Algebra · Variables and Equations · Lesson 1

Writing Equations from Word Problems (and Back Again)

Many everyday puzzles — sharing money among friends, working out someone's age, finding the price of one item — can be written as a short mathematical sentence called an equation. In this lesson you will learn to turn words into equations, use a flag diagram to undo the steps of a puzzle, and turn an equation back into a story.

NaCCA B7.2.3.1.1 ⏱ about 45 min

By the end you will be able to

✔Translate key words such as 'more than', 'less than', 'twice' and 'the result is' into algebraic symbols.
✔Write a linear equation in one variable for a given word problem.
✔Use a flag diagram and inverse operations to find the unknown number.
✔Write a word problem that matches a given linear equation.

Key words

Variable: A letter, such as x or n, that stands for a number we do not yet know.
Expression: Numbers and letters joined by operations, with no equals sign, e.g. $2 x - 7$ .
Equation: A mathematical sentence saying two things are equal, e.g. $2 x - 7 = 41$ .
Linear equation: An equation in which the variable has power 1 only (no $x^{2}$ ), such as $3 x + 5 = 32$ .
Inverse operation: The operation that undoes another: + undoes −, × undoes ÷, and the other way round.
Flag diagram: A chain of boxes and arrows showing the operations done to a number, which we can run backwards to find the number.

Curriculum reference

Content standard: B7.2.3.1 Demonstrate an understanding of linear equations of the form x + a = b (where a and b are integers) by modelling problems as a linear equation and solving the problems concretely, pictorially, and symbolically.

Indicator: Translate word problems to linear equations in one variable and vice versa.

Starter

Ama's number trick

At break time in a school in Cape Coast, Ama says to Kofi: "Think of a number, double it and subtract 7. Tell me your answer." Kofi says "41." In two seconds Ama shouts, "Your number was 24!" Is Ama a magician? No — she simply wrote the puzzle as an equation in her head and worked backwards. By the end of this lesson you can do the same trick.

1. From words to symbols

In algebra we use a letter to stand for a number we do not know yet. The letter is called a variable or an unknown. Any letter will do — $x$ , $n$ , $a$ — but always say clearly what it stands for, for example *"let $x$ be the number Kofi thought of"*.

Word problems use certain words again and again. Each one tells you which operation to use. Learn this table well; it is the dictionary for translating English into algebra.

Key words and the symbols they become
Words	Operation	In symbols (number = x)
5 more than a number / a number increased by 5	add	$x + 5$
5 less than a number / a number decreased by 5	subtract	$x - 5$
a number subtracted from 20	subtract	$20 - x$
twice / double a number	multiply by 2	$2 x$
three times a number	multiply by 3	$3 x$
half of a number / a number divided by 2	divide by 2	$\frac{x}{2}$
two-thirds of a number	multiply by $\frac{2}{3}$	$\frac{2}{3} x$
the result is / gives / equals / is	equals sign	$=$

Notice the difference between an expression and an equation. "Twice a number, decreased by 7" is the expression $2 x - 7$ : it has no equals sign and we cannot solve it. "Twice a number, decreased by 7, is 41" is the equation $2 x - 7 = 41$ : now we have a fact we can use to find $x$ .

2. Flag diagrams: doing and undoing

A flag diagram shows what happens to the unknown number step by step, like a trotro moving from stop to stop. To find the starting number, we travel back along the same road, doing the inverse (opposite) of each operation, in reverse order.

▶ Watch the top row build the equation 2x − 7 = 41. Then follow the bottom row backwards: every operation is undone by its inverse, in reverse order.

1Let the unknown number be $x$ and draw a box for it.
2Draw an arrow for each operation in the order the problem gives them, and write the result at the end.
3Read off the equation, e.g. $2 x - 7 = 41$ .
4Start from the result and go backwards, using the inverse of each operation.
5Check your answer by putting it back into the original words.

3. Writing equations for real-life problems

Real problems are not always "think of a number". They talk about ages, money, oranges or marks. The secret is the same every time: choose one quantity to be $x$ , write **every other quantity in terms of $x$ , then find the sentence that tells you something is equal**.

Ages: The sum of the ages of two friends is 25, and the older one is 4 times as old as the younger one. Let the younger one's age be $x$ years. Then the older one is $4 x$ years. Their ages add up to 25, so $x + 4 x = 25$ , which is $5 x = 25$ .

Sharing: Adaako and Afrakoma shared 40 oranges. Afrakoma had 6 more than Adaako. Let Adaako's share be $x$ . Afrakoma's share is $x + 6$ . Together they have 40, so $x + (x + 6) = 40$ , which simplifies to $2 x + 6 = 40$ .

