---
title: "Proportionality and Data"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 44
exercises: 12
source: https://one-course.com/books/math/1/en/chapter/44-proportionality-and-data
---

# Chapter 44 — Proportionality and Data

If three notebooks cost $6$ euros, six notebooks cost $12$: double the notebooks, double the price. Quantities behaving like this are *[proportional](#def-g6-propdata-def)* — one of the most useful ideas in all of mathematics, developed further in [Chapter 49](https://one-course.com/books/math/1/en/chapter/49-proportionality#ch-g7-prop). The chapter ends with reading and drawing simple data charts.

## 44.1 Proportional quantities

**Definition 44.1 (Proportionality).**

Two quantities are *proportional* when the values of one are obtained from the values of the other by multiplying always by the *same* number, called the *proportionality coefficient*.

**Example 44.2.**

Notebooks at $2$ euros each:

| notebooks | $3$ | $5$ | $8$ | $12$ |
| --- | --- | --- | --- | --- |
| price (euros) | $6$ | $10$ | $16$ | $24$ |

Each price is the number of notebooks $\times 2$: the coefficient is $2$ (the unit price). A quick check that a table is [proportional](#def-g6-propdata-def): all the “column [quotients](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder)” $\frac{6}{3}, \frac{10}{5}, \frac{16}{8},
\frac{24}{12}$ must be equal — here they all equal $2$.

**Example 44.3 (A non-example).**

Age and height are not [proportional](#def-g6-propdata-def): a $12$-year-old is not twice as tall as a $6$-year-old. Check on numbers: $1.50$ m at $12$ years and $1.15$ m at $6$ years give [quotients](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $\frac{1.50}{12} = 0.125$ and $\frac{1.15}{6} \approx 0.19$ — not equal.

**Method 44.4 (Completing a proportionality table).**

Three ways, to be chosen freely:

1. *coefficient* : find the multiplier from one complete column, apply it to the others;
2. *columns* : a column can be multiplied by a number, or two columns can be added, to produce a new column;
3. *back to the unit* : find the value for *one* item first, then multiply.

**Example 44.5.**

Five identical mugs cost $15$ euros; how much do eight mugs cost?

*Back to the unit*: one mug costs $15 \div 5 = 3$ euros, so eight mugs cost $8 \times 3 = 24$ euros.

*Columns*: $8 = 5 + 3$; three mugs cost $\frac{3}{5}$ of $15$, i.e. $9$; so eight mugs cost $15 + 9 = 24$ euros. Same answer, as it must be.

## 44.2 Percentages

**Definition 44.6 (Percentage).**

“$t\,\%$ of a quantity” means the [fraction](https://one-course.com/books/math/1/en/chapter/39-fractions-first-steps#def-g6-fractions-def) $\frac{t}{100}$ of it: $25\,\%$ of $60$ is $\frac{25}{100} \times 60 = 15$. A percentage is a [proportionality](#def-g6-propdata-def) with coefficient $\frac{t}{100}$.

**Example 44.7.**

A class of $30$ students contains $40\,\%$ girls. Number of girls, step by step: $40\,\%$ means $\frac{40}{100} = 0.4$, and $0.4 \times 30 = 12$ girls. Useful landmarks: $50\,\%$ is one [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half), $25\,\%$ one [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half), $10\,\%$ one tenth — so $10\,\%$ of $30$ is $3$, and $40\,\% = 4 \times 10\,\%$ is $4 \times 3 = 12$: same answer, computed mentally.

## 44.3 Reading and drawing charts

**Method 44.8 (Reading a table or chart).**

Whatever the picture — table, bar chart, line graph:

1. read the title and the units on both axes first;
2. to answer a question, locate the right bar/column, then read the value carefully against the graduation;
3. to compare, compare bars by height — but check the axis starts at $0$ , or the picture can mislead.

**Example 44.9.**

The bar chart below shows the number of books borrowed at the school library during one week.

