---
title: "Literal Expressions"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 48
exercises: 11
source: https://one-course.com/books/math/1/en/chapter/48-literal-expressions
---

# Chapter 48 — Literal Expressions

A letter in a computation stands for a number we do not know yet, or for *all* numbers at once: writing $P = 4c$ gives the [perimeter](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-perimeter) of every square in one formula. This chapter teaches the writing conventions, how to substitute values, and how to simplify simple expressions — the start of algebra, continued in [Chapter 57](https://one-course.com/books/math/1/en/chapter/57-literal-calculation-and-equations#ch-g8-equations).

## 48.1 Writing with letters

**Definition 48.1 (Literal expression).**

A *literal expression* is a computation containing one or more letters, each standing for a number. Writing conventions:

- the sign $\times$ is dropped before a letter or a bracket: $3 \times a = 3a$ , $a \times b = ab$ , $2 \times (x + 5) = 2(x + 5)$ ;
- but it stays between two numbers: $3 \times 5$ cannot be written $35$ !
- $a \times a$ is written $a^2$ (“ $a$ squared”), $a \times a \times a = a^3$ ;
- $1a$ is written $a$ , and $0a$ is $0$ .

**Example 48.2.**

Simplify the writing:

$$
5 \times x + 3 \times y = 5x + 3y,
\qquad
a \times 7 = 7a \ \text{(number first)},
\qquad
x \times x \times 4 = 4x^2 .
$$

**Example 48.3 (Formulas are literal expressions).**

For a rectangle of length $L$ and width $w$: [perimeter](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-perimeter) $P = 2(L + w)$, [area](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area) $A = Lw$. One formula summarizes the infinitely many rectangles at once — that is the whole point of letters.

![Two formulas, valid for every rectangle: substitute the values of L and w to get numbers.](https://one-course.com/images/onecourse/chapters/math-1/g7-literal/fig-6b29894197b6.svg)

*Two formulas, valid for every rectangle: substitute the values of $L$ and $w$ to get numbers.*

## 48.2 Substituting

**Method 48.4 (Evaluating an expression).**

To evaluate an expression for given values of its letters:

1. rewrite the expression putting the hidden $\times$ signs back: $3a + a^2$ is $3 \times a + a \times a$ ;
2. replace each letter by its value, keeping brackets around it if needed;
3. compute, respecting the priorities of [Chapter 45](https://one-course.com/books/math/1/en/chapter/45-priorities-of-operations#ch-g7-priorities) .

**Example 48.5.**

Evaluate $E = 3x + x^2$ for $x = 5$:

$$
E = 3 \times 5 + 5 \times 5 = 15 + 25 = 40 .
$$

Evaluate $F = 2(a + 3b)$ for $a = 4$ and $b = 0.5$:

$$
F = 2 \times (4 + 3 \times 0.5) = 2 \times (4 + 1.5) = 2 \times 5.5
= 11 .
$$

## 48.3 Reducing expressions

**Proposition 48.6 (Collecting like terms).**

Terms containing the same letter part can be collected by adding their number parts (this is distributivity, [Theorem 45.6](https://one-course.com/books/math/1/en/chapter/45-priorities-of-operations#thm-g7-priorities-distributivity), read backwards):

$$
5x + 3x = (5 + 3)x = 8x,
\qquad
7a - 2a = 5a .
$$

Terms with different letter parts (like $3x$ and $5y$, or $x$ and $x^2$) cannot be collected.

**Example 48.7.**

Reduce $E = 4x + 7 + 3x + 2$, sorting the terms:

$$
E = (4x + 3x) + (7 + 2) = 7x + 9 .
$$

Careful: $7x + 9$ is *not* $16x$! Test with $x = 1$: $4 + 7 + 3 + 2 = 16$ and $7 \times 1 + 9 = 16$ agree, but $16x$ would also be $16$ — test with $x = 2$: the original gives $8 + 7 + 6 + 2 = 23$, and $7 \times 2 + 9 = 23$, while $16 \times 2 = 32$. The reduction $7x + 9$ is right, $16x$ is wrong.

**Method 48.8 (Testing an equality).**

To check whether two expressions might be equal for every value:

1. substitute the same value in both, at least twice (e.g. $x = 2$ and $x = 10$ ; avoid $0$ and $1$ , they hide too many [differences](https://one-course.com/books/math/1/en/chapter/3-subtraction-first-steps#ex-g1-subtraction-difference) );
2. if the results differ even once, the expressions are *not* equal — one counterexample kills a general claim;
3. if the results agree, the equality is only *plausible* : a proof needs algebra (expanding, reducing).

**Example 48.9.**

Is $2(x + 3)$ equal to $2x + 6$ for every $x$? Expanding proves it: $2(x+3) = 2x + 2 \times 3 = 2x + 6$. Is $(x + 1)^2$ equal to $x^2 + 1$? Test $x = 2$: $(2+1)^2 = 9$ but $2^2 + 1 = 5$. No — and one counterexample is a complete proof of “no”.

## 48.4 Producing expressions

**Example 48.10 (From situation to expression).**

A taxi charges $4$ euros plus $2$ euros per kilometer. For a trip of $k$ kilometers the price is

$$
p = 4 + 2k \quad\text{euros}.
$$

For $k = 7$: $p = 4 + 14 = 18$ euros. The expression *is* the general answer; substituting produces every particular one.

**Example 48.11 (Number tricks).**

“Think of a number, double it, add $10$, divide by $2$, subtract your number.” Call the number $n$ and follow the steps:

$$
n \ \to\ 2n \ \to\ 2n + 10 \ \to\ \frac{2n + 10}{2} = n + 5 \ \to\
n + 5 - n = 5 .
$$

Everyone gets $5$, whatever $n$ was — the letters explain the magic.

## 48.5 Exercises

**Exercise 48.1 ★.**

Simplify the writing:

$$
7 \times x, \qquad
y \times 3, \qquad
a \times b \times 5, \qquad
x \times x, \qquad
2 \times (a + 4).
$$

**Solution of Exercise 48.1.**

$7x$; $3y$; $5ab$; $x^2$; $2(a + 4)$.

**Exercise 48.2 ★.**

Put the $\times$ signs back:

$$
5ab, \qquad
3(x + 2), \qquad
x^2 + 4x, \qquad
2a^3 .
$$

**Solution of Exercise 48.2.**

$5ab = 5 \times a \times b$; $3(x+2) = 3 \times (x + 2)$; $x^2 + 4x = x \times x + 4 \times x$; $2a^3 = 2 \times a \times a \times a$.

**Exercise 48.3 ★.**

Evaluate for $x = 3$: $5x + 1$; $x^2$; $2x^2 - x$; $10 - 2x$.

**Solution of Exercise 48.3.**

For $x = 3$: $5 \times 3 + 1 = 16$; $3 \times 3 = 9$; $2 \times 9 - 3 = 15$; $10 - 6 = 4$.

**Exercise 48.4 ★.**

Evaluate $3a + 2b$ and $a b + 5$ for $a = 4$, $b = 1.5$.

**Solution of Exercise 48.4.**

$3a + 2b = 12 + 3 = 15$; $ab + 5 = 4 \times 1.5 + 5 = 6 + 5 = 11$.

**Exercise 48.5 ★.**

Reduce:

$$
8x + 5x, \qquad
9a - 4a + a, \qquad
6y + 2 + 3y + 7, \qquad
5t + 3 - 2t - 3 .
$$

**Solution of Exercise 48.5.**

$8x + 5x = 13x$; $9a - 4a + a = 6a$; $6y + 2 + 3y + 7 = 9y + 9$; $5t + 3 - 2t - 3 = 3t$.

**Exercise 48.6 ★.**

Give a formula for: the [perimeter](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-perimeter) of a square of side $c$; the [perimeter](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-perimeter) of an [isosceles triangle](https://one-course.com/books/math/1/en/chapter/41-triangles-and-quadrilaterals#def-g6-shapes-triangles) with base $b$ and equal sides $s$; the price of $n$ breads at $1.20$ euros each.

