---
title: "Fractions: First Steps"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 39
exercises: 12
source: https://one-course.com/books/math/1/en/chapter/39-fractions-first-steps
---

# Chapter 39 — Fractions: First Steps

Sharing three pizzas fairly among four people gives each person … not a whole number of pizzas. [Fractions](#def-g6-fractions-def) are the numbers invented for sharing. This chapter builds the picture: a [fraction](#def-g6-fractions-def) as a part of a whole, as a point on the number line, and as the exact result of a division.

## 39.1 What a fraction means

**Definition 39.1 (Fraction).**

Cut a unit into $b$ equal parts, and take $a$ of them: the quantity obtained is the *fraction*

$$
\frac{a}{b}
\qquad
\begin{array}{l}
a \text{: the \emph{numerator} --- how many parts we take;}\\
b \text{: the \emph{denominator} --- into how many parts the unit is cut.}
\end{array}
$$

![Two fractions of the same bar. The denominator tells how fine the cutting is, the numerator how much we take.](https://one-course.com/images/onecourse/chapters/math-1/g6-fractions/fig-fa1daf41ee45.svg)

*Two [fractions](#def-g6-fractions-def) of the same bar. The [denominator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) tells how fine the cutting is, the [numerator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) how much we take.*

**Example 39.2.**

A [fraction](#def-g6-fractions-def) can be bigger than $1$: $\frac74$ means seven [quarters](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) — one whole unit (four [quarters](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half)) and three [quarters](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) more:

$$
\frac74 = 1 + \frac34 .
$$

**Proposition 39.3 (Fractions on the number line).**

To place $\frac ab$ on the number line, cut each unit interval into $b$ equal parts and count $a$ parts from $0$.

![The line graduated in quarters: 3/4 is before 1, 7/4 is between 1 and 2, and 22/4 = 5 + 2/4 is halfway between 5 and 6.](https://one-course.com/images/onecourse/chapters/math-1/g6-fractions/fig-b24e42c46fdc.svg)

*The line graduated in [quarters](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half): $\frac34$ is before $1$, $\frac74$ is between $1$ and $2$, and $\frac{22}{4} = 5 + \frac24$ is halfway between $5$ and $6$.*

**Theorem 39.4 (A fraction is a division).**

The [fraction](#def-g6-fractions-def) $\frac ab$ is the number which, multiplied by $b$, gives $a$:

$$
\frac ab \times b = a .
$$

In other words, $\frac ab$ is the exact [quotient](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) of $a$ by $b$ — even when the division “does not come out even”.

**Proof.** *Admitted at this level.* ∎

**Example 39.5.**

Three pizzas for four people: each person gets $\frac34$ of a pizza. Check with the theorem: $4$ shares of $\frac34$ make $\frac34 \times 4 = 3$ pizzas. Some [fractions](#def-g6-fractions-def) are [decimal numbers](https://one-course.com/books/math/1/en/chapter/38-decimal-numbers#def-g6-decimals-places) ($\frac34 = 0.75$), others are not: $\frac13 = 0.333\dots$ never stops. The [fraction](#def-g6-fractions-def) is the *exact* value; $0.33$ is only an approximation.

## 39.2 Equal fractions

**Proposition 39.6 (Equivalent fractions).**

A [fraction](#def-g6-fractions-def) does not change when its [numerator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) and [denominator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) are both multiplied (or divided) by the same nonzero number:

$$
\frac{a}{b} = \frac{a \times k}{b \times k} .
$$

**Idea of proof.** Cutting each part of the sharing into $k$ smaller pieces multiplies both the number of pieces taken and the total number of pieces by $k$, but the quantity taken is the same. A picture says it best: ∎

![2/3 = 4/6: cutting each third in two doubles the counts, top and bottom, without changing the shaded amount.](https://one-course.com/images/onecourse/chapters/math-1/g6-fractions/fig-1cfba25eec39.svg)

*$\frac23 = \frac46$: cutting each third in two doubles the counts, top and bottom, without changing the shaded amount.*

**Method 39.7 (Simplifying a fraction).**

Look for a number dividing the [numerator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) *and* the [denominator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def), and divide both. Repeat until no common [divisor](https://one-course.com/books/math/1/en/chapter/32-division-and-multiples#def-g5-division-multiple) remains:

$$
\frac{12}{18} = \frac{12 \div 2}{18 \div 2} = \frac{6}{9}
= \frac{6 \div 3}{9 \div 3} = \frac{2}{3}.
$$

## 39.3 Taking a fraction of a quantity

**Method 39.8 (Fraction of a quantity).**

To compute $\frac ab$ of a quantity, divide the quantity by $b$ (one part), then multiply by $a$ (the number of parts):

$$
\frac ab \text{ of } Q = (Q \div b) \times a .
$$

(Multiplying first by $a$ and then dividing by $b$ gives the same result — choose the order that makes the numbers nicer.)

