Equations from word problems

A word problem is just a story whose ending is a number you have to discover. The trick is to translate the story carefully into an equation, solve the equation, and then translate the answer back into a sentence.

Idea

A reliable four-step recipe:

1. Read the problem twice. What is given, and what is asked?
2. Name the unknown. "Let $x$ be the number of …" or "Let $y$ be the age …".
3. Translate each English phrase into algebra.
   • "twice $x$" $\to 2x$
   • "$5$ more than $x$" $\to x + 5$
   • "$3$ less than twice $x$" $\to 2x - 3$
   • "the sum of $x$ and $y$" $\to x + y$
   • "split into $4$ equal parts" $\to \tfrac{x}{4}$
4. Set up the equality and solve. Then write the answer in a full sentence.

Common categories.

Age problems. "Riya is $5$ years older than Anu. The sum of their ages is $25$. Find their ages." Let Anu's age be $x$, then Riya's age is $x+5$. Equation: $x + (x+5) = 25$, giving $2x + 5 = 25$, so $x = 10$. Anu is $10$; Riya is $15$.

Money / cost problems. "Pens cost $\rupee 5$ each, pencils $\rupee 2$ each. Sara buys $4$ pens and some pencils for $\rupee 30$. How many pencils?" Let $p$ be the number of pencils. Equation: $20 + 2p = 30$, so $p = 5$.

Number problems. "Three times a number minus $4$ equals $11$." Let the number be $n$. Equation: $3n - 4 = 11$, so $n = 5$.

Geometry problems. "A rectangle has length $3$ cm more than twice its breadth. Its perimeter is $30$ cm. Find the dimensions." Let breadth $= b$; length $= 2b + 3$. Perimeter $= 2(b + 2b + 3) = 30$. So $6b + 6 = 30$, giving $b = 4$; length $= 11$ cm.

Sanity check. After solving, ask: "Does this answer make sense?" A person's age cannot be negative. A length cannot be zero. The number of pencils must be a whole number. If your equation gives an impossible answer, re-read the problem , you may have framed it incorrectly.

Worked examples

Example 1. "A number is doubled and increased by $7$; the result is $19$. Find the number."

Let the number be $x$. Equation: $2x + 7 = 19$. Solving: $2x = 12$, so $x = 6$. The number is $\boxed{6}$.

Example 2. "Anil's father is $4$ times as old as Anil. Their ages add up to $50$ years. Find Anil's age."

Let Anil's age $= a$. Father's age $= 4a$. Equation: $a + 4a = 50 \Rightarrow 5a = 50 \Rightarrow a = 10$. Anil is $10$ years old (father is $40$).

Example 3. "Tickets to a school play cost $\rupee 20$ for students and $\rupee 50$ for adults. The school sold $80$ tickets and collected $\rupee 2500$. How many adult tickets were sold?"

Let adult tickets $= y$; student tickets $= 80 - y$. Equation: $50y + 20(80 - y) = 2500$. Open: $50y + 1600 - 20y = 2500 \Rightarrow 30y = 900 \Rightarrow y = 30$. So $30$ adult tickets and $50$ student tickets.

Example 4. "The length of a rectangle is $6$ cm more than its breadth. The perimeter is $40$ cm. Find the area."

Let breadth $= b$; length $= b + 6$. Perimeter $= 2(b + b + 6) = 40 \Rightarrow 4b + 12 = 40 \Rightarrow b = 7$. Length $= 13$ cm. Area $= 7 \times 13 = 91 \text{ cm}^2$.

Try it yourself

1. The sum of two consecutive numbers is $35$. Find them.
2. A number is increased by $5$ and the result is multiplied by $2$ to give $24$. Find the number.
3. Five times a number minus $3$ equals twice the number plus $9$. Find the number.
4. Reena is $3$ years older than Meena. Their ages total $25$. Find each age.
5. A rectangle has length $5$ cm more than its breadth and perimeter $34$ cm. Find the area.
6. Pens cost $\rupee 7$ each and erasers $\rupee 3$ each. Buying $5$ pens and some erasers cost $\rupee 50$. How many erasers?
7. The sum of three consecutive even integers is $48$. Find them.
8. A father's age is $3$ years more than three times his son's age. The sum of their ages is $55$. Find both ages.

Activity

Open today's newspaper. Find one news item that includes numbers (e.g. cricket scores, ticket sales, election results). Make up a "find the unknown" word problem based on it. Trade with a partner. Solve each other's problems!