Translating Verbal Relationships into Algebraic Expressions and Equations

Hello. In the previous lesson, you practised reading symbolic expressions accurately: grouping symbols, fraction bars, operation order, and signs. That discipline now becomes the tool for a different task: turning words into symbols without changing the meaning.

This opening module builds the algebra base needed before IPMAT-style equations, arithmetic word problems, and quantitative reasoning become fast rather than fragile. In this lesson, you will learn to distinguish expressions from equations, assign a variable, preserve the order implied by language, and translate verbal relationships into one-variable equations.

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Expressions state quantities; equations state relationships

An algebraic expression represents a quantity but does not claim that it equals anything.

$$3x+5$$

can mean “five more than three times a number.” It is an expression.

An equation states that two quantities have the same value.

$$3x+5=26$$

can mean “five more than three times a number is $26$.” The word is supplies the equal sign.

This distinction is fundamental:

Words given	Mathematical form	Type
“Five more than three times a number”	$3x+5$	Expression
“Five more than three times a number is $26$”	$3x+5=26$	Equation
“The cost of $x$ notebooks at ₹18 each”	$18x$	Expression
“The cost of $x$ notebooks at ₹18 each is ₹180”	$18x=180$	Equation

Think of an equation as a claim that must stay balanced. The two sides may look different, but they represent equal values.

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Start with a variable and a “target”

In a word problem, do not start by hunting for operation keywords alone. First decide:

1. What quantity is unknown?
2. What symbol will represent it?
3. What quantity is the statement describing or asking about?

Write a tiny legend when the setting has several quantities:

$$x=\text{number of mock tests taken}$$

Suppose a test series charges a one-time registration fee of ₹150 and ₹80 per mock test. If $x$ is the number of mocks, then:

$$150+80x$$

is the total cost.

If the learner spent ₹950, the complete relationship is:

$$150+80x=950$$

Notice the structure:

• ₹150 is fixed, so it is a constant.
• $80x$ means ₹80 for each of $x$ tests.
• The word “spent” does not automatically create an equation; the stated total of ₹950 does.

A useful test-day habit is to write the target in words before symbols:

Total cost = registration fee + cost of all mock tests

Only then write:

$$950=150+80x$$

or, equivalently,

$$150+80x=950$$

Both equations say the same thing.

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Translate the operation, but preserve the order

Certain words strongly suggest operations.

Verbal phrase	Typical algebraic form
sum, total, increased by, more than	addition
difference, decreased by, less than, fewer than	subtraction
product, times, twice, of, each	multiplication
quotient, ratio, per, divided by	division
is, equals, gives, results in	equal sign

The table helps, but word order is more important than keyword memorisation. Addition and multiplication may be reordered without changing value; subtraction and division generally may not.

The “less than” trap

Compare these phrases carefully:

$$\text{6 less than }x=x-6$$ $$\text{6 subtracted from }x=x-6$$ $$\text{the difference between }x\text{ and }6=x-6$$

The phrase after than or from is the starting quantity. So “6 less than five times a number” means:

$$5x-6$$

It does not mean:

$$6-5x$$

A quick numerical check exposes the error. If $x=4$, then “6 less than five times a number” means 6 less than $5(4)=20$, which is $14$. Only $5x-6$ gives $14$.

The same reversal occurs with division:

$$\text{the quotient of }x\text{ and }4=\frac{x}{4}$$ $$\text{4 divided into }x=\frac{x}{4}$$

The quantity being divided goes in the numerator; the divisor goes in the denominator.

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Group complete quantities before operating on them

Previous work on brackets now has a practical purpose: brackets preserve the meaning of a verbal group.

Compare:

Three times the sum of a number and $4$

First form the sum:

$$x+4$$

Then multiply that complete sum by $3$:

$$3(x+4)$$

This is different from:

The sum of three times a number and $4$

$$3x+4$$

The phrases sound similar but describe different quantities. Test with $x=2$:

$$3(x+4)=3(6)=18$$ $$3x+4=6+4=10$$

The brackets are therefore not decoration. They encode which words belong together.

