From Words to Algebra

A taxi charges \pounds 3 just to get in, then \pounds 2 for every mile — but you don't yet know how far you are going. Everyday situations like this are full of numbers we don't know yet, and algebra lets us give each one a letter so we can still write down a tidy rule for the total.

In a letter simply stands for a number we don't know yet. Once we have a letter, we can take a short phrase written in plain words and rewrite it as a tidy expression — a little recipe made of numbers, letters and operations.

The trick is always the same. Read the phrase slowly and ask: what is happening to the unknown number? Then write that down with symbols. If we call the unknown number n, here are the four moves you meet most often:

“5 more than n” — add five — n + 5
“double n” — two of them — 2n
“3 less than n” — take three away — n - 3
“n less than 10” — start at ten, take n away — 10 - n

Look hard at the last two: both say “less than”, but the order is flipped. “3 less than n” starts at n and removes 3, so it is n - 3; “n less than 10” starts at 10 and removes n, so it is 10 - n. With subtraction the order matters, so the words tell you which number goes first.

Notice too that 2n drops the \times sign: in algebra, writing a number and a letter side by side already means multiply, so 2n is just a tidy way to write 2 \times n. (When several operations meet, the usual still applies.)

Three worked examples

Take each phrase apart one word at a time.

“7 more than a number.” Call the number n. “More than” means add, and “7 more” means add seven:

n \;\longrightarrow\; n + 7

“Double a number.” “Double” means two of it — two copies added together, n + n — which we write as two lots of n:

n \;\longrightarrow\; 2n

“6 less than a number.” Start at the number and take six away. The number comes first because we are removing 6 from it:

n \;\longrightarrow\; n - 6

Each time the recipe is the same: name the unknown with a letter, then translate the words into one small piece of algebra.

Two traps that catch everybody at first:

Order matters in subtraction. “3 less than n” is n - 3, not 3 - n. The amount you take away goes second.
2n means 2 \times n, not “twenty-something”. The 2 and the n are multiplied — they are not two digits pushed together to make a two-digit number.

Picture a sealed bag holding some sweets. You can't count them, so give that number a name: n sweets. Now 3 bags hold three lots of n — that is 3n sweets altogether — and if a friend hands you 2 loose sweets as well, you have 3n + 2. The letter lets us talk about the total even though nobody has opened a single bag.

sweet sweet sweet

Not at all. n is just a popular choice (it reminds us of “number”), but x, a or t would do the same job. “5 more than the number of apples” could be a + 5. What matters is that you say once what the letter stands for, then use the very same letter all the way through.

apple apple

See it: a box of n, plus a few loose

Here is the same idea as a picture. The box holds n cookies — we can't see inside, so that count is unknown. Next to it sit a few loose cookies we can count. Altogether that is n plus however many are loose. Press Refresh for a new number of loose cookies and watch the expression change.

Watch a phrase turn into its expression. The words appear first; then the matching algebra grows underneath. The letter is just a stand-in for a number we haven't been told yet.

Khan Academy walks through writing expressions from words here: