--- title: "Problems and Charts" book: "Primary & Middle School Mathematics" subject: math language: en chapter: 36 exercises: 11 source: https://one-course.com/books/math/1/en/chapter/36-problems-and-charts --- # Chapter 36 — Problems 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](https://one-course.com/books/math/1/en/chapter/44-proportionality-and-data#ch-g6-propdata). ## 36.1 Multi-step problems **Method 36.1 (Taming a long problem).** 1. Read to the end; say what the final question is; 2. plan backwards: to answer it, what do I need first? 3. solve the steps in order, writing one operation and one short sentence per step; 4. 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?” 1. Books: $4 \times 25 = 100$ books. 2. Book cost: $100 \times 6 = 600$ . 3. 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](https://one-course.com/books/math/1/en/chapter/44-proportionality-and-data#ch-g6-propdata). **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? 1. Pens: $2 \times 2.30 = 4.60$ . 2. Spent so far: $12.40 + 4.60 = 17$ . 3. Remaining: $30 - 17 = 13$ — exactly enough for the game, with nothing left. ## 36.2 Reading charts **Method 36.5 (Reading any chart).** 1. Read the title: what is being counted? 2. Read the axes or the key: what does one graduation, one bar, one symbol stand for? 3. 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.](https://one-course.com/images/onecourse/chapters/math-1/g5-data/fig-e82eaf903746.svg) *A bar chart of rainfall. One graduation $= 20$ mm.* Readings: the wettest month is May ($80$ mm); June got [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of April’s rain ($30$ against $70$? no — [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of $70$ is $35$, so *not quite* [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-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.](https://one-course.com/images/onecourse/chapters/math-1/g5-data/fig-d8a843474517.svg) *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 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 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 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 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](#ex-g5-data-chart): which months got more than $50$ mm? How much more rain fell in May than in June? **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](#ex-g5-data-line): how tall was the plant in week $2$? Between which weeks did it grow by exactly $4$ cm? **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 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 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](https://one-course.com/books/math/1/en/chapter/3-subtraction-first-steps#ex-g1-subtraction-difference)* in tickets — and check that the answers agree. **Solution of Exercise 36.9.** Way 1: Saturday $148 \times 9.50 = 1\,406$; Sunday $95 \times 9.50 = 902.50$; [difference](https://one-course.com/books/math/1/en/chapter/3-subtraction-first-steps#ex-g1-subtraction-difference) $503.50$. Way 2: $148 - 95 = 53$ more tickets, each worth $9.50$: $53 \times 9.50 = 503.50$. Same answer — the [difference](https://one-course.com/books/math/1/en/chapter/3-subtraction-first-steps#ex-g1-subtraction-difference) of the takings is the taking of the [difference](https://one-course.com/books/math/1/en/chapter/3-subtraction-first-steps#ex-g1-subtraction-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 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](#met-g5-data-steps), and test it on a classmate. **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 $12.50 - 7 = 5.50$.