---
title: "Proportionality"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 49
exercises: 11
source: https://one-course.com/books/math/1/en/chapter/49-proportionality
---

# Chapter 49 — Proportionality

Grade 6 met [proportional](https://one-course.com/books/math/1/en/chapter/44-proportionality-and-data#def-g6-propdata-def) tables ([Chapter 44](https://one-course.com/books/math/1/en/chapter/44-proportionality-and-data#ch-g6-propdata)); this chapter turns [proportionality](#def-g7-prop-table) into a fully-fledged tool: the cross rule for finding a fourth value, percentages, and [scales](#def-g7-prop-scale) of maps and models. Speed, the most famous [proportionality](#def-g7-prop-table) of all, is studied in [Chapter 61](https://one-course.com/books/math/1/en/chapter/61-proportionality-speed-and-averages#ch-g8-speed).

## 49.1 Recognizing and completing tables

**Definition 49.1 (Proportionality table).**

A table of two rows is a *proportionality table* when the numbers of the second row are those of the first multiplied by one fixed number, the *coefficient*. Equivalently: all column [quotients](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $\frac{\text{bottom}}{\text{top}}$ are equal.

**Example 49.2.**

| top | $4$ | $10$ | $14$ |
| --- | --- | --- | --- |
| bottom | $6$ | $15$ | $21$ |

[Quotients](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder): $\frac64 = 1.5$, $\frac{15}{10} = 1.5$, $\frac{21}{14} = 1.5$: [proportional](https://one-course.com/books/math/1/en/chapter/44-proportionality-and-data#def-g6-propdata-def), coefficient $1.5$.

**Theorem 49.3 (Cross rule).**

In a [proportionality table](#def-g7-prop-table) with columns $\begin{pmatrix} a \\ b
\end{pmatrix}$ and $\begin{pmatrix} c \\ d \end{pmatrix}$, the cross [products](https://one-course.com/books/math/1/en/chapter/10-multiplication-first-steps#def-g2-mult-def) are equal:

$$
a \times d = b \times c .
$$

Consequently the fourth value can be computed from the three others: $d = \dfrac{b \times c}{a}$.

**Proof.** Let $k$ be the coefficient: $b = ka$ and $d = kc$. Then

$$
a \times d = a \times kc = k \times ac,
\qquad
b \times c = ka \times c = k \times ac :
$$

both cross [products](https://one-course.com/books/math/1/en/chapter/10-multiplication-first-steps#def-g2-mult-def) equal $k \times ac$. Dividing $ad = bc$ by $a$ gives the formula for $d$. ∎

**Example 49.4 (Fourth proportional).**

If $7$ identical books weigh $2.8$ kg, how much do $12$ such books weigh? Table and cross rule:

| books | $7$ | $12$ |
| --- | --- | --- |
| kg | $2.8$ | $m$ |

$$
7 \times m = 2.8 \times 12
\quad\Longrightarrow\quad
m = \frac{2.8 \times 12}{7} = \frac{33.6}{7} = 4.8 \text{ kg}.
$$

Check by the unit: one book weighs $0.4$ kg, twelve weigh $4.8$ kg.

## 49.2 Percentages

**Method 49.5 (Applying and finding percentages).**

1. *Apply* $t\,\%$ : multiply by $\frac{t}{100}$ ( $12\,\%$ of $350$ is $0.12 \times 350 = 42$ );
2. *find* the percentage that a part represents: compute $\frac{\text{part}}{\text{whole}}$ and rewrite over $100$ ( $21$ out of $60$ : $\frac{21}{60} = \frac{35}{100} = 35\,\%$ );
3. always ask: *a percentage of what?* The reference whole matters.

**Example 49.6.**

In a school, $180$ of the $400$ students are girls. Percentage:

$$
\frac{180}{400} = \frac{180 \div 4}{400 \div 4} = \frac{45}{100}
= 45\,\% .
$$

The other way: how many are the $55\,\%$ boys? $0.55 \times 400 = 220$. Check: $180 + 220 = 400$.