![A bar chart: one bar per day, height equal to the count.](https://one-course.com/images/onecourse/chapters/math-1/g6-propdata/fig-63e83d377e08.svg)

*A bar chart: one bar per day, height equal to the count.*

Readings: the busiest day is Wednesday ($17$ books). Tuesday and Thursday together account for $8 + 6 = 14$ books — fewer than Friday alone ($15$). Total for the week: $12 + 8 + 17 + 6 + 15 = 58$ books.

![Proportional quantities have a very recognizable graph: the points line up on a straight line through the origin (here notebooks at 3 euros each; the dashes read the price of 6 notebooks: 18 euros).](https://one-course.com/images/onecourse/chapters/math-1/g6-propdata/fig-76f95b21379e.svg)

*[Proportional](#def-g6-propdata-def) quantities have a very recognizable graph: the points line up on a straight line through the origin (here notebooks at $3$ euros each; the dashes read the price of $6$ notebooks: $18$ euros).*

## 44.4 Exercises

**Exercise 44.1 ★.**

Is the table [proportional](#def-g6-propdata-def)? Justify by computing [quotients](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder).

| kg of apples | $2$ | $3$ | $5$ |
| --- | --- | --- | --- |
| price (euros) | $5$ | $7.5$ | $12.5$ |

| age (years) | $2$ | $4$ | $8$ |
| --- | --- | --- | --- |
| height (cm) | $85$ | $103$ | $130$ |

**Solution of Exercise 44.1.**

Apples: [quotients](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $\frac52 = 2.5$, $\frac{7.5}{3} = 2.5$, $\frac{12.5}{5} = 2.5$ — all equal: [proportional](#def-g6-propdata-def) (price $2.50$ per kg).

Age/height: $\frac{85}{2} = 42.5$ but $\frac{103}{4} = 25.75$ — not equal: not [proportional](#def-g6-propdata-def).

**Exercise 44.2 ★.**

Four croissants cost $4.80$ euros. Find the price of one croissant, then of seven croissants.

**Solution of Exercise 44.2.**

One croissant: $4.80 \div 4 = 1.20$ euros. Seven: $7 \times 1.20 = 8.40$ euros.

**Exercise 44.3 ★.**

Complete the [proportionality](#def-g6-propdata-def) table (coefficient first!):

| liters of fuel | $10$ | $25$ | $?$ | $60$ |
| --- | --- | --- | --- | --- |
| price (euros) | $18$ | $?$ | $72$ | $?$ |

**Solution of Exercise 44.3.**

Coefficient: $18 \div 10 = 1.8$ euros per liter. Then $25$ L cost $25 \times 1.8 = 45$ euros; $72$ euros buy $72 \div 1.8 = 40$ L; $60$ L cost $60 \times 1.8 = 108$ euros.

**Exercise 44.4 ★.**

A car uses $6$ L of fuel per $100$ km. How much fuel for $250$ km? For $350$ km? How far can it go with $27$ L?

**Solution of Exercise 44.4.**

For $250$ km: $2.5$ times the fuel of $100$ km, so $2.5 \times 6 = 15$ L. For $350$ km: $3.5 \times 6 = 21$ L. With $27$ L: $27 \div 6 = 4.5$ hundreds of km, i.e. $450$ km.

**Exercise 44.5 ★.**

Compute mentally, using the landmarks of [Example 44.7](#ex-g6-propdata-percent): $50\,\%$ of $84$; $25\,\%$ of $200$; $10\,\%$ of $63$; $30\,\%$ of $70$.

**Solution of Exercise 44.5.**

$50\,\%$ of $84$: [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half), $42$. $25\,\%$ of $200$: a [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half), $50$. $10\,\%$ of $63$: a tenth, $6.3$. $30\,\%$ of $70$: $3 \times 7 = 21$.

**Exercise 44.6 ★.**

In a school of $450$ students, $60\,\%$ eat at the cafeteria. How many students is that? How many do not?

