Expressions Using Letter-Numbers

Why This Matters

Think about your age and your friend’s age.

Suppose your friend Aftab is some age. His sister Shabnam is always 3 years older than him. If Aftab is 10, Shabnam is 13. If Aftab is 18, Shabnam is 21.

Now here is the question. Can you write one short rule that works for every age Aftab could be? You don’t want to make a fresh sum each time. You want one rule that handles all cases at once.

That is exactly what this chapter teaches. We will learn a clever trick: use a letter to stand for a number. With this trick, a whole pattern fits into a tiny piece like a + 3.

This is your first step into algebra. It might feel new and strange. But it is just shorthand. It is a faster, smarter way to say things you already know. Once you get it, you will use it in maths for the rest of your life. Let’s go slow and make it click.

The Big Idea

When we don’t know a number yet, or when we want one rule to work for many numbers at once, we use a letter to stand for the number. A short maths phrase built with such a letter — like a + 3 or 2n — is called an expression. The letter is just a box waiting for a number. Put a number in, and the expression turns into an ordinary answer. One small expression can capture a pattern that goes on forever.

Let’s Break It Down

A letter that stands for a number (a variable) — and WHY this is useful

Here is the new idea. We are going to let a letter stand for a number.

For example, we can write the letter n and say: “n is some number.” We don’t fix what n is right now. It could be 1. It could be 2. It could be 100. The letter n is like an empty box. We can drop any number we like into that box.

A letter used this way — to stand for a number — is called a letter-number. (You may also hear the word variable. It means the same thing. It is called a variable because its value can vary, that is, change.)

The picture below shows the idea. The letter n is one box, and we can put many different numbers inside it.

Tap to zoom

So why is this useful? Why not just use plain numbers?

Here is the reason. Go back to Aftab and Shabnam. Shabnam is always 3 years older. Without letters, you would have to write a separate sum for each age:

• Aftab is 4 → Shabnam is 4 + 3
• Aftab is 10 → Shabnam is 10 + 3
• Aftab is 23 → Shabnam is 23 + 3

That is a never-ending list. But look at the right-hand side. Every line is “Aftab’s age, then + 3.” Only the first number changes. The ”+ 3” never changes.

So let’s use the letter a for Aftab’s age. Then Shabnam’s age, for any value of a, is just:

a + 3

That one short thing replaces the whole endless list. That is the power of a letter-number. It captures the whole pattern in one go. Figure 4.2 below shows the same table, ending in the letter row.

Tap to zoom

An expression that contains a letter-number, like a + 3, is called an algebraic expression. “Algebraic” just means “uses letters for numbers.” Don’t let the long word scare you — you already understand the idea.

We can pick any letter we like. We used a for Aftab’s age. We could have used x, or k, or t. The letter is just a name for the unknown number. People often pick a letter that reminds them of the thing — a for age, l for length, n for a number of things.

Writing an expression from a situation or a pattern

Once you can use a letter for a number, you can turn a real-life situation or a picture pattern into a short expression. The trick is always the same:

1. Describe the rule in plain words first.
2. Then choose a letter for the unknown number.
3. Then write the rule using that letter.

Let’s try it with a matchstick pattern. Parthiv makes L shapes out of matchsticks. Each L uses 2 matchsticks. He places them in a row.

• 1 L needs 2 matchsticks (that is 2 × 1).
• 2 Ls need 4 matchsticks (that is 2 × 2).
• 3 Ls need 6 matchsticks (that is 2 × 3).

In words: the number of matchsticks is 2 times the number of Ls. Figure 4.3 below shows this growing pattern.

Tap to zoom

Now choose a letter. Let n stand for the number of Ls. Then:

Number of matchsticks = 2 × n

This little expression, 2 × n, tells us the matchsticks needed for any number of Ls. Need 45 Ls? Just put 45 in place of n: 2 × 45 = 90 sticks. You didn’t have to count.

Here is another situation. Krithika buys coconuts and jaggery. A coconut costs ₹35. One kilogram of jaggery costs ₹60. If she buys some coconuts and some jaggery, what is the total cost?

Describe it in words: cost of coconuts plus cost of jaggery. Cost of coconuts is (number of coconuts × 35). Cost of jaggery is (number of kg × 60).

Now choose letters. Let c be the number of coconuts and j be the number of kg of jaggery. The total cost is:

c × 35 + j × 60

Put in any values and you get the bill. For 7 coconuts and 4 kg jaggery: 7 × 35 + 4 × 60 = 245 + 240 = ₹485. A rule like this, which describes a real relationship, is also called a formula.

What 3n, n + 2 and 2n − 1 mean

You will see expressions like 3n, n + 2, and 2n − 1 all the time. Let’s read each one carefully, slowly, so there is no confusion.

First, a key shortcut. When we multiply a number and a letter, we drop the multiplication sign. We write the number first, then the letter, stuck together.

So 3 × n is written simply as 3n. The number and the letter sit side by side. This does not mean “3 and then n” like a code. It means 3 times n — nothing else.

Now let’s read three common expressions:

• 3n means 3 × n → “three times the number n.” If n is 5, then 3n = 3 × 5 = 15.
• n + 2 means “the number n, with 2 added on” → 2 more than n. If n is 5, then n + 2 = 7.
• 2n − 1 means (2 × n) − 1 → “double the number, then take away 1.” If n is 5, then 2n − 1 = 10 − 1 = 9.

