Word problems using linear equations

The Five-Step Method

Every word problem follows this systematic approach:

What: Choose what to represent with $x$ .
How: Pick the quantity the problem asks you to find, OR the simplest quantity from which you can compute the answer.
Why: Labeling the unknown makes the invisible visible. Without $x$ , you're jugling vague ideas. With $x$ , you have a handle.

Step 2: Express Other Quantities in Terms of $x$

What: Write every other relevant quantity using $x$ .
How: Use the relationships given in the problem. If John has 5 more aples than Mary and Mary has $x$ apples, then John has $x+5$ .
Why: This step unifies all information under one symbol, letting you write one equation instead of many disconnected facts.

Step 3: Set Up the Equation

What: Translate the main relationship into an equation.
How: Find the sentence that relates quantities with "is," "equals," "total," or "same as." That's your equation.
Why: The equation is the mathematical form of the constraint. It's where the problem's logic lives.

Step 4: Solve the Equation

What: Use algebraic techniques to isolate $x$ .
How: Apply inverse operations: if $x$ is added to something, subtract it from both sides; if $x$ is multiplied, divide both sides.
Why: Solving reveals the numerical value that satisfies all relationships simultaneously.

Step 5: Verify and Interpret

What: Check your answer makes sense in the original problem.
How: Substitute your value back into the problem's conditions. Does it satisfy all statements? Are units correct? Is the number reasonable?
Why: This catches arithmetic errors and nonsensical answers (like negative ages or fractional people).

Figure — Word problems using linear equations

Common Problem Types & Translation Patterns

Type 1: Age Problems

Pattern: Relate ages at different times using $+$ or $-$ for time shifts.

Example Structure: "In 3 years, Sarah will be twice as old as Tom is now."

Let Sarah's current age = $x$
Sarah in 3 years = $x + 3$
Equation: $x + 3 = 2(\text{Tom's current age})$

Type 2: Number Problems

Pattern: Consecutive integers, digit problems, or relationships between numbers.

Key Translations:

Consecutive integers: $x$ , $x+1$ , $x+2$ , ...
Consecutive even/odd: $x$ , $x+2$ , $x+4$ , ...
"Sum of" → use $+$ between terms
"Product of" → use $\times$ between terms

Type 3: Money/Mixture Problems

Pattern: Different quantities at different rates, total value = sum of individual values.

Formula Pattern: If you have quantity $q_1$ at rate $r_1$ and quantity $q_2$ at rate $r_2$ : $\text{Total Value} = q_1 \cdot r_1 + q_2 \cdot r_2$

Why: Value is extensive (adds up). Each component contributes independently.

Type 4: Distance/RateTime Problems

Pattern: Use $\text{Distance} = \text{Rate} \times \text{Time}$

Key Insight: If multiple objects travel, set their distances/times/rates equal based on the constraint (meeting point, same time, etc.).

Type 5: Work Problems

Pattern: If someone completes a job in $t$ hours, their rate is $\frac{1}{t}$ jobs per hour.

Combined Work Formula: If person A completes job in $a$ hours and person B in $b$ hours: $\text{Combined rate} = \frac{1}{a} + \frac{1}{b} = \frac{1}{t_{\text{together}}}$

Derivation: Rate measures "fraction of job done per unit time." If A does $\frac{1}{a}$ per hour and B does $\frac{1}{b}$ per hour, together they do $\frac{1}{a} + \frac{1}{b}$ per hour. To find time $t$ to complete 1 job: $t \cdot (\frac{1}{a} + \frac{1}{b}) = 1$ , so $t = \frac{1}{\frac{1}{a} + \frac{1}{b}} = \frac{ab}{a+b}$ .

Worked example Worked Example 1: Age Problem

Problem: A father is 3 times as old as his son. In 12 years, he will be twice as old as his son. How old is the son now?

Step 1 — Identify Unknown:
Let son's current age = $x$ years.
Why $x$ for son? The problem asks for son's age, so that's our target.

Step 2 — Express Other Quantities:
Father's current age = $3x$ years (father is 3 times as old)
Son's age in 12 years = $x + 12$ years
Father's age in 12 years = $3x + 12$ years
Why these expressions? "In 12 years" means add 12 to current age. "3 times as old" means multiply by 3.

Step 3 — Set Up Equation:
The relationship "in 12 years, father will be twice son's age" gives: $3x + 12 = 2(x + 12)$ Why this equation? Left side = father's future age, right side = twice son's future age. "Will be" means equals.

Step 4 — Solve: $3x + 12 = 2x + 24$ Why subtract $2x$ from both sides? To collect $x$ terms on one side. $3x - 2x = 24 - 12$ $x = 12$ Why subtract 12 from both sides? To isolate $x$ . Step 5 — Verify:
Son now: 12 years
Father now: $3(12) = 36$ years (✓ three times)
In 12 years: Son = 24, Father = 48(✓ 48 = 2×24)

Answer: Son is currently 12 years old.

