Primary & Middle School Mathematics · Grades 1–9

36Problems and Charts

The last chapter of the year puts everything to work: multi-step problems with money, quantities and durations, and the charts that present information at a glance. The star idea is “per one” — finding the price of a single item — which will grow into proportionality in Chapter 44.

36.1 Multi-step problems

Method 36.1 (Taming a long problem)

Read to the end; say what the final question is;
plan backwards: to answer it, what do I need first?
solve the steps in order, writing one operation and one short sentence per step;
check the final answer against common sense and against an estimate.

Example 36.2

“A school orders $4$ boxes of $25$ books at $6$ each, and pays $180$ for delivery on top. What is the total cost?”

Books: $4 \times 25 = 100$ books.
Book cost: $100 \times 6 = 600$ .
Total: $600 + 180 = 780$ .

Estimate check: about a hundred books at $6$ is about $600$ , plus about $200$ : about $800$ — the answer $780$ is plausible.

Example 36.3 (Per one)

“ $5$ identical mugs cost $12.50$ ; what do $8$ mugs cost?” Find the price of one mug first:

12.50 \div 5 = 2.50, \qquad\text{then}\qquad 8 \times 2.50 = 20 .

Through one, everything becomes reachable — this two-step pattern solves a whole family of problems (recipes, speeds, prices), and is the seed of Chapter 44.

Example 36.4 (Change and budget)

Zoe has $30$ . She buys a book at $12.40$ and two pens at $2.30$ each. Can she also afford a $13$ board game?

Pens: $2 \times 2.30 = 4.60$ .
Spent so far: $12.40 + 4.60 = 17$ .
Remaining: $30 - 17 = 13$ — exactly enough for the game, with nothing left.

36.2 Reading charts

Method 36.5 (Reading any chart)

Read the title: what is being counted?
Read the axes or the key: what does one graduation, one bar, one symbol stand for?
Only then answer questions — pointing at the chart with a ruler helps read heights exactly.

Example 36.6

Rainfall in a town, month by month:

A bar chart of rainfall. One graduation = 20 mm. — A bar chart of rainfall. One graduation $= 20$ mm.

Readings: the wettest month is May ( $80$ mm); June got half of April’s rain ( $30$ against $70$ ? no — half of $70$ is $35$ , so not quite half: reading precisely matters!); total for the six months: $60 + 45 + 55 + 70 + 80 + 30 = 340$ mm.

Example 36.7 (A changing quantity)

A plant’s height, measured every week:

Joining the measurement points shows the growth: the steeper the segment, the faster the plant grew that week. — Joining the measurement points shows the *growth*: the steeper the segment, the faster the plant grew that week.

The plant grew fastest between weeks $3$ and $4$ ( $+5$ cm) and slowed down at the end ( $+2$ cm in week $6$ ).

36.3 Exercises

Exercise 36.1 ★

Solve in steps: “A club rents a bus for $240$ and buys $32$ museum tickets at $7$ each. What is the total cost of the trip?”

Solution

Solution of Exercise 36.1.

Tickets: $32 \times 7 = 224$ . Total: $240 + 224 = 464$ .

Exercise 36.2 ★

Solve with “per one”: “ $3$ kg of apples cost $7.50$ . What do $5$ kg cost?”

Solution

Solution of Exercise 36.2.

Per one kg: $7.50 \div 3 = 2.50$ . Five kg: $5 \times 2.50 = 12.50$ .

Exercise 36.3 ★

Solve: “Six friends share equally the $87$ cost of a picnic. How much does each pay?”

Solution

Solution of Exercise 36.3.

$87 \div 6 = 14.5$ : each pays $14.50$ .

Exercise 36.4 ★

Sam has $25$ . He buys a T-shirt at $13.90$ and a cap at $8.50$ . How much money has he left?

Solution

Solution of Exercise 36.4.

Spent: $13.90 + 8.50 = 22.40$ . Left: $25 - 22.40 = 2.60$ .

Exercise 36.5 ★

Using the rainfall chart of Example 36.6: which months got more than $50$ mm? How much more rain fell in May than in June?

Solution

Solution of Exercise 36.5.

More than $50$ mm: January ( $60$ ), March ( $55$ ), April ( $70$ ), May ( $80$ ). May minus June: $80 - 30 = 50$ mm.

Exercise 36.6 ★

Using the plant chart of Example 36.7: how tall was the plant in week $2$ ? Between which weeks did it grow by exactly $4$ cm?

Solution

Solution of Exercise 36.6.

Week $2$ : $6$ cm. Growth of $4$ cm: between weeks $2$ and $3$ ( $6 \to 10$ ), and between weeks $4$ and $5$ ( $15 \to 19$ ).

Exercise 36.7 ★

Draw a bar chart for the number of goals a team scored: September $4$ , October $7$ , November $3$ , December $6$ . Choose the graduation, and give the total.

Solution

Solution of Exercise 36.7.

Four bars of heights $4$ , $7$ , $3$ , $6$ (graduation every $1$ or $2$ goals). Total: $4 + 7 + 3 + 6 = 20$ goals.

Exercise 36.8 ★

A pack of $8$ yogurts costs $3.20$ ; sold alone, one yogurt costs $0.50$ . Compute the “per one” price in the pack, and how much one saves per yogurt by buying the pack.

Solution

Solution of Exercise 36.8.

Pack per one: $3.20 \div 8 = 0.40$ . Saving per yogurt: $0.50 - 0.40 = 0.10$ (ten cents).

Exercise 36.9 ★★

“A cinema sold $148$ tickets at $9.50$ on Saturday and $95$ tickets at the same price on Sunday. How much more money did Saturday bring than Sunday?” Solve it two ways: computing both days’ takings, or computing with the difference in tickets — and check that the answers agree.

Solution

Solution of Exercise 36.9.

Way 1: Saturday $148 \times 9.50 = 1\,406$ ; Sunday $95 \times 9.50 = 902.50$ ; difference $503.50$ .

Way 2: $148 - 95 = 53$ more tickets, each worth $9.50$ : $53 \times 9.50 = 503.50$ . Same answer — the difference of the takings is the taking of the difference.

Exercise 36.10 ★★

A recipe for $4$ people needs $300$ g of rice. Aline cooks for $10$ people. How much rice does she need? (Per one person first — decimals allowed.)

Solution

Solution of Exercise 36.10.

Per person: $300 \div 4 = 75$ g. For ten: $10 \times 75 = 750$ g of rice.

Exercise 36.11 ★★

Invent a three-step problem using money whose final answer is $5.50$ , write its full solution in the style of Method 36.1, and test it on a classmate.

Solution

Solution of Exercise 36.11.

Many correct problems — for instance: “Milo buys $3$ juices at $1.20$ each and a sandwich at $3.40$ . He pays with a $12.50$ voucher. How much of the voucher is left?” Solution: juices $3 \times 1.20 = 3.60$ ; total spent $3.60 + 3.40 = 7$ ; left $12.50 - 7 = 5.50$ .