Primary & Middle School Mathematics · Grades 1–9

13Solving 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?)

putting together, in all: add ( $+$ );
taking away, how many left, how many more: subtract ( $-$ );
so many equal groups: multiply ( $\times$ );
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.

Example 13.2 (A two-step problem)

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

Groups: $3 \times 6 = 18$ stickers;
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. — 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

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

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

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

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

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

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: how many tarts sold over the two days? How many cakes of all kinds on Saturday?

Solution

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: how many fruits in all? Which colour has the most?

Solution

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

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 — or count the trays first!)

Solution

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 $8 \times 4 = 32$ .)