---
title: "Solving Problems"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 13
exercises: 10
source: https://one-course.com/books/math/1/en/chapter/13-solving-problems
---

# Chapter 13 — Solving Problems

A story with numbers hides a question: add? take away? multiply? This chapter gives a way to decide, uses simple bar pictures to *see* the operation, and reads information from tables and pictograms.

## 13.1 Choosing the operation

**Method 13.1 (Which operation?).**

1. *putting together, in all* : add ( $+$ );
2. *taking away, how many left, how many more* : subtract ( $-$ );
3. *so many equal groups* : multiply ( $\times$ );
4. draw a quick picture when unsure, and always answer with a sentence.

![Bar pictures: two bars joined mean an addition; two bars compared mean a subtraction.](https://one-course.com/images/onecourse/chapters/math-1/g2-problems/fig-182045189f52.svg)

*Bar pictures: two bars joined mean an addition; two bars compared mean a subtraction.*

**Example 13.2 (A two-step problem).**

“Sam buys $3$ packs of $6$ stickers, then gives $5$ away. How many does he keep?”

1. Groups: $3 \times 6 = 18$ stickers;
2. give away: $18 - 5 = 13$ .

Sam keeps $13$ stickers. Reread the question: is the story really finished? Here yes.

**Example 13.3 (A hidden extra number).**

“A $210$-page book …” — careful, not every number in a story is needed. “Ben has $8$ red and $6$ blue marbles and is $7$ years old. How many marbles?” The age is a decoy: $8 + 6 = 14$ marbles. Sort the numbers before computing.

## 13.2 Tables and pictograms

**Example 13.4 (A table).**

Cakes sold at a stall:

|  | Saturday | Sunday | total |
| --- | --- | --- | --- |
| muffins | $12$ | $9$ | $21$ |
| tarts | $7$ | $10$ | $17$ |

Reading: Sunday tarts, $10$. Computing: muffins over the two days, $12 + 9 = 21$; more tarts on Sunday than Saturday by $10 - 7 = 3$.

**Example 13.5 (A pictogram).**

Read the key first: here one apple stands for $2$ pieces of fruit.

![A pictogram: red shows 4 dots, so 4 × 2 = 8 red fruits; green 2 dots, 4 fruits; yellow 3 dots, 6 fruits.](https://one-course.com/images/onecourse/chapters/math-1/g2-problems/fig-81f89bce9ca7.svg)

*A pictogram: red shows $4$ dots, so $4 \times 2 = 8$ red fruits; green $2$ dots, $4$ fruits; yellow $3$ dots, $6$ fruits.*

## 13.3 Exercises

**Exercise 13.1 ★.**

Name the operation (do not compute): “$34$ and $28$ marbles put together”; “$50$ sweets, $12$ eaten”; “$4$ boxes of $5$ apples”; “Ana has $30$, Leo has $18$: how many more has Ana?”.

**Solution of Exercise 13.1.**

Put together: addition. $50$ sweets, $12$ eaten: subtraction. $4$ boxes of $5$: multiplication. How many more: subtraction.

**Exercise 13.2 ★.**

Solve: “A car park has $156$ cars; $48$ leave. How many stay?”

**Solution of Exercise 13.2.**

$156 - 48 = 108$. There stay $108$ cars.

**Exercise 13.3 ★.**

Solve: “$5$ children each pick $4$ flowers. How many flowers in all?”

**Solution of Exercise 13.3.**

$5 \times 4 = 20$ flowers.

**Exercise 13.4 ★.**

Draw a bar picture, then solve: “Tom has $45$ cards, Mia has $17$ fewer. How many has Mia?”

**Solution of Exercise 13.4.**

Comparison bar: Tom $45$, Mia shorter by $17$: $45 - 17 = 28$ cards.

**Exercise 13.5 ★.**

Two steps: “A crate holds $6$ bottles. A shop gets $4$ crates and sells $9$ bottles. How many bottles remain?”

**Solution of Exercise 13.5.**

Bottles received: $4 \times 6 = 24$; remaining: $24 - 9 = 15$ bottles.

**Exercise 13.6 ★.**

Find and cross out the useless number, then solve: “Ben is $8$ years old and has $23$ marbles. He wins $15$ more. How many marbles now?”

**Solution of Exercise 13.6.**

The useless number is the age ($8$). Marbles: $23 + 15 = 38$.

**Exercise 13.7 ★.**

Using the table of [Example 13.4](#ex-g2-problems-table): how many tarts sold over the two days? How many cakes of all kinds on Saturday?

**Solution of Exercise 13.7.**

Tarts: $7 + 10 = 17$. Saturday, all cakes: $12 + 7 = 19$.

**Exercise 13.8 ★.**

Using the pictogram of [Example 13.5](#ex-g2-problems-pictogram): how many fruits in all? Which colour has the most?

**Solution of Exercise 13.8.**

Red $8$, green $4$, yellow $6$: in all $8 + 4 + 6 = 18$ fruits. Red has the most.

**Exercise 13.9 ★.**

Solve: “A ribbon is $80$ cm long. Lea cuts off $35$ cm. How long is the piece that remains?”

**Solution of Exercise 13.9.**

$80 - 35 = 45$ cm remain.

**Exercise 13.10 ★★.**

“A baker makes $5$ trays of $4$ cakes in the morning and $3$ trays of $4$ cakes in the afternoon. How many cakes that day?” (Two multiplications, then a [sum](https://one-course.com/books/math/1/en/chapter/2-addition-first-steps#def-g1-addition-def) — or count the trays first!)

**Solution of Exercise 13.10.**

Morning: $5 \times 4 = 20$; afternoon: $3 \times 4 = 12$; total $20 + 12 = 32$ cakes. (Or: $5 + 3 = 8$ trays of $4$, and $8 \times 4 = 32$.)