Worked example Worked Example 2: Money Problem

Problem: A pigy bank contains ₹155 in₹5 and ₹10 notes. There are 23 notes in total. How many notes of each type are there?

Step 1 — Identify Unknown:
Let number of ₹5 notes = $x$
Why? We could choose either note type;₹5 is arbitrary but fine.

Step 2 — Express Other Quantities:
Number of ₹10 notes = $23 - x$ (since total notes = 23)
Why? If $x$ notes are ₹5, the remaining must be ₹10.

Step 3 — Set Up Equation:
Total value equation: $5x + 10(23 - x) = 155$ Why this equation? (value per note) × (quantity) for each type, sum to total value.
Why not use "number of notes"? We already used that to express quantities. Value is our independent constraint.

Step 4 — Solve: $5x + 230 - 10x = 155$ Why expand? Distribute 10 into the parentheses. $-5x + 230 = 155$ $-5x = 155 - 230 = -75$ $x = \frac{-75}{-5} = 15$ Why divide by -5? Isolate $x$ .

Number of ₹10 notes = $23 - 15 = 8$ Step 5 — Verify:
Total notes: $15 + 8 = 23$ ✓
Total value: $5(15) + 10(8) = 75 + 80 = 155$ ✓

Answer: 15 notes of ₹5 and 8 notes of ₹10.

Worked example Worked Example 3: Work Problem

Problem: Pipe A can fill a tank in 6 hours. Pipe B can fill it in 9 hours. If both pipes are opened together, how long will it take to fill the tank? Step 1 — Identify Unknown:
Let time to fill together = $t$ hours
Why? That's what the problem asks for directly.

Step 2 — Understand Rates:
Pipe A's rate = $\frac{1}{6}$ tank per hour (fills $\frac{1}{6}$ of tank each hour)
Pipe B's rate = $\frac{1}{9}$ tank per hour
Why rates? Work problems need rates to add contributions.

Step 3 — Set Up Equation:
Combined rate = $\frac{1}{6} + \frac{1}{9}$ tank per hour
To fill 1 complete tank: $t \left(\frac{1}{6} + \frac{1}{9}\right) = 1$ Why multiply rate by time? Rate × Time = Work done. We want work done =1 tank.

Step 4 — Solve: $\frac{1}{6} + \frac{1}{9} = \frac{3}{18} + \frac{2}{18} = \frac{5}{18}$ Why find common denominator? To add fractions. LCM(6,9) = 18. $t \cdot \frac{5}{18} = 1$ $t = \frac{18}{5} = 3.6 \text{ hours} = 3 \text{ hours } 36 \text{ minutes}$ Why invert and multiply? Dividing by a fraction = multiplying by its reciprocal.

Step 5 — Verify:
In 3.6 hours, Pipe A fills: $3.6 \times \frac{1}{6} = 0.6$ tank
In 3.6 hours, Pipe B fills: $3.6 \times \frac{1}{9} = 0.4$ tank
Total: $0.6 + 0.4 = 1.0$ tank ✓

Answer: 3 hours 36 minutes (or $\frac{18}{5}$ hours).

Worked example Worked Example 4: Consecutive Integer Problem

Problem: The sum of three consecutive odd numbers is 63. Find the numbers.

Step 1 — Identify Unknown:
Let first odd number = $x$
Why odd, not even? The problem specifies odd. We'll verify our answer is odd.

Step 2 — Express Other Quantities:
Second odd number = $x + 2$ (odd numbers differ by 2)
Third odd number = $x + 4$
Why add 2, not 1? Consecutive odd:1,3,5,... or 7,9,11,... The gap is 2.

Step 3 — Set Up Equation:
Sum =63: $x + (x+2) + (x+4) = 63$ Why this equation? Direct translation of "sum of three numbers is 63."

Step 4 — Solve: $3x + 6 = 63$ Why combine? $x + x = 3x$ and $2+ 4 = 6$ . $3x = 57$ $x = 19$ The three numbers: 19, 21, 23 Step 5 — Verify:
Are they odd? 19, 21, 23 are all odd ✓
Are they consecutive? 21-19=2, 23-21=2 ✓
Sum: $19+21+23 = 63$ ✓

Answer: The numbers are 19, 21, and 23.

Formula Master Translation Table

English Phrase	Mathematical Symbol	Example
is, equals, is the same as	$=$	"Age is 15" → $x = 15$
more than, increased by, sum	$+$	"5 more than $x$ " → $x+5$
less than, decreased by, difference	$-$	"5 less than $x$ " → $x-5$
times, product of (multiplication)	$\times$ or $\cdot$	"3 times $x$ " → $3x$
per, ratio, quotient	$\div$ or $/$	"km per hour" → $\frac{\text{km}}{\text{hour}}$
total, combined, altogether	$+$ (sum all)	"Total of $x$ and $y$ " → $x+y$
consecutive integers	$x, x+1, x+2, \ldots$	Three consecutive: $x$ , $x+1$ , $x+2$
consecutive odd/even	$x, x+2, x+4, \ldots$	Consecutive odd: $x$ , $x+2$ , $x+4$

Recall Explain to a 12-Year-Old (Feynman Technique)

Imagine you have a mystery box with some marbles inside. You don't know how many, but someone gives you clues: "If you add 5 more marbles, you'll have 20 total."

