Writing Equations from Word Problems

The translation dictionary

“Increased by” and “more than” mean addition. “Decreased by” and “fewer than” mean subtraction. “Times”, “twice”, and “product” mean multiplication. “Is”, “was”, and “equals” all mark the equals sign.

“A number” means the variable. So “a number increased by $8$ is $21$” translates piece by piece: $x + 8 = 21$.

The “less than” trap

“Seven less than twice a number” is $2x - 7$, not $7 - 2x$. “Less than” flips the order: the $7$ is taken away from the $2x$. Read it as “twice a number, minus seven.”

Subtraction order always matters. “A number decreased by $5$” is $x - 5$, but “$5$ decreased by a number” is $5 - x$ — two different expressions with different solutions.

Fee-plus-rate problems

Many word problems combine a one-time amount with a repeating amount: a pickup fee plus a per-mile charge, or a joining fee plus a monthly cost. The repeating amount gets multiplied by the variable; the one-time amount is added once.

A taxi that charges a 3-dollar pickup fee plus 2 dollars per mile costs $2m + 3$ dollars for a ride of $m$ miles. Set that equal to the total spent and solve.

Worked examples

Try one yourself

Common questions