Choose x, write the rest in terms of x, then find what is equal
Problem	Let x be…	Other quantity	Equation
Ages add to 25; older is 4 times younger	younger's age	older = $4 x$	$x + 4 x = 25$
40 oranges; Afrakoma has 6 more than Adaako	Adaako's share	Afrakoma = $x + 6$	$x + (x + 6) = 40$
3 pens and a GH₵2 eraser cost GH₵20	price of 1 pen (GH₵)	3 pens = $3 x$	$3 x + 2 = 20$

4. From an equation back to a story

The indicator also asks you to go the other way: given an equation, write a word problem for it. Read the equation from left to right and say in words what happens to the unknown number. There are many correct stories — any story whose sentence gives exactly the same equation is right.

Equation	A matching word problem
$x + x = 15$	The sum of two equal numbers is 15.
$2 x - 4 = 12$	When 4 is taken away from two times a certain number, the result is 12.
$\frac{2}{3} x = 4$	Two-thirds of a certain number is 4.
$3 x + 2 = 20$	Kwame bought 3 pens of the same price and an eraser for GH₵2. He paid GH₵20 altogether.

Worked examples

Try each one yourself first, then open the solution and compare every step.

Example 1

Translating a simple sentence

When 9 is added to a number, the result is 23. Write an equation and find the number.

Show the solution

Step 1Let the number be $x$ .
Step 2"9 is added to a number" gives $x + 9$ ; "the result is 23" gives $= 23$ .
Step 3Equation: $x + 9 = 23$ .
Step 4Undo + 9 by subtracting 9: $x = 23 - 9 = 14$ .
Step 5Check: 14 + 9 = 23 ✓

Answer: $x + 9 = 23$ ; the number is 14

Example 2

Ama's number trick with a flag diagram

Think of a number, double it and subtract 7. The result is 41. What was the original number?

Show the solution

Step 1Flag diagram: $x \times 2 2 x - 7 41$ , so the equation is $2 x - 7 = 41$ .
Step 2Go backwards. Undo − 7 by adding 7: 41 + 7 = 48, so $2 x = 48$ .
Step 3Undo × 2 by dividing by 2: 48 ÷ 2 = 24, so $x = 24$ .
Step 4Check: 2 × 24 − 7 = 48 − 7 = 41 ✓

Answer: 24

Example 3

An age problem

The sum of the ages of two friends is 25, and the older one is 4 times as old as the younger one. Write an equation and find both ages.

Show the solution

Step 1Let the younger friend's age be $x$ years. The older friend is $4 x$ years.
Step 2Their ages add to 25: $x + 4 x = 25$ , so $5 x = 25$ .
Step 3Divide by 5: $x = 5$ .
Step 4Older friend: 4 × 5 = 20 years. Check: 5 + 20 = 25 ✓ and 20 = 4 × 5 ✓

Answer: $x + 4 x = 25$ ; the friends are 5 and 20 years old

Example 4

Sharing oranges

Adaako and Afrakoma shared 40 oranges. Afrakoma had 6 more than Adaako. Write an equation and find each share.

Show the solution

Step 1Let Adaako's share be $x$ . Afrakoma's share is $x + 6$ .
Step 2Together: $x + (x + 6) = 40$ , so $2 x + 6 = 40$ .
Step 3Subtract 6: $2 x = 34$ . Divide by 2: $x = 17$ .
Step 4Afrakoma: 17 + 6 = 23. Check: 17 + 23 = 40 ✓

Answer: Adaako 17 oranges, Afrakoma 23 oranges

Example 5

Writing a story for an equation

Write a word problem for

3 x + 2 = 20

and solve it.

Show the solution

Step 1 $3 x$ means three equal amounts; + 2 is an extra 2; = 20 is the total.
Step 2Story: Kwame bought 3 pens of the same price and an eraser for GH₵2. He paid GH₵20. Find the price of one pen.
Step 3Solve: subtract 2 to get $3 x = 18$ , then divide by 3: $x = 6$ .
Step 4Check: 3 × 6 + 2 = 18 + 2 = 20 ✓

Answer: One possible story is above; each pen costs GH₵6

Summary

A variable is a letter that stands for an unknown number; always say what it stands for.
Key words become symbols: more than → +, less than → −, twice → × 2, the result is → =.
"5 less than x" is $x - 5$ , not $5 - x$ .
A flag diagram shows the operations in order; undo them with inverse operations in reverse order.
For real problems, let x be one quantity (usually the smaller), write the others in terms of x, then find what is equal.
Many different stories can match one equation; check that your story gives exactly that equation.

Ama's number trick

1. From words to symbols

2. Flag diagrams: doing and undoing

3. Writing equations for real-life problems

4. From an equation back to a story

Worked examples

Translating a simple sentence

Ama's number trick with a flag diagram

An age problem

Sharing oranges

Writing a story for an equation

Summary

Practice: 15 questions