## 49.3 Scales

**Definition 49.7 (Scale).**

On a map or model at *scale* $\frac{1}{n}$ (written $1 : n$), the distances on the map are [proportional](https://one-course.com/books/math/1/en/chapter/44-proportionality-and-data#def-g6-propdata-def) to the real distances, with coefficient $\frac1n$:

$$
\text{map distance} = \frac{1}{n} \times \text{real distance},
$$

both measured in the *same unit*.

**Example 49.8.**

On a $1 : 50\,000$ map, two towns are $7$ cm apart. Real distance:

$$
7 \times 50\,000 = 350\,000 \text{ cm} = 3\,500 \text{ m} = 3.5
\text{ km}.
$$

Conversely, a $12$ km hiking trail measures on the map $12$ km $= 1\,200\,000$ cm, so $1\,200\,000 \div 50\,000 = 24$ cm.

![A scale is a proportionality: map distances against real distances line up on a straight line through the origin (here 1 cm 4 km; the dashes read 7 cm 28 km).](https://one-course.com/images/onecourse/chapters/math-1/g7-prop/fig-01e42bcb344f.svg)

*A [scale](#def-g7-prop-scale) is a [proportionality](#def-g7-prop-table): map distances against real distances line up on a straight line through the origin (here $1$ cm $\leftrightarrow$ $4$ km; the dashes read $7$ cm $\leftrightarrow$ $28$ km).*

**Remark 49.9.**

Graphs are the quickest test of [proportionality](#def-g7-prop-table) ([Chapter 44](https://one-course.com/books/math/1/en/chapter/44-proportionality-and-data#ch-g6-propdata)): points on a straight line *through the origin* mean [proportional](https://one-course.com/books/math/1/en/chapter/44-proportionality-and-data#def-g6-propdata-def) quantities; a line missing the origin (like a taxi fare with a fixed charge) means *not* [proportional](https://one-course.com/books/math/1/en/chapter/44-proportionality-and-data#def-g6-propdata-def), even though it is a straight line. Affine functions, in [Chapter 67](https://one-course.com/books/math/1/en/chapter/67-linear-and-affine-functions#ch-g9-linfunc), will make this precise.

## 49.4 Exercises

**Exercise 49.1 ★.**

Which tables are [proportionality tables](#def-g7-prop-table)? Give the coefficient when it exists.

|  | $3$ | $7$ | $11$ |
| --- | --- | --- | --- |
|  | $7.5$ | $17.5$ | $27.5$ |

|  | $2$ | $5$ | $9$ |
| --- | --- | --- | --- |
|  | $6$ | $15$ | $28$ |

**Solution of Exercise 49.1.**

First table: [quotients](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $\frac{7.5}{3} = 2.5$, $\frac{17.5}{7} = 2.5$, $\frac{27.5}{11} = 2.5$: [proportional](https://one-course.com/books/math/1/en/chapter/44-proportionality-and-data#def-g6-propdata-def), coefficient $2.5$.

Second table: $\frac62 = 3$ and $\frac{15}{5} = 3$, but $\frac{28}{9} \approx 3.1$: not [proportional](https://one-course.com/books/math/1/en/chapter/44-proportionality-and-data#def-g6-propdata-def).

**Exercise 49.2 ★.**

Complete using the cross rule, writing the computation: $5$ kg of potatoes cost $6$ euros; how much do $8$ kg cost?

**Solution of Exercise 49.2.**

With the cross rule: $5 \times p = 6 \times 8$, so $p = \frac{48}{5} = 9.60$ euros.

**Exercise 49.3 ★.**

A printer prints $36$ pages in $3$ minutes. At the same rate, how many pages in $5$ minutes? How long for $96$ pages?

**Solution of Exercise 49.3.**

Rate: $36 \div 3 = 12$ pages per minute. In $5$ minutes: $60$ pages. For $96$ pages: $96 \div 12 = 8$ minutes.

**Exercise 49.4 ★.**

Compute: $30\,\%$ of $250$; $15\,\%$ of $60$; $t\,\%$ such that $t\,\%$ of $80$ is $20$.

**Solution of Exercise 49.4.**

$30\,\%$ of $250$: $0.3 \times 250 = 75$. $15\,\%$ of $60$: $0.15 \times 60 = 9$. $t\,\%$ of $80$ is $20$: $\frac{20}{80} = \frac{25}{100}$, so $t = 25$.

