Primary & Middle School Mathematics · Grades 1–9

20Solving 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)

Read the problem twice; say in your own words what is asked, and what is known.
Choose the operation: draw a quick sketch or a bar picture if the choice is not obvious.
Compute, in columns or mentally, with an estimate as a safety net.
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.

Example 20.3 (A two-step problem)

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

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

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

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

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

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

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

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: how many science books were borrowed in the two weeks together? How many more novels in week 2 than in week 1?

Solution

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: how many cakes were sold in the three days together? On which day were the most cakes sold?

Solution

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

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

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

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 $120 - 20 = 100$ .