**Solution of Exercise 48.6.**

Square: $P = 4c$. [Isosceles triangle](https://one-course.com/books/math/1/en/chapter/41-triangles-and-quadrilaterals#def-g6-shapes-triangles): $P = b + 2s$. Breads: $p = 1.20\,n$ euros.

**Exercise 48.7 ★.**

Match each expression to a description: $2n$; $n + 2$; $n^2$; $\frac n2$. Descriptions: “the [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of $n$”; “the double of $n$”; “the square of $n$”; “two more than $n$”.

**Solution of Exercise 48.7.**

$2n$: the double; $n + 2$: two more than $n$; $n^2$: the square; $\frac n2$: the [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half).

**Exercise 48.8 ★★.**

Using [Method 48.8](#met-g7-literal-test), decide (with proof or counterexample) whether these equalities hold for every $x$:

$$
3(x + 4) = 3x + 12; \qquad
2x + 5 = 7x; \qquad
x + x = x^2 .
$$

**Solution of Exercise 48.8.**

$3(x+4) = 3x + 12$: *true* for every $x$ (distributivity — this is a proof, no test needed).

$2x + 5 = 7x$: test $x = 2$: left $9$, right $14$ — *false* in general (it holds only for $x = 1$).

$x + x = x^2$: test $x = 3$: left $6$, right $9$ — *false* in general ($x + x = 2x$).

**Exercise 48.9 ★★.**

A rectangle has width $w$ and its length is $3$ cm more than its width.

1. Express the length, then the [perimeter](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-perimeter) , in terms of $w$ , and reduce.
2. Compute the [perimeter](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-perimeter) for $w = 5$ cm, directly and with your reduced formula. Same answer?

**Solution of Exercise 48.9.**

*1.* Length: $w + 3$. [Perimeter](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-perimeter):

$$
P = 2\bigl(w + (w + 3)\bigr) = 2(2w + 3) = 4w + 6 .
$$

*2.* For $w = 5$: directly, the sides are $5$ and $8$, so $P = 2 \times 13 = 26$ cm; with the formula, $4 \times 5 + 6 = 26$ cm. They agree.

**Exercise 48.10 ★★.**

Follow the trick of [Example 48.11](#ex-g7-literal-trick) with letters: “think of a number, add $6$, multiply by $3$, subtract $18$, divide by your starting number” (assume it is not [zero](https://one-course.com/books/math/1/en/chapter/1-counting-to-20#def-g1-counting-zero)). What does everyone get?

**Solution of Exercise 48.10.**

Following the steps with the letter $n$:

$$
n \ \to\ n + 6 \ \to\ 3(n + 6) = 3n + 18 \ \to\ 3n + 18 - 18 = 3n
\ \to\ \frac{3n}{n} = 3 .
$$

Everyone gets $3$.

**Exercise 48.11 ★★★.**

The figure below is a square of side $a$ with a square corner of side $b$ removed ($b$ smaller than $a$). Write two different expressions for its [area](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area) (one as a [difference](https://one-course.com/books/math/1/en/chapter/3-subtraction-first-steps#ex-g1-subtraction-difference), one by cutting the L-shape into two rectangles), and check they agree for $a = 5$, $b = 2$.

**Solution of Exercise 48.11.**

As a [difference](https://one-course.com/books/math/1/en/chapter/3-subtraction-first-steps#ex-g1-subtraction-difference): $A = a^2 - b^2$. By cutting the L into two rectangles: a full-width strip of height $a - b$ and, on top of it, a narrower rectangle:

$$
A = a(a - b) + b(a - b)
$$

(bottom rectangle $a$ wide and $a - b$ tall, top rectangle $a - b$ wide and $b$ tall gives the variant $a(a-b) + (a-b)b$ — the same). For $a = 5$, $b = 2$: $a^2 - b^2 = 25 - 4 = 21$, and $5 \times 3 + 2 \times 3 = 15 + 6 = 21$. Both expressions agree — they always do: this is the identity $a^2 - b^2 = (a + b)(a - b)$ of [Chapter 66](https://one-course.com/books/math/1/en/chapter/66-algebra-and-equations#ch-g9-algebra) in disguise.

