Word problems and real-life expressions

A word problem is just a sentence in disguise. Behind every sentence sits an arithmetic expression waiting to be uncovered.

Idea

To turn a sentence into an expression, scan for operation words:

• "and", "plus", "added to", "total", "altogether" → $+$
• "minus", "difference", "less than", "left over" → $-$
• "of", "times", "product" → $\times$
• "shared equally", "per", "each" (in a ratio sense), "quotient" → $\div$

Then identify the order the sentence suggests. Sometimes a sentence forces you to add first, sometimes to multiply first. Brackets help record this.

For example, "buy $3$ pens at $\rupee 10$ each, and a notebook at $\rupee 25$" becomes $3 \times 10 + 25 = 55$. But "twice the sum of $\rupee 8$ and $\rupee 12$" becomes $2 \times (8 + 12) = 40$, with a bracket around the sum.

Once you have the expression, simplify by BODMAS. Always re-read the sentence to check that your expression matches the meaning. A common test: plug in tiny numbers and see whether the answer "feels right".

Real-life expressions hide everywhere: in shopping bills, in distance calculations ($d = v \times t$), in cooking recipes, in age puzzles. Practice spotting them.

Worked examples

Example 1. "A taxi charges $\rupee 50$ as base fare plus $\rupee 12$ per kilometre. What is the fare for $15$ km?"

Expression: $50 + 12 \times 15$. By BODMAS, do multiplication first: $12 \times 15 = 180$. Then $50 + 180 = \rupee 230$.

Example 2. "Riya bought $4$ books at $\rupee 80$ each and a school bag at $\rupee 350$. She paid with a $\rupee 1000$ note. How much change did she get?"

Cost: $4 \times 80 + 350 = 320 + 350 = \rupee 670$. Change: $1000 - 670 = \rupee 330$.

Example 3. "Twice the sum of $9$ and $4$, minus thrice the difference of $7$ and $2$."

Sum of $9$ and $4$: $9 + 4 = 13$. Twice that: $2 \times 13 = 26$. Difference of $7$ and $2$: $7 - 2 = 5$. Thrice that: $3 \times 5 = 15$. Final: $26 - 15 = 11$. As one expression: $2(9 + 4) - 3(7 - 2) = 26 - 15 = 11$.

Example 4. "$30$ students are divided into $5$ teams. Each team plays $4$ matches and scores $3$ goals per match. How many goals do all teams together score?"

Each team scores $4 \times 3 = 12$ goals. Total: $5 \times 12 = 60$. As an expression: $5 \times 4 \times 3 = 60$. (The number $30$ is a distractor , not needed for total goals.)

Try it yourself

1. Write an expression for "$5$ times the sum of $6$ and $4$" and evaluate.
2. A book costs $\rupee 65$ and a pen $\rupee 18$. Find the cost of $3$ books and $4$ pens.
3. A rectangular field is $25$ m by $14$ m. What is its area and perimeter?
4. "Eight more than thrice $7$" , write and evaluate.
5. A movie ticket costs $\rupee 150$. A group of $5$ adults and $3$ children (half price) goes. Total cost?
6. From a $\rupee 500$ note you buy a $\rupee 120$ shirt and two $\rupee 80$ pencil-boxes. Change?
7. "The sum of $20$ and $4$, divided by the difference of $9$ and $3$" , write and evaluate.
8. A train travels $60$ km per hour for $3$ hours, then $80$ km per hour for $2$ hours. Total distance?

Activity

Keep a "real expression" journal for one week. Every time you spot a real-life arithmetic problem (shopping, distance, time), write down the expression in your journal. At the end of the week, simplify all of them and share with classmates.