**Solution of Exercise 44.6.**

$60\,\%$ of $450$: $\frac{60}{100} \times 450 = 270$ students eat at the cafeteria; $450 - 270 = 180$ do not.

**Exercise 44.7 ★.**

Using the bar chart of [Example 44.9](#ex-g6-propdata-chart): on which days were fewer than $10$ books borrowed? How many more books were borrowed on Wednesday than on Thursday?

**Solution of Exercise 44.7.**

Fewer than $10$ books: Tuesday ($8$) and Thursday ($6$). Wednesday minus Thursday: $17 - 6 = 11$ more books.

**Exercise 44.8 ★.**

The temperatures at noon from Monday to Friday were $14$, $16$, $13$, $17$, $20$ degrees. Draw a bar chart of these data (choose a sensible graduation), and read off the warmest day.

**Solution of Exercise 44.8.**

Bar chart with five bars of heights $14$, $16$, $13$, $17$, $20$ (graduation every $2$ or $5$ degrees works well). Warmest day: Friday ($20$ degrees).

**Exercise 44.9 ★★.**

A recipe for $6$ people needs $450$ g of flour and $3$ eggs. Adapt it for $10$ people. (For the eggs, think before writing a [decimal number](https://one-course.com/books/math/1/en/chapter/38-decimal-numbers#def-g6-decimals-places) of eggs!)

**Solution of Exercise 44.9.**

For $10$ people, multiply by $\frac{10}{6} = \frac53$: flour $450 \times \frac{10}{6} = 750$ g. Eggs: $3 \times \frac{10}{6} = 5$ — luckily a whole number. (If it were not, one would [round](https://one-course.com/books/math/1/en/chapter/21-large-numbers#def-g4-numbers-round) *up* to have enough.)

**Exercise 44.10 ★★.**

At a constant speed, a cyclist rides $24$ km in $1$ hour.

1. How far does she ride in $2$ h? In [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) an hour? In $1$ h $30$ min?
2. How long does she need for $60$ km?

**Solution of Exercise 44.10.**

*1.* In $2$ h: $48$ km. In [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) an hour: $12$ km. In $1$ h $30$: $24 + 12 = 36$ km.

*2.* $60 \div 24 = 2.5$ hours, i.e. $2$ h $30$ min.

**Exercise 44.11 ★★.**

A shop offers “$20\,\%$ off everything”. Compute the discount and the new price for: a ball at $15$ euros; a racket at $40$ euros. Is the new price [proportional](#def-g6-propdata-def) to the old price? What is the coefficient?

**Solution of Exercise 44.11.**

Ball: discount $20\,\%$ of $15 = 3$ euros, new price $12$ euros. Racket: discount $8$ euros, new price $32$ euros. The new price is $80\,\%$ of the old one in every case: [proportional](#def-g6-propdata-def), coefficient $0.8$.

**Exercise 44.12 ★★★.**

Two candles of the same height are lit at the same time. The thick one burns down completely in $6$ hours, the thin one in $4$ hours, each at its own constant rate. After how much time is the thin candle exactly [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) as tall as the thick one? (Express the remaining heights after $t$ hours as [fractions](https://one-course.com/books/math/1/en/chapter/39-fractions-first-steps#def-g6-fractions-def) of the initial height.)

**Solution of Exercise 44.12.**

After $t$ hours, the thick candle has burned $\frac t6$ of its height: it stands at $1 - \frac t6$ of the initial height; the thin one at $1 - \frac t4$. We want

$$
1 - \frac t4 = \frac12 \left(1 - \frac t6\right)
\quad\text{i.e.}\quad
1 - \frac t4 = \frac12 - \frac t{12}.
$$

Then $\frac12 = \frac t4 - \frac t{12} = \frac{3t - t}{12} =
\frac{t}{6}$, so $t = 3$ hours. Check: after $3$ h the thick candle stands at $1 - \frac36 = \frac12$ and the thin one at $1 - \frac34 = \frac14$ — indeed [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of it.