**Exercise 49.5 ★.**

Out of $25$ shots, a basketball player scored $18$. What is her success percentage? (Rewrite the [fraction](https://one-course.com/books/math/1/en/chapter/39-fractions-first-steps#def-g6-fractions-def) over $100$.)

**Solution of Exercise 49.5.**

$\frac{18}{25} = \frac{18 \times 4}{25 \times 4} = \frac{72}{100} =
72\,\%$.

**Exercise 49.6 ★.**

On a $1 : 25\,000$ map, a path measures $9$ cm. What is its real length in km? A lake is $2$ km long: how long is it on the map?

**Solution of Exercise 49.6.**

Path: $9 \times 25\,000 = 225\,000$ cm $= 2.25$ km. Lake: $2$ km $= 200\,000$ cm on the ground, so $200\,000 \div 25\,000 = 8$ cm on the map.

**Exercise 49.7 ★.**

A model car is built at [scale](#def-g7-prop-scale) $1 : 43$. The real car is $4.3$ m long. How long is the model, in cm?

**Solution of Exercise 49.7.**

$4.3$ m $= 430$ cm, and $430 \div 43 = 10$ cm.

**Exercise 49.8 ★★.**

A recipe uses $240$ g of chocolate for $8$ servings. Aline has $300$ g of chocolate. For how many servings is that enough (whole number of servings)?

**Solution of Exercise 49.8.**

Chocolate per serving: $240 \div 8 = 30$ g. With $300$ g: $300 \div 30 = 10$ servings exactly.

**Exercise 49.9 ★★.**

The price of a jacket drops from $80$ to $60$ euros.

1. What is the discount in euros? In percent of the original price?
2. Later the price goes back up from $60$ to $80$ euros. Explain why this rise is *not* $25\,\%$ — compute the correct percentage of increase.

**Solution of Exercise 49.9.**

*1.* Discount: $20$ euros, i.e. $\frac{20}{80} = 25\,\%$ of the original price.

*2.* The rise of $20$ euros is now compared with the *new* reference $60$: $\frac{20}{60} = \frac13 \approx 33\,\%$. A percentage always refers to a whole; the whole changed, so the percentage does too.

**Exercise 49.10 ★★.**

Two rows of a table are [proportional](https://one-course.com/books/math/1/en/chapter/44-proportionality-and-data#def-g6-propdata-def):

|  | $4$ | $x$ | $10$ |
| --- | --- | --- | --- |
|  | $y$ | $10.5$ | $17.5$ |

Find the coefficient, then $x$ and $y$.

**Solution of Exercise 49.10.**

Coefficient from the complete column: $\frac{17.5}{10} = 1.75$. Then $y = 4 \times 1.75 = 7$ and $x = \frac{10.5}{1.75} = 6$.

**Exercise 49.11 ★★★.**

A photocopier reduces documents to $80\,\%$ of their size. A [segment](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-objects) of $10$ cm is copied, then the *copy* is copied again with the same setting.

1. How long is the [segment](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-objects) after one copy? After two?
2. Why is the answer after two copies not $60\,\%$ of the original? What single percentage corresponds to two copies?

**Solution of Exercise 49.11.**

*1.* After one copy: $10 \times 0.8 = 8$ cm. After two: $8 \times 0.8 = 6.4$ cm.

*2.* The second reduction applies to the already-reduced copy, not to the original: the factors multiply, $0.8 \times 0.8 = 0.64$, so two copies reduce to $64\,\%$ of the original — not $60\,\%$.

## 49.5 Problem: Measuring the Earth with a stick

**Problem 49.1.**

Weekend problem — shadows are a proportionality, and how Eratosthenes computed the circumference of the Earth in 240 BC

Twenty-two centuries ago, with no telescope, no satellite and no calculator, the librarian of Alexandria measured the Earth — using a stick, a well, a camel caravan and this chapter’s mathematics. This problem retraces his steps: first the [proportionality](#def-g7-prop-table) of shadows, then the famous computation itself, and finally what his answer looks like at human [scale](#def-g7-prop-scale).