## 48.6 Problem: The matchstick oracle

**Problem 48.1.**

Weekend problem — growing patterns and their $n$-th term: three formulas for one pattern, and the famous pattern that breaks at the sixth step

Lay matchsticks in a row of squares: one square, then two, then three … How many matches will the hundredth figure need? Counting them one by one would take all weekend; a *formula in $n$* answers instantly for every figure at once — that is exactly what letters are for ([Example 48.3](#ex-g7-literal-formulas)). But beware: this problem ends with a celebrated pattern that plays fair for five steps and then betrays everyone who trusted it.

**Part I — A row of squares.** Figure $1$ is one matchstick square ($4$ matches); each following figure adds one more square sharing a side with the previous one.

1. Draw figures $1$ to $4$ and count their matches.
2. Explain why each new square costs exactly $3$ extra matches, and use this to predict the count for figure $10$ without any formula.
3. Justify the formula: figure $n$ needs $3n + 1$ matches. (Where does the lonely “ $+1$ ” come from?) Check it against question 1.
4. Amir reasons differently: “ $4$ matches for the first square, then $3$ for each of the $n - 1$ others”, giving $4 + 3(n - 1)$ . Expand and reduce his expression ( [Proposition 48.6](#prop-g7-literal-reduce) ): does he agree with question 3?
5. Lena counts a third way: “ $2n$ horizontal matches, and $n + 1$ vertical ones.” Write her formula, reduce it, and conclude. Then answer the opening question: how many matches for figure $100$ ?

**Part II — Triangles, tables and going backwards.**

6. A row of matchstick triangles (each new triangle leans on the previous one): count the matches for $1$ , $2$ , $3$ triangles, then justify the formula $2n + 1$ for $n$ triangles.
7. Going backwards: with exactly $49$ matches, how many triangles can the row have? (Undo the formula step by step.)
8. Square banquet tables seat one guest per free side. When $n$ tables are pushed into one long row, explain why the seating formula is $2n + 2$ , and check it for $1$ , $2$ , $3$ tables.
9. A party expects $30$ guests. How many tables does one row need? The caterer instead splits the same tables into *two* rows of seven: how many guests fit now? Explain the gain with the formula.
10. Dot pattern: figure $n$ is an L of dots, $n$ dots tall and $n$ dots wide (the corner counted once). Count the dots of figures $1$ , $2$ , $3$ , find the formula, and name the numbers it produces. (Stacked together, these Ls tile a square — a story continued in [Problem 57.1](https://one-course.com/books/math/1/en/chapter/57-literal-calculation-and-equations#pb-g8-equations-1) .)

**Part III — Trust, test, prove.**

11. A classmate claims figure $n$ of Part I needs $3(n + 2) - 5$ matches. Settle the claim by expanding and reducing — no testing needed.
12. “Squaring makes numbers bigger, so $n^2 \geq n$ for every number $n$ .” Test the claim for $n = 2$ , $3$ , $10$ ; then find a number for which it fails. What did the three successful tests prove, and what does the single failure prove ( [Method 48.8](#met-g7-literal-test) )?
13. Mark points on a circle and draw *every* [segment](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-objects) between them, then count the regions inside the circle. Do it for $2$ , $3$ , $4$ and $5$ points (place the points unevenly, so that no three [segments](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-objects) cross at one interior point). Record the counts. What formula does the pattern whisper?
14. Now $6$ points, drawn large and unevenly, counting with great care (there are many small regions). How many regions do you find — and what happened to the conjecture?
15. The moral, in two sentences: what would have been needed to *prove* the doubling conjecture, and why were the matchstick formulas of Parts I and II never in such danger? Celebrate with a proved prediction: how many matches for the row of $2\,026$ squares?