## 44.5 Problem: Maps, plans, and how to lie with a chart

**Problem 44.1.**

Weekend problem — the scale of a map is a proportionality: lengths follow the coefficient, areas follow its square, and badly drawn charts fool the eye

Every map, floor plan and model obeys one rule: real lengths and drawn lengths are *[proportional](#def-g6-propdata-def)* ([Definition 44.1](#def-g6-propdata-def)). The coefficient has a famous name — the *scale* — and it hides two traps that this problem springs deliberately: [areas](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area) do *not* follow the scale, and percentages, like charts, depend entirely on what they are measured against.

**Part I — Reading a map.** A hiking map is at scale $1:100\,000$: every centimeter on the map stands for $100\,000$ centimeters on the ground.

1. Convert $100\,000$ cm into kilometers. What does $1$ cm of this map represent, in km?
2. Two villages lie $7.5$ cm apart on the map; the shore of a lake is a $12$ cm curve. Give the real distances.
3. A dead-straight Roman road runs $20$ km. How long is it on the map?
4. A second map announces “ $1$ cm for $5$ km”. Write its scale in the form $1: \,?$ . Which of the two maps shows more detail for the same region?
5. Make a small table (map distance $1$ , $2$ , $7.5$ cm against real distance) for the hiking map, and explain why a scale is exactly a [proportionality](#def-g6-propdata-def) in the sense of [Definition 44.1](#def-g6-propdata-def) . What is the coefficient that turns centimeters-on-the-map into kilometers?

**Part II — Zoe’s bedroom plan.** Zoe draws a plan of her bedroom — a rectangle of $4$ m by $3$ m — at scale $1:50$.

6. What are the dimensions of the room on the plan?
7. Her bed measures $2$ m by $0.9$ m, and the doorway is $80$ cm wide. Give all three measurements on the plan.
8. Now the trap. Compute the real [area](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area) of the room in m $^2$ , then the [area](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area) of its plan in cm $^2$ . Convert the real [area](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area) into cm $^2$ ( [Definition 43.5](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area) ) and divide by the plan [area](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area) : is the [quotient](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $50$ ?
9. Explain the number you found: on a $1:50$ plan, each square centimeter of paper represents a real square of $50$ cm by $50$ cm. How many real cm $^2$ is that? State the rule: when lengths are divided by $50$ , [areas](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area) are divided by …
10. A [round](https://one-course.com/books/math/1/en/chapter/21-large-numbers#def-g4-numbers-round) carpet covers $6$ cm $^2$ on the plan. What real [area](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area) does it cover, in cm $^2$ and then in m $^2$ ?

**Part III — How to lie with a chart (and with a percentage).**

11. A shop sold $105$ euros’ worth on Saturday and $110$ on Sunday. The manager draws a bar chart whose vertical axis starts at $100$ : the Saturday bar rises $5$ small units, the Sunday bar $10$ . What does the picture suggest about Sunday, and what is the truth? (Compute Sunday’s increase as a percentage of Saturday, to the nearest percent.) Which warning of [Method 44.8](#met-g6-propdata-read) did the manager ignore?
12. Redraw (or describe) the honest chart, with the axis starting at $0$ : how do the two bars compare now?
13. A collection grows from $40$ to $60$ stamps: compute the increase as a percentage of the starting value. It then shrinks back from $60$ to $40$ : compute the decrease as a percentage of *its* starting value. Why are the two answers different, for the same $20$ stamps?
14. Sale season: a coat at $100$ euros gets “ $20\,\%$ off”, and at the till an extra “ $10\,\%$ off the reduced price”. Compute the final price step by step. Is the total discount $30\,\%$ ? Explain to the shopper what it really is.
15. Finale, back on the map of question 4 ( $1$ cm for $5$ km): a forest occupies $3$ cm $^2$ of that map. What is its real [area](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area) , in km $^2$ ? Conclude with the rule of this whole problem: lengths follow the scale, [areas](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area) follow its …

**Solution of Problem 44.1.**

**1.** $100\,000$ cm $= 1\,000$ m $= 1$ km: one centimeter on the map is one kilometer on the ground.

**2.** $7.5$ cm $\rightarrow 7.5$ km between the villages; $12$ cm $\rightarrow 12$ km of lake shore.

**3.** $20$ km $\rightarrow 20$ cm on the map.

**4.** $5$ km $= 500\,000$ cm, so the scale is $1:500\,000$. The hiking map ($1:100\,000$) is the more detailed: it uses $5$ cm of paper where the other spends only $1$ cm.