**Part I — The shadow of a stick.** The sun is so far away that its rays reach us [parallel](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp) to one another. At a given moment, all vertical objects and their shadows are therefore [proportional](https://one-course.com/books/math/1/en/chapter/44-proportionality-and-data#def-g6-propdata-def).

1. At noon, a vertical stick of $1$ m casts a shadow of $0.4$ m, and a tree’s shadow measures $3.2$ m. How tall is the tree ( [Theorem 49.3](#thm-g7-prop-cross) )?
2. At the same moment, a tower casts a $26$ m shadow: how tall is the tower? And how long is the shadow of a child $1.5$ m tall?
3. Explain in one sentence why all these computations are valid only at the *same moment* of the day.
4. Legend says Thales measured the Great [Pyramid](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) by waiting for the moment when *his own shadow was exactly as long as he was tall* . What is the [proportionality](#def-g7-prop-table) coefficient at that moment, and how tall is the [pyramid](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) if the tip of its shadow lies $147$ m from the point of the ground directly below its apex?
5. Summarize Part I: at one given moment, which quantity is the same for the stick, the tree, the tower and the [pyramid](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) ( [Definition 49.1](#def-g7-prop-table) )?

**Part II — Eratosthenes’ computation.** Eratosthenes knew two facts. In the town of Syene, at noon on the summer solstice, the sun stood *exactly overhead*: sunlight reached the bottom of the deepest wells, and vertical sticks cast no shadow. In Alexandria, $800$ km due north, at the same moment, a vertical stick *did* cast a shadow, and the sun’s rays made an angle of $7.2^\circ$ with the vertical.

6. Explain why these two observations together prove that the ground of Syene and the ground of Alexandria are not [parallel](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp) — that is, the Earth’s surface is curved. (What would [parallel](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp) rays do to two sticks on a flat Earth?)
7. On a drawing of the [round](https://one-course.com/books/math/1/en/chapter/21-large-numbers#def-g4-numbers-round) Earth with [parallel](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp) sun rays, the $7.2^\circ$ in Alexandria reappears at the center of the Earth, as the angle between the directions of the two cities. Make the drawing and convince yourself (the clean justification uses the equal-angle pairs of [Chapter 51](https://one-course.com/books/math/1/en/chapter/51-triangles-and-angles#ch-g7-triangles) ).
8. What [fraction](https://one-course.com/books/math/1/en/chapter/39-fractions-first-steps#def-g6-fractions-def) of a full turn is $7.2^\circ$ ?
9. The arc from Syene to Alexandria ( $800$ km) corresponds to $7.2^\circ$ ; the whole circumference corresponds to $360^\circ$ . Set up the [proportionality](#def-g7-prop-table) and compute the circumference of the Earth.
10. Deduce the [diameter](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-circle) of the Earth, using $\pi \approx  3.14$ ( [Proposition 43.3](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#prop-g6-measure-circle) ) and [rounding](https://one-course.com/books/math/1/en/chapter/38-decimal-numbers#def-g6-decimals-rounding) to the nearest hundred kilometers. (The modern value is $12\,742$ km — how close was a man with a stick in 240 BC?)

**Part III — The Earth at human [scale](#def-g7-prop-scale).**

11. A globe is built at [scale](#def-g7-prop-scale) $1 : 40\,000\,000$ . Using the circumference found in question 9, show that the globe’s circumference is exactly $1$ m. What is its [diameter](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-circle) , to the nearest centimeter?
12. Mont Blanc rises about $4.8$ km. Convert $4.8$ km to centimeters, divide by $40\,000\,000$ , and express the mountain’s height on the globe in millimeters. What does the answer say about how “bumpy” the Earth really is?
13. A walker covers $40$ km per day. At that pace, how many days for Eratosthenes’ full circumference — and roughly how many years is that?
14. The breathable atmosphere is concentrated in roughly the first $10$ km above the ground. What percentage of the Earth’s [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) (about $6\,370$ km) is that ( [Method 49.5](#met-g7-prop-percent) )? Round to the nearest tenth of a percent.
15. Historians estimate that Eratosthenes’ announced value, converted to modern units, may have been about $42\,000$ km. Taking $40\,000$ km as the true value, compute his percentage of error. Conclude in one sentence about sticks, wells and [proportionality](#def-g7-prop-table) .

