Solving Equations Mentally
How to read a simple equation as a question and answer it in your head, with a substitution check to confirm the result.
--
Simple equations do not always need written working. If you can hear the question an equation is asking, the answer often arrives on its own. This is a habit worth building early: understanding beats memorising a procedure.
Concept
An equation is a question
Every equation hides a plain-language question. Once you say that question out loud, the arithmetic you already know supplies the answer.
Concept
Addition and subtraction equations
When the variable has something added to it, ask what must be added. When something is taken away, ask what it started from.
Notice that the second one grows rather than shrinks. The question form protects you from reaching for subtraction out of habit.
Concept
Multiplication and division equations
The same approach works one level up, using times tables instead of number bonds.
Concept
The four questions
Each type of simple equation has its own phrasing:
Addition — what number was added?
Subtraction — what number was it taken from?
Multiplication — what number was multiplied?
Division — what number was divided?
Identify the operation, ask the matching question, and answer it.
Concept
Always check by substituting
Mental work is quick, which makes a quick check worthwhile. Put the answer back into the original equation and confirm both sides agree.
The check takes a second and catches the slips that speed introduces.
Example
The price of a pen
Five identical pens cost 25 riyals in total. What does one pen cost?
Let 
be the price of one pen, so five pens cost 
.
The equation is 
.
The question: what times 5 gives 25?
Check: 
.
⟹ one pen costs 5 riyals
You could also have divided 25 by 5 directly. Both routes agree — which is exactly the point: the equation is a written form of reasoning you can already do.
Example
Four equations in a row
Solve ,
,
and
.
What adds to 9 to give 13? 
.
What becomes 12 after losing 7? 
.
What times 5 gives 25? 
.
What divided by 3 gives 6? 
.
⟹ 4, 19, 5 and 18
Note
Mistakes to avoid
Subtracting when the equation calls for addition: 
gives 19, not 5.
Skipping the substitution check because the equation looked easy.
Trying to do it mentally when the numbers are large — write those out.
Reciting steps without knowing what question is being answered.
Forgetting to state what the variable represents in a word problem.
Summary
- An equation is a question written in symbols.
- Say the question aloud and the answer usually follows.
- Each operation has its own phrasing: what was added, subtracted, multiplied or divided.
- Substituting the answer back is a fast and reliable check.
- Mental methods suit small numbers; write out the larger ones.