**5.**

| map (cm) | $1$ | $2$ | $7.5$ |
| --- | --- | --- | --- |
| real (km) | $1$ | $2$ | $7.5$ |

Real distance $=$ map distance $\times 1$ (in these units): always the *same* multiplier, which is exactly the definition of [proportionality](#def-g6-propdata-def) ([Definition 44.1](#def-g6-propdata-def)). The coefficient here is $1$ kilometer per centimeter.

**6.** $4$ m $= 400$ cm and $400 \div 50 = 8$; $3$ m $= 300$ cm and $300 \div 50 = 6$: the plan shows an $8$ cm $\times$ $6$ cm rectangle.

**7.** Bed: $200 \div 50 = 4$ cm by $90 \div 50 = 1.8$ cm. Doorway: $80 \div 50 = 1.6$ cm.

**8.** Real [area](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area): $4 \times 3 = 12$ m$^2$. Plan [area](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area): $8 \times 6 = 48$ cm$^2$. Converting: $12$ m$^2 = 12 \times 10\,000 = 120\,000$ cm$^2$, and

$$
120\,000 \div 48 = 2\,500 .
$$

The [quotient](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) is not $50$ but $2\,500 = 50 \times 50$.

**9.** One cm$^2$ of paper stands for a real square of $50$ cm by $50$ cm, which contains $50 \times 50 = 2\,500$ cm$^2$. So when lengths are divided by $50$, [areas](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area) are divided by $50 \times 50 = 2\,500$: [areas](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area) follow the *square* of the scale.

**10.** $6 \times 2\,500 = 15\,000$ cm$^2$, and $15\,000 \div 10\,000 = 1.5$ m$^2$.

**11.** The picture suggests Sunday sold *twice* as much (a bar twice as tall). Truth: the increase is $5$ euros out of $105$, and $5 \div 105 \approx 0.048$: about $5\,\%$ more, not $100\,\%$ more. The manager ignored the warning to check that the axis starts at $0$ ([Method 44.8](#met-g6-propdata-read)).

**12.** With the axis from $0$, the bars rise to $105$ and $110$ small units: two bars of nearly the same height, the honest picture of a $5\,\%$ [difference](https://one-course.com/books/math/1/en/chapter/3-subtraction-first-steps#ex-g1-subtraction-difference).

**13.** Up: the increase is $20$ stamps from a start of $40$: $\frac{20}{40} = 50\,\%$. Down: the decrease is $20$ stamps from a start of $60$: $\frac{20}{60} = \frac13 \approx
33\,\%$. Same $20$ stamps, different starting values — a percentage is always a [fraction](https://one-course.com/books/math/1/en/chapter/39-fractions-first-steps#def-g6-fractions-def) *of something*, and the “something” changed.

**14.** After $20\,\%$ off: $100 - 20 = 80$ euros. The extra $10\,\%$ applies to $80$: discount $8$ euros, final price $72$ euros. Total discount: $28$ euros out of $100$, so $28\,\%$ — not $30\,\%$. The second discount acted on the already-reduced price, so its euros are smaller: percentages of different quantities do not add.

**15.** On that map, $1$ cm stands for $5$ km, so $1$ cm$^2$ stands for $5 \times 5 = 25$ km$^2$ (question 9’s rule). The forest: $3 \times 25 = 75$ km$^2$. Lengths follow the scale; [areas](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area) follow its *square*.