Notice the order of operations in 2n − 1. We do the multiply first (2 × n), then subtract 1. We do not subtract first.

A nice way to picture an expression is as a little machine. You feed a number in. The machine does its job. An answer comes out. Figure 4.4 below shows the machine for 2n − 1.

Tap to zoom

Finding the value of an expression for a given number

An expression is not a fixed number on its own. 3n could be 6, or 15, or 300 — it depends on what n is. But the moment you are told what the letter equals, the expression turns into one definite number.

The way to find it is called substitution. That is a big word for a simple action: replace the letter with the given number, then work out the sum.

Look at Figure 4.5 below. We replace the letter m by the number 2, everywhere it appears, and then just do the arithmetic.

Tap to zoom

Let’s do one fully, step by step.

We want the value of 5m + 3 when m = 2. Remember that 5m means 5 × m, and we do the multiply before the add.

Let’s try one more, with a subtraction. Find the value of 2r + 1 when r = 8.

Like terms and simplifying

A term is one piece of an expression. In 18c + 11d, there are two terms: 18c and 11d. Terms are joined by + or − signs.

Now, some terms can be combined to make the expression shorter and tidier. Making an expression shorter like this is called simplifying.

Here is the key rule. Like terms are terms that use the same letter. They can be added together. Unlike terms use different letters, and they cannot be added into one term.

Think about it with fruit. 5 mangoes + 3 mangoes = 8 mangoes — easy, same fruit. But 8 mangoes + 11 apples cannot become “19 somethings.” They are different fruits. You just leave it as “8 mangoes and 11 apples.”

Letters work the same way. Suppose a shop sells pencils. Each pencil costs c rupees. On three days it sells 5, then 3, then 10 pencils. The money from pencils is:

5c + 3c + 10c

All three are groups of c — they are like terms. So we just add the counts: 5 + 3 + 10 = 18. The total is 18c. We turned three terms into one.

But what if the shop also sells erasers at d rupees each, earning 11d in total? Then the full expression is 18c + 11d. Here c and d are different letters — unlike terms. We cannot join them. So 18c + 11d is already as short as it gets. Figure 4.6 below shows both cases side by side.

Tap to zoom

So simplifying just means: gather the like terms and add them up, and leave unlike terms separate.

Let’s simplify a longer one step by step: 7p + 8p + 6p.

Equal (equivalent) expressions

Sometimes two expressions look different but are really the same. They give the same answer for every value of the letter. We call them equal or equivalent expressions.

Here is a clear example. The perimeter of a rectangle is length + breadth + length + breadth. Using l for length and b for breadth, the perimeter is:

l + b + l + b

But l + l = 2l, and b + b = 2b. So this is the same as:

2l + 2b

These two look different. But they are equal. Let’s check by putting in numbers. Take l = 3 and b = 4:

• l + b + l + b = 3 + 4 + 3 + 4 = 14
• 2l + 2b = 2 × 3 + 2 × 4 = 6 + 8 = 14

Same answer. Try any other l and b — they will always match. So l + b + l + b and 2l + 2b are equal expressions. The shorter one, 2l + 2b, is the simplified form.

But be careful — not everything that looks similar is equal. Take 5u and 5 + u. These are not equal. 5u means 5 times u. 5 + u means 5 added to u. Check with u = 2:

• 5u = 5 × 2 = 10
• 5 + u = 5 + 2 = 7

Different answers, so they are different expressions. Two expressions are equal only when they match for every value of the letter, not just for one lucky value.

Common Mistakes

These are the slip-ups almost every student makes at first. Spot them now and you will avoid them forever.

Quick Check

Test yourself. Read each question, decide your answer, then tap to see if you’re right.

Practice Problems

Try each problem on your own first. Then reveal the full solution to check your steps.

Easy

Medium

Challenge

Summary

• A letter-number (also called a variable) is a letter used to stand for a number. It is like an empty box: any number can go inside it.
• We use letters because one short rule with a letter can describe a whole pattern at once — for example, Shabnam’s age is a + 3 for any value of Aftab’s age a.
• An algebraic expression is a maths phrase built with letter-numbers, like a + 3, 2n, or 3x + 2y.
• When a number sits next to a letter, the hidden operation is multiply: 3n means 3 × n, not “3 and then n”, and not 3 + n.
• To find the value of an expression, substitute (replace the letter with the given number) and then do the arithmetic — multiplications before additions or subtractions.
• Like terms use the same letter and can be added together (5c + 3c = 8c). Unlike terms use different letters and cannot be joined (18c + 11d stays as it is). Combining like terms is called simplifying.
• Two expressions are equal (equivalent) when they give the same value for every number — like l + b + l + b and 2l + 2b. But 5u and 5 + u are not equal, because they do different things.

What’s Next

You’ve taken your first big step into algebra. Letters standing for numbers will come back again and again, so the comfort you built here will pay off for years.

Next, in Chapter 5 — Parallel and Intersecting Lines, we switch from numbers to shapes. You’ll explore how lines meet, cross, or run side by side forever, and the special angles that appear when they do. It’s a fresh, visual change of pace — see you there.