🔮 Word Problems: The Secret Language of Math
Welcome to the World of Algebra Detectives!
Imagine you’re a detective. Someone hands you a mystery note that says: “I’m thinking of a number. If you double it and add 5, you get 17. What’s my number?”
How do you solve this? You become an algebra detective! You use a secret weapon called variables (letters like x) to catch the mystery number.
Let’s learn how to translate everyday puzzles into algebra—and solve them like a pro!
🎯 The Big Picture: What Are Algebraic Word Problems?
Word problems are just stories with hidden math. Your job is to:
- Read the story carefully
- Find the unknown (what’s missing?)
- Translate words into math symbols
- Solve the equation
- Check your answer makes sense
Think of it like this: Words are just math wearing a costume. Your job is to see through the disguise!
graph TD A[📖 Read the Story] --> B[🔍 Find the Unknown] B --> C[✏️ Write an Equation] C --> D[🧮 Solve It] D --> E[✅ Check Your Answer]
📝 Translating Words to Algebra
This is your secret decoder ring! Here’s how to turn English into math:
The Translation Dictionary
| English Words | Math Symbol |
|---|---|
| is, equals, gives | = |
| more than, added to, plus | + |
| less than, minus, decreased by | − |
| times, of, multiplied by | × |
| divided by, per, out of | ÷ |
| a number, unknown | x (or any letter) |
| twice, double | 2× |
| half of | ÷2 or ×½ |
🌟 Example: Translating a Sentence
Story: “Five more than a number is twelve”
Let’s break it down:
- “a number” → x
- “five more than” → + 5
- “is” → =
- “twelve” → 12
Translation: x + 5 = 12
Solution: x = 12 - 5 = 7 ✨
🔢 Number Problems
Number problems are like riddles about mystery numbers. They’re the friendliest word problems to start with!
The Pattern
Someone describes a number using clues. You find it!
🌟 Example 1: The Mystery Number
Story: “I’m thinking of a number. If you triple it and subtract 4, you get 11. What’s my number?”
Step 1: Let the mystery number = x
Step 2: Translate:
- “triple it” → 3x
- “subtract 4” → −4
- “you get 11” → = 11
Equation: 3x − 4 = 11
Step 3: Solve:
- Add 4 to both sides: 3x = 15
- Divide by 3: x = 5
Check: 3(5) − 4 = 15 − 4 = 11 ✓
🌟 Example 2: Consecutive Numbers
Story: “The sum of three consecutive numbers is 24. What are they?”
Consecutive means numbers in a row (like 1, 2, 3).
Let: First number = x
- Second number = x + 1
- Third number = x + 2
Equation: x + (x+1) + (x+2) = 24
Solve:
- 3x + 3 = 24
- 3x = 21
- x = 7
Answer: The numbers are 7, 8, and 9 ✨
Check: 7 + 8 + 9 = 24 ✓
👨👩👧 Age Problems
Age problems are like time travel puzzles! They compare ages now, in the past, or in the future.
The Golden Rule
Everyone ages at the same rate! If 5 years pass:
- You get 5 years older
- Your friend gets 5 years older
- Everyone adds the same number!
🌟 Example 1: Father and Son
Story: “A father is 3 times as old as his son. In 10 years, he’ll be twice as old as his son. How old are they now?”
Let: Son’s age now = x
- Father’s age now = 3x
In 10 years:
- Son’s age = x + 10
- Father’s age = 3x + 10
The condition: Father will be twice the son’s age
Equation: 3x + 10 = 2(x + 10)
Solve:
- 3x + 10 = 2x + 20
- 3x − 2x = 20 − 10
- x = 10
Answer:
- Son is 10 years old
- Father is 30 years old
Check in 10 years:
- Son: 20, Father: 40
- Is 40 = 2 × 20? Yes! ✓
🌟 Example 2: Looking Back in Time
Story: “Maya is 16. Four years ago, she was twice as old as her brother is now. How old is her brother?”
Let: Brother’s current age = x
Maya’s age 4 years ago: 16 − 4 = 12
Equation: 12 = 2x
Solve: x = 6
Answer: Her brother is 6 years old ✨
⚖️ Ratio and Proportion
A ratio compares two quantities. A proportion says two ratios are equal.
Think of it like a recipe. If a recipe for 4 cookies needs 2 cups of flour, how much flour for 10 cookies? That’s proportion!