**Solution of Problem 49.1.**

**1.** Heights and shadows are [proportional](https://one-course.com/books/math/1/en/chapter/44-proportionality-and-data#def-g6-propdata-def), so by the cross rule ([Theorem 49.3](#thm-g7-prop-cross)), with the stick’s column $(1,\ 0.4)$ and the tree’s $(h,\ 3.2)$: $0.4 \times h = 1 \times 3.2$, hence $h = \frac{3.2}{0.4} = 8$ m.

**2.** Tower: $\frac{26}{0.4} = 65$ m. Child’s shadow: $1.5 \times 0.4 = 0.6$ m (shadow $=$ height $\times$ the moment’s coefficient $0.4$).

**3.** As the sun moves across the sky, the coefficient linking heights to shadows changes; only measurements taken at the same moment share the same coefficient.

**4.** At that moment the coefficient is $1$: every height equals its shadow. The [pyramid](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def)’s apex is therefore $147$ m high — the height of the Great [Pyramid](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) (its shadow’s tip, measured from the point below the apex, is all one needs).

**5.** The column [quotient](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $\frac{\text{shadow}}{\text{height}}$ — the [proportionality](#def-g7-prop-table) coefficient of the moment — is common to every vertical object: that is exactly what makes the table of heights and shadows a [proportionality table](#def-g7-prop-table) ([Definition 49.1](#def-g7-prop-table)).

**6.** Sun rays arrive [parallel](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp). On a *flat* Earth, two [parallel](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp) rays would strike two vertical sticks at the same angle: both would cast [proportional](https://one-course.com/books/math/1/en/chapter/44-proportionality-and-data#def-g6-propdata-def) shadows — either both no shadow, or both a shadow. One stick with no shadow (Syene) and one with a shadow (Alexandria), at the same instant, is impossible on a flat Earth: the two verticals must point in different directions — the surface curves.

**7.** On the drawing, the Syene vertical points straight at the sun; the Alexandria vertical is tilted by the angle between the two city directions, seen from the center. Since the rays are [parallel](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp), that tilt is exactly the $7.2^\circ$ measured between ray and stick in Alexandria.

**8.** $\frac{7.2}{360} = \frac{72}{3600} = \frac{1}{50}$: one fiftieth of a full turn.

**9.** Arc lengths are [proportional](https://one-course.com/books/math/1/en/chapter/44-proportionality-and-data#def-g6-propdata-def) to angles: if $7.2^\circ$ — one fiftieth of the turn — corresponds to $800$ km, the full turn corresponds to

$$
50 \times 800 = 40\,000 \text{ km}.
$$

**10.** [Diameter](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-circle) $= \frac{\text{circumference}}{\pi}
\approx \frac{40\,000}{3.14} \approx 12\,700$ km — against the modern $12\,742$ km: correct to well within one percent, with a stick.

**11.** $40\,000$ km $= 4\,000\,000\,000$ cm, and $4\,000\,000\,000 \div 40\,000\,000 = 100$ cm $= 1$ m of circumference. [Diameter](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-circle): $100 \div 3.14 \approx 32$ cm — a handsome desk globe.

**12.** $4.8$ km $= 480\,000$ cm, and $480\,000 \div 40\,000\,000 = 0.012$ cm $= 0.12$ mm. The highest mountain of the Alps is a tenth of a millimeter on a [meter-round](https://one-course.com/books/math/1/en/chapter/21-large-numbers#def-g4-numbers-round) globe: a speck of dust. At this [scale](#def-g7-prop-scale) the Earth is smoother than most polished balls.

**13.** $40\,000 \div 40 = 1\,000$ days, and $1\,000 \div 365 \approx 2.7$: nearly three years of walking.

**14.** $\frac{10}{6\,370} \approx 0.0016$, that is about $0.2\,\%$ (more precisely $0.16\,\%$): the atmosphere is a whisper-thin skin on the planet.

**15.** Error: $42\,000 - 40\,000 = 2\,000$ km, and $\frac{2\,000}{40\,000} = \frac{5}{100} = 5\,\%$. One sentence: with a stick, a well, a measured road and one [proportionality](#def-g7-prop-table), Eratosthenes measured a planet to within a few percent — mathematics travels far on very little.