Here are three high-value translation patterns:

Verbal relationship	Build it in layers	Expression
Five more than three times a number	three times $x$, then add 5	$3x+5$
The difference between five times a number and 6	first $5x$, then subtract 6	$5x-6$
The quotient of 18 and the difference between a number and 2	denominator is $x-2$	$\frac{18}{x-2}$

For the final example, parentheses are essential if you use a slash:

$$\frac{18}{x-2}$$

Writing $18/x-2$ would normally mean a different expression:

$$\frac{18}{x}-2$$

On entrance exams, this difference is often exactly where a distractor is designed to catch rushed reading.

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From a verbal relationship to a one-variable equation

A one-variable equation has one unknown quantity, commonly $x$, and enough information to relate it to a known value.

Use this translation routine.

1. Choose the unknown. State what $x$ represents.
2. Locate the core quantity. Build the phrase around the unknown first.
3. Package inner groups. A sum, difference, or quotient may be a complete unit.
4. Add outer operations. Apply “twice,” “five less than,” and similar wording to the correct unit.
5. Find the equality statement. Words such as “is,” “equals,” or “totals” connect the two sides.
6. Read the equation back in English. If it no longer matches the sentence, repair it before solving.

Example 1: a direct relationship

Five less than three times a number is $22$.

Let the number be $x$.

• Three times the number: $3x$
• Five less than that: $3x-5$
• Is $22$: attach the equality

$$3x-5=22$$

Do not write $5-3x=22$. That would mean “five minus three times a number.”

Example 2: consecutive numbers

The sum of a number and the next integer is $41$.

Let the first number be $x$. The next integer is not a new unrelated unknown; it is:

$$x+1$$

So the equation is:

$$x+(x+1)=41$$

This is a major modelling idea: whenever the words say its next, one more than it, or three less than it, express the related quantity using the same variable.

Example 3: an outer operation applied to a group

Half the difference between a number and $6$ is $9$.

Let the number be $x$.

The difference between the number and 6 is:

$$x-6$$

Half of that entire difference is:

$$\frac{x-6}{2}$$

The equation is:

$$\frac{x-6}{2}=9$$

A common incorrect translation is:

$$x-\frac{6}{2}=9$$

That says “the number minus half of 6,” not “half of the difference.”

Example 4: a relationship involving “each”

A learner pays ₹40 per sectional test and a fixed ₹120 platform fee. The total bill is ₹520.

Let $x$ be the number of sectional tests.

The test cost is:

$$40x$$

Including the fixed fee gives:

$$40x+120$$

Since the total bill is ₹520:

$$40x+120=520$$

The phrase per test signals multiplication, while the fixed fee is added only once.

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Translation checks that prevent IPMAT-style errors

Before solving any word-problem equation, spend ten seconds on these checks:

• Equal sign check: Is there a stated equality, total, or known result? If not, you may only need an expression.
• Reference check: What does “than” or “from” refer to? For subtraction, write the base quantity first.
• Grouping check: Does a multiplier apply to a single term or to an entire sum or difference?
• Same-variable check: If two quantities are connected by “next,” “twice,” “less than,” or “more than,” have you expressed the second in terms of the first?
• Unit check: Are you adding like units? For example, total rupees can equal fixed rupees plus rupees per test times number of tests.
• Read-back check: Say your equation in words. It should reproduce the original statement precisely.

For example, read

$$3(x+4)-5=22$$

as:

Five less than three times the sum of a number and four is twenty-two.

If your read-back instead says “three times a number plus four,” you have lost the bracket structure.

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Key takeaways

Words are not converted symbol by symbol; they are converted by identifying the structure of the relationship.

• An expression represents a quantity; an equation asserts equality between two quantities.
• Define the unknown clearly before modelling.
• Translate “less than” and “subtracted from” with special care: subtraction order matters.
• Use parentheses whenever a word such as “sum,” “difference,” or “quotient” creates a group that another operation acts upon.
• Build the relationship in layers, then read the equation back to verify it.

The next lesson develops another compact symbolic language: exponent laws, including zero, negative, and fractional exponents.