Instead of guessing, you create a nickname for the mystery number: call it $x$ marbles. Now translate the clue: $x + 5 = 20$ . To find $x$ , you "undo" the adding of 5 by subtracting 5 from both sides: $x = 15$ . You had 15 marbles!

Word problems are the same. The problem gives you clues in English. You:

Pick a nickname ( $x$ ) for the unknown number
Translate the clues into a math equation
Solve the equation like a puzzle (do the opposite operations)
Check your answer fits all the clues

It's like being a detective: the equation is already hidden in the words, you just reveal it. Every "is" becomes an equals sign. Every "more than" becomes a plus. Once you see the pattern, word problems are just translation practice.

Connections

Linear Equations in One Variable — the foundational equation form
Algebraic Expressions — building expressions for the "express in terms of $x$ " step
Inverse Operations — the solving techniques (add/subtract, multiply/divide)
Ratio and Proportion — many word problems involve proportional relationships
Applications of Algebra — broader context of modeling real situations
Simultaneous Linear Equations — when one equation isn't enough

#flashcards/maths

What are the five steps to solve word problems using linear equations? :: 1) Identify the unknown (choose $x$ ), 2) Express other quantities in terms of $x$ , 3) Set up the equation, 4) Solve the equation, 5) Verify and interpret the answer.

How do you represent "5 less than a number

x

x - 5

(NOT

5-x

). "Less than" means subtract FROM the number.

What is the formula for consecutive odd integers starting from

x

x

x+2

x+4

, ... (they differ by 2, not 1).

In work problems, if a job takes 8 hours to complete, what is the rate?

\frac{1}{8}

jobs per hour (rate is the reciprocal of time).

If Pipe A fills a tank in

a

hours and Pipe B fills it in

b

hours, what is the time to fill together?

t = \frac{ab}{a+b}

hours. Derive: combined rate

= \frac{1}{a} + \frac{1}{b} = \frac{a+b}{ab}

, so time

= \frac{1}{\text{rate}} = \frac{ab}{a+b}

How do you translate "is, equals, is the same as" into math?

The equals sign

=

. It shows two expressions have the same value.

In an age problem, "in 5 years" means what operation on current age

x

Add 5:

x + 5

. Future age = current age + years passed.

What is the key mistake when defining variables in word problems?

Defining

x

then contradicting yourself by re-expressing the same quantity differently. Stay consistent with your original definition.

In a money problem with two types of items totaling

n

items, if

x

are Type A, how many are Type B?

n - x

. The sum must equal total, so Type B = Total - Type A.

What does "three times as old as" translate to mathematically?

Multiplication by 3. If son's age is

x

, "three times as old" means

3x

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Word problems ko samajhna ek translation game hai, bhai. Problem mein har sentence ke andar ek mathematical relationship chhipi hoti hai—tumhara kaam hai us hidden equation ko dhoondhna. Jab tum "is" padho, socho "equals" ( $=$ ). "More than" dekho toh "plus" samjho ( $+$ ). "Times as much" matlab "multiply" ( $\times$ ). Yeh patterns ek baar pakad liye, toh phir koi bhi word problem mushkil nahi.

Sabse pehla step: unknown ko pehchano aur use $x$ se label karo. Jaise agar "son ki age kitni hai?" poocha, toh let $x$ = son ki current age. Phir baki sab chezein is $x$ ke through express karo—father ki age, future ages, sab kuch. Isse ek unified view milta hai, aur tumhe sirf ek hi equation banana padta hai.

Iske baad main relationship ko equation mein convert karo. Problem mein jahan "total," "equals," ya "is the same as" likha ho, wahi tumhari equation ban jayegi. Phir algebraic techniques use karke $x$ ko isolate karo—add, subtract, multiply, divide jo bhi inverse operation chahiye. Last mein apne answer ko problem mein wapas daal ke verify karo.Agar sab conditions satisfy ho rahi hain, toh tumhara jawab sahi hai. Y

The Five-Step Method

Step 1: Identify the Unknown

Step 2: Express Other Quantities in Terms of $x$

Step 3: Set Up the Equation

Step 4: Solve the Equation

Step 5: Verify and Interpret

Common Problem Types & Translation Patterns

Type 1: Age Problems

Type 2: Number Problems

Type 3: Money/Mixture Problems

Type 4: Distance/RateTime Problems

Type 5: Work Problems

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Algebra — Introduction & Intermediate

Connections