**Solution of Problem 48.1.**

**1.** Figures $1$ to $4$ need $4$, $7$, $10$, $13$ matches.

**2.** A new square shares one side with the previous figure, so only $3$ of its $4$ sides are new matches. From figure $4$ ($13$ matches), six more squares cost $6 \times 3 = 18$: figure $10$ needs $13 + 18 = 31$ matches.

**3.** Reading the row from the left: it starts with one vertical match, and each of the $n$ squares then contributes a top, a bottom and a right side — $3$ matches each. Total: $3n + 1$; the “$+1$” is the very first vertical stick. Check: $n = 1, 2, 3, 4$ give $4, 7, 10, 13$.

**4.** $4 + 3(n - 1) = 4 + 3n - 3 = 3n + 1$: Amir’s formula expands to exactly the same reduced expression. Full agreement.

**5.** Lena: $2n + (n + 1) = 3n + 1$ — the same again. Three ways of seeing, one reduced form. Figure $100$: $3 \times 100 + 1 = 301$ matches.

**6.** Counts: $3$, $5$, $7$ matches. Each new triangle leans on an existing side and adds only $2$ matches, starting from $3$: figure $n$ needs $3 + 2(n - 1) = 2n + 1$ matches.

**7.** Undo $2n + 1 = 49$: removing the first match leaves $48$, and each triangle then accounts for $2$: $48 \div 2 = 24$ triangles.

**8.** In a row of $n$ tables, each table offers its top and bottom side ($2n$ seats), and the two end tables offer one extra side each ($+2$): $2n + 2$ seats. Check: $4$, $6$, $8$ for $n = 1, 2, 3$.

**9.** One row: $2n + 2 \geq 30$ needs $2n \geq 28$, so $14$ tables. Splitting the same $14$ tables into two rows of $7$: each row seats $2 \times 7 + 2 = 16$, together $32$ — two more than the single row, because every new row brings two new *ends*. (Each split gains $2$ seats; caterers know.)

**10.** The L of size $n$ has $n$ dots going up and $n$ going across, the corner dot shared: $n + n - 1 = 2n - 1$ dots. Figures $1$, $2$, $3$: $1$, $3$, $5$ dots — the *[odd numbers](https://one-course.com/books/math/1/en/chapter/14-numbers-up-to-10-000#def-g3-numbers-evenodd)*. Nested one around the next, the Ls fill an $n \times n$ square of dots: the story of [Problem 57.1](https://one-course.com/books/math/1/en/chapter/57-literal-calculation-and-equations#pb-g8-equations-1).

**11.** $3(n + 2) - 5 = 3n + 6 - 5 = 3n + 1$: the claim reduces to the proved formula, so it is right for every $n$ — established by algebra, not by examples.

**12.** Tests: $4 \geq 2$, $9 \geq 3$, $100 \geq 10$: all pass. But $n = 0.5$ gives $n^2 = 0.25 < 0.5$: the claim fails. The three tests proved nothing about *all* numbers (they only failed to find trouble); the single counterexample *proves* the general claim false ([Method 48.8](#met-g7-literal-test)). A letter stands for every number — decimals included.

**13.** Regions: $2$ points $\to 2$; $3$ points $\to 4$; $4$ points $\to 8$; $5$ points $\to 16$. The pattern whispers “each point doubles the count”: $32$ regions expected for $6$ points.

**14.** A careful count gives $31$ regions — not $32$. The doubling conjecture is dead: it survived four checks and failed the fifth. (This counterexample is famous enough to have a name, “the circle-regions trap”; no evenly-spaced shortcuts — three chords meeting at one point would merge regions and spoil the count.)

**15.** A proof would have needed a *structural* reason why adding a point doubles the regions — and there is none: the doubling was a coincidence of small cases. The matchstick formulas were safe because each was read off the construction itself (each square visibly costs $3$ matches, each table visibly offers $2$ seats): structure, not testing. Proved prediction: $3 \times 2\,026 + 1 = 6\,079$ matches.