Understanding Ratios
Ratio 3:5 means “for every 3 of one thing, there are 5 of another”
If marbles are in ratio 3:5 and total is 40:
- Parts = 3 + 5 = 8
- Each part = 40 ÷ 8 = 5
- First group = 3 × 5 = 15
- Second group = 5 × 5 = 25
🌟 Example: Sharing Money
Story: “Divide $120 between Ali and Ben in the ratio 2:3”
Step 1: Total parts = 2 + 3 = 5
Step 2: Value of one part = 120 ÷ 5 = $24
Step 3:
- Ali gets: 2 × 24 = $48
- Ben gets: 3 × 24 = $72
Check: $48 + $72 = $120 ✓
✖️ Cross Multiplication
This is your superpower for solving proportions!
When you have: a/b = c/d
Cross multiply to get: a × d = b × c
It’s like making an X across the equal sign!
graph TD A["a/b = c/d"] --> B["a × d = b × c"] B --> C["Solve for unknown!"]
🌟 Example 1: Finding the Unknown
Solve: x/6 = 8/12
Cross multiply: x × 12 = 6 × 8
- 12x = 48
- x = 4
Check: 4/6 = 2/3 and 8/12 = 2/3 ✓
🌟 Example 2: Real Life Problem
Story: “If 5 notebooks cost $15, how much do 8 notebooks cost?”
Set up proportion:
- 5 notebooks / $15 = 8 notebooks / $x
- 5/15 = 8/x
Cross multiply:
- 5x = 15 × 8
- 5x = 120
- x = $24
🔄 Componendo and Dividendo
This is an advanced trick that makes hard problems easy!
The Magic Formula
If a/b = c/d, then:
Componendo: (a+b)/b = (c+d)/d
Dividendo: (a−b)/b = (c−d)/d
Componendo-Dividendo (Together):
(a+b)/(a−b) = (c+d)/(c−d)
When to Use It
Use this when you see something like:
- (x + y)/(x − y) = something
- Variables added and subtracted together
🌟 Example
Given: (x+2)/(x−2) = 3
Apply Componendo-Dividendo backwards:
If (a+b)/(a−b) = 3/1, then a/b = (3+1)/(3−1) = 4/2 = 2
So: x/2 = 2
- x = 4
Check:
- (4+2)/(4−2) = 6/2 = 3 ✓
Another Way to Solve
Direct approach:
- x + 2 = 3(x − 2)
- x + 2 = 3x − 6
- 2 + 6 = 3x − x
- 8 = 2x
- x = 4 ✓
💯 Percentage Problems
Percentages are fractions with a fancy name! Percent means “per hundred.”
50% = 50/100 = 0.5 = half
The Magic Formula
Part = Percentage × Whole
Or rearranged:
- Percentage = Part ÷ Whole × 100
- Whole = Part ÷ (Percentage/100)
🌟 Example 1: Finding the Part
Story: “What is 25% of 80?”
Solve: 25/100 × 80 = 0.25 × 80 = 20
🌟 Example 2: Finding the Whole
Story: “30 is 60% of what number?”
Equation: 30 = 0.60 × x
- x = 30 ÷ 0.60
- x = 50
🌟 Example 3: Percentage Increase
Story: “A shirt’s price increased from $40 to $50. What’s the percentage increase?”
Step 1: Find the change = 50 − 40 = $10
Step 2: Percentage = (Change/Original) × 100
- = (10/40) × 100
- = 0.25 × 100
- = 25%
🌟 Example 4: Word Problem
Story: “After a 20% discount, a bag costs $64. What was the original price?”
Think: If 20% is removed, 80% remains.
- 80% of original = $64
- 0.80 × x = 64
- x = 64 ÷ 0.80
- x = $80
Check: 20% of $80 = $16. $80 − $16 = $64 ✓
🎯 Problem-Solving Strategy: The 5-Step Method
graph TD A["1️⃣ READ carefully<br/>What's the story about?"] --> B["2️⃣ IDENTIFY unknowns<br/>Let x = what you need"] B --> C["3️⃣ TRANSLATE to equation<br/>Use the decoder ring!"] C --> D["4️⃣ SOLVE the equation<br/>Do the math"] D --> E["5️⃣ CHECK your answer<br/>Does it make sense?"]
Pro Tips 🌟
- Underline key words - numbers and relationship words
- Draw a picture when possible
- Label everything with units ($, years, etc.)
- Check if your answer makes sense - a person can’t be −5 years old!
🎮 You’re Ready!
You now have all the tools to tackle algebraic word problems:
✅ Translating - Turn words into equations
✅ Number Problems - Find mystery numbers
✅ Age Problems - Time travel with math
✅ Ratios & Proportions - Compare fairly
✅ Cross Multiplication - Your proportion superpower
✅ Componendo-Dividendo - The advanced trick
✅ Percentages - Master the “per hundred”
Remember: Every word problem is just a story waiting to be solved. You’re the detective. The equation is your clue. Now go solve some mysteries! 🔍✨