more" $\rightarrow + 5$
"He has 12 in total" $\rightarrow = 12$
The Equation: $x + 5 = 12$

Step 3: Solve the Equation

To find $x$, we need to get it all by itself. We do this by doing the opposite of what is happening to it.
Since the equation says plus 5, we subtract 5 from the other side.
$x = 12 - 5$
$x = 7$

Step 4: Check and Answer

Does it make sense? If he had 7 and got 5 more, is that 12? Yes! $7 + 5 = 12$.
Answer: Peter had 7 candies at the start.

Key Takeaway:

Always start by writing "Let $x$ be..." to clearly define what you are looking for.

Section 4: Common Patterns in HKAT Problems

Here are a few common types of problems you might see on the Hong Kong Attainment Test:

1. Multiplication Patterns

Example: "A pencil costs \$4. If Miss Wong buys $y$ pencils for \$48, what is $y$?"
Equation: $4y = 48$
Solve: $y = 48 \div 4$
Result: $y = 12$

2. Problems with Brackets

Example: "There are 3 boxes of oranges. Each box has the same number of oranges. After eating 2 oranges from each box, there are 24 oranges left in total."
Let $x$ be the original number of oranges in each box.
Equation: $3(x - 2) = 24$
Solve:
Divide by 3 first: $x - 2 = 24 \div 3$
$x - 2 = 8$
Add 2: $x = 8 + 2$
Result: $x = 10$

Key Takeaway:

If an action (like eating 2 oranges) happens to multiple groups, use brackets in your equation.

Section 5: Avoiding Common Mistakes

Even the best math detectives make mistakes! Watch out for these "traps":

The "Less Than" Flip: If a problem says "5 less than $x$", it is written as $x - 5$, NOT $5 - x$. Subtraction order matters!
Forgetting Units: If the question asks for "money," make sure your final answer has the \$ sign. However, in the equation itself, we usually leave the units out to keep it clean.
Not Solving Completely: Sometimes $x$ isn't the final answer. Re-read the question: are they asking for $x$, or for $x + 10$?

Quick Review Box

1. Let $x$ = the unknown.
2. Translate words to symbols.
3. Opposite Operations:
- Opposite of $+$ is $-$
- Opposite of $-$ is $+$
- Opposite of $\times$ is $\div$
- Opposite of $\div$ is $\times$
4. Double check your answer!

You’ve got this! Algebra is just a puzzle, and now you have the tools to solve it. Keep practicing, and it will become second nature!

Quick check

Can you answer these now?

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

What is a variable in algebra?

A variable is a letter (like $x$, $y$, or $a$) used to represent an unknown number.

Which keywords indicate addition and subtraction in word problems?

Addition keywords include "sum," "plus," and "increased by"; subtraction keywords include "difference," "minus," "decreased by," and "less than."

How do we represent multiplication between a number and a variable in algebra?

In algebra, we write the number and the variable side-by-side without a multiplication symbol, such as writing $3x$ for $3 \times x$.

What is the purpose of a "Let" statement in a word problem?

A "Let" statement identifies and defines exactly what the variable represents in the problem.

How do you build an equation from a word problem?

Translate the story line-by-line into math symbols, using the variable for the unknown and an equals sign for words like "is" or "results in."

What is the standard method for solving for a variable in an equation?

Isolate the variable by performing the opposite operation on the other side of the equation.

When are brackets necessary in an algebraic equation?

Brackets are used when an operation applies to multiple items or groups, such as when an action is repeated for several identical boxes.

What is the correct way to write "5 less than $x$" in algebra?

It is written as $x - 5$, not $5 - x$.

Should units (like dollars or cm) be included inside the algebraic equation?

No, units should be left out of the equation to keep it clean, but they must be included in the final answer.

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