---
title: "Solving Problems"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 20
exercises: 11
source: https://one-course.com/books/math/1/en/chapter/20-solving-problems
---

# Chapter 20 — Solving Problems

Knowing the four operations is one thing; knowing *which one* a story needs is another. This chapter gives a method for word problems — the same four steps every time — and teaches how to read information from tables and pictograms.

## 20.1 The four steps

**Method 20.1 (Solving a word problem).**

1. *Read* the problem twice; say in your own words what is asked, and what is known.
2. *Choose* the operation: draw a quick sketch or a bar picture if the choice is not obvious.
3. *Compute* , in columns or mentally, with an estimate as a safety net.
4. *Answer* with a full sentence, including the unit — and check the answer makes sense (a child does not weigh $300$ kg).

**Example 20.2 (Which operation?).**

The words hint at the operation, the sketch decides:

- *putting together / in all* : addition;
- *taking away / how many more / what is missing* : subtraction;
- *so many groups of so many / so many times* : multiplication;
- *sharing fairly / how many groups* : division.

![Bar pictures make the operation visible: parts side by side call for an addition; two bars compared call for a subtraction.](https://one-course.com/images/onecourse/chapters/math-1/g3-problems/fig-f739750cf4d0.svg)

*Bar pictures make the operation visible: parts side by side call for an addition; two bars compared call for a subtraction.*

**Example 20.3 (A two-step problem).**

“Maya buys $3$ packs of $6$ bottles and drinks $2$ bottles. How many bottles are left?”

1. What is asked: the number of bottles remaining. Known: $3$ packs of $6$ ; $2$ drunk.
2. Sketch: $3$ groups of $6$ , then take $2$ away. Two operations!
3. Compute: $3 \times 6 = 18$ , then $18 - 2 = 16$ .
4. Sentence: $16$ bottles are left.

Never stop after the first operation: reread the question to see whether the story is finished.

**Example 20.4 (A useless number).**

“A baker, who is $52$ years old, bakes $120$ rolls and sells $85$. How many rolls are left?” The age is there to distract you: the answer only needs $120 - 85 = 35$ rolls. Before computing, always sort the numbers: which ones does the question really use?

## 20.2 Reading tables and pictograms

**Example 20.5 (A table).**

Books borrowed at the library:

|  | comics | novels | science | total |
| --- | --- | --- | --- | --- |
| week 1 | $14$ | $9$ | $6$ | $29$ |
| week 2 | $11$ | $12$ | $8$ | $31$ |

Reading: week 2, novels: $12$. Computing from the table: in the two weeks together, $14 + 11 = 25$ comics were borrowed; the busiest week was week 2 ($31 > 29$).

**Example 20.6 (A pictogram).**

In a pictogram, one symbol stands for a fixed amount — read the key first!

![A pictogram: Monday shows 3 dots, so 3 × 4 = 12 cakes; Tuesday 5 dots, 20 cakes; Wednesday 3 dots, 12 cakes.](https://one-course.com/images/onecourse/chapters/math-1/g3-problems/fig-abd4ac749946.svg)

*A pictogram: Monday shows $3$ dots, so $3 \times 4 = 12$ cakes; Tuesday $5$ dots, $20$ cakes; Wednesday $3$ dots, $12$ cakes.*

## 20.3 Exercises

**Exercise 20.1 ★.**

For each story, just *name* the operation (do not compute): a flock of $48$ sheep joins a flock of $37$; $56$ cherries shared among $7$ children; $9$ boxes of $8$ pencils; Ali had $71$ cards and gave away $18$.

**Solution of Exercise 20.1.**

Flocks joining: addition. Sharing cherries: division. Boxes of pencils: multiplication. Cards given away: subtraction.

**Exercise 20.2 ★.**

Solve with the four steps: “A parking lot has $246$ cars in the morning; $158$ more arrive at noon. How many cars are there now?”

**Solution of Exercise 20.2.**

Asked: the number of cars now. Known: $246$, then $158$ more. Operation: addition. Compute: $246 + 158 = 404$. Sentence: there are now $404$ cars in the parking lot.

**Exercise 20.3 ★.**

Solve: “A rope of $90$ m is cut into pieces of $9$ m. How many pieces?”

**Solution of Exercise 20.3.**

Grouping division: $90 \div 9 = 10$. The rope gives $10$ pieces.

**Exercise 20.4 ★.**

Solve: “Tickets cost $8$ each. What do $26$ tickets cost?”

**Solution of Exercise 20.4.**

Multiplication: $26 \times 8 = 208$. The tickets cost $208$.

**Exercise 20.5 ★.**

Draw a bar picture, then solve: “Emma has $64$ stickers, Hugo has $29$ fewer. How many has Hugo?”

**Solution of Exercise 20.5.**

Comparison bar: Emma’s bar $64$, Hugo’s bar shorter by $29$. Subtraction: $64 - 29 = 35$. Hugo has $35$ stickers.

**Exercise 20.6 ★.**

Two steps: “A crate holds $45$ apples. A grocer receives $6$ crates and sells $178$ apples. How many apples remain?”

**Solution of Exercise 20.6.**

Received: $6 \times 45 = 270$ apples. Remaining: $270 - 178 = 92$ apples.

**Exercise 20.7 ★.**

Using the table of [Example 20.5](#ex-g3-problems-table): how many science books were borrowed in the two weeks together? How many more novels in week 2 than in week 1?

**Solution of Exercise 20.7.**

Science books: $6 + 8 = 14$. Novels: $12 - 9 = 3$ more in week 2.

**Exercise 20.8 ★.**

Using the pictogram of [Example 20.6](#ex-g3-problems-pictogram): how many cakes were sold in the three days together? On which day were the most cakes sold?

**Solution of Exercise 20.8.**

Monday $12$, Tuesday $20$, Wednesday $12$: total $12 + 20 + 12 = 44$ cakes. The best day was Tuesday.

**Exercise 20.9 ★.**

This problem has a useless number — find it, then solve: “A $210$-page book costs $12$. Leo reads $35$ pages a day. How many days does he need to finish the book?”

**Solution of Exercise 20.9.**

The useless number is the price ($12$). Division: $210 \div 35 = 6$ (since $35 \times 6 = 210$). Leo needs $6$ days.

**Exercise 20.10 ★★.**

“A family of $2$ adults and $3$ children visits the museum. Adult tickets cost $9$, children tickets $4$. They pay with a $50$ bill. How much change do they get?” (Three steps: adults, children, then the change.)

**Solution of Exercise 20.10.**

Adults: $2 \times 9 = 18$. Children: $3 \times 4 = 12$. Total price: $18 + 12 = 30$. Change: $50 - 30 = 20$. They get $20$ back.

**Exercise 20.11 ★★.**

Make up your own two-step problem whose answer is $100$, write its solution with the four steps, and test it on a classmate.

**Solution of Exercise 20.11.**

Many correct answers — for instance: “A florist makes $8$ bunches of $15$ flowers, then removes $20$ wilted flowers. How many flowers remain?” Solution: $8 \times 15 = 120$, then $120 - 20 = 100$.