**Example 39.9.**

Compute $\frac35$ of $40$ euros, step by step:

1. one fifth of $40$ : $40 \div 5 = 8$ ;
2. three fifths: $8 \times 3 = 24$ .

So $\frac35$ of $40$ euros is $24$ euros. Check: the remaining $\frac25$ is $16$ euros, and $24 + 16 = 40$.

## 39.4 Exercises

**Exercise 39.1 ★.**

For each picture description, write the [fraction](#def-g6-fractions-def): a cake cut in $8$ slices, $3$ eaten; a chocolate bar of $12$ squares, $7$ eaten; a pie cut in $6$, all $6$ eaten.

**Solution of Exercise 39.1.**

Cake: $\frac38$ eaten. Chocolate: $\frac{7}{12}$. Pie: $\frac66 = 1$ (the whole pie).

**Exercise 39.2 ★.**

Draw a bar cut into $5$ equal parts and color $\frac45$ of it. Is $\frac45$ smaller or larger than $1$? And $\frac65$?

**Solution of Exercise 39.2.**

Four parts out of five are colored. $\frac45 < 1$ (four parts out of five is less than the whole bar); $\frac65 > 1$ (six fifths is one whole bar and one fifth more).

**Exercise 39.3 ★.**

Place on a number line graduated in thirds: $\frac13$; $\frac53$; $2$; $\frac73$. Which two of these numbers are equal distances from $2$?

**Solution of Exercise 39.3.**

On the line graduated in thirds: $\frac13$ is one tick after $0$; $\frac53$ is one tick before $2$ (since $2 = \frac63$); $\frac73$ is one tick after $2$. So $\frac53$ and $\frac73$ are both at distance $\frac13$ from $2$.

**Exercise 39.4 ★.**

Write each [fraction](#def-g6-fractions-def) as a whole number plus a [fraction](#def-g6-fractions-def) smaller than $1$ (as in $\frac74 = 1 + \frac34$):

$$
\frac{9}{4}, \qquad \frac{13}{5}, \qquad \frac{12}{6}, \qquad
\frac{25}{8} .
$$

**Solution of Exercise 39.4.**

$\frac94 = 2 + \frac14$; $\frac{13}{5} = 2 + \frac35$; $\frac{12}{6} = 2$ (exactly); $\frac{25}{8} = 3 + \frac18$.

**Exercise 39.5 ★.**

Complete so the [fractions](#def-g6-fractions-def) are equal:

$$
\frac{2}{3} = \frac{?}{12}, \qquad
\frac{15}{20} = \frac{3}{?}, \qquad
\frac{7}{10} = \frac{21}{?}, \qquad
\frac{?}{6} = \frac{10}{12}.
$$

**Solution of Exercise 39.5.**

$\frac23 = \frac{8}{12}$ (multiply by $4$); $\frac{15}{20} = \frac34$ (divide by $5$); $\frac{7}{10} = \frac{21}{30}$ (multiply by $3$); $\frac56 = \frac{10}{12}$ (divide by $2$).

**Exercise 39.6 ★.**

Simplify as much as possible: $\dfrac{6}{8}$; $\dfrac{15}{25}$; $\dfrac{18}{24}$; $\dfrac{35}{7}$.

**Solution of Exercise 39.6.**

$\frac68 = \frac34$; $\frac{15}{25} = \frac35$; $\frac{18}{24} = \frac{9}{12} = \frac34$; $\frac{35}{7} = 5$ (a whole number).

**Exercise 39.7 ★.**

Compute: $\frac12$ of $86$; $\frac34$ of $60$; $\frac25$ of $35$; $\frac{7}{10}$ of $250$.

**Solution of Exercise 39.7.**

$\frac12$ of $86$: $86 \div 2 = 43$.

$\frac34$ of $60$: $60 \div 4 = 15$, then $15 \times 3 = 45$.

$\frac25$ of $35$: $35 \div 5 = 7$, then $7 \times 2 = 14$.

$\frac{7}{10}$ of $250$: $250 \div 10 = 25$, then $25 \times 7 = 175$.

**Exercise 39.8 ★.**

Which is larger: $\frac12$ or $\frac13$ (of the same cake)? $\frac25$ or $\frac35$? $\frac34$ or $\frac78$ (write both with [denominator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) $8$)?

**Solution of Exercise 39.8.**

$\frac12 > \frac13$: halves are bigger parts than thirds. $\frac35 > \frac25$: same parts, more of them. $\frac34 = \frac68 < \frac78$: seven eighths beat six eighths.

**Exercise 39.9 ★★.**

Give the decimal value of the [fractions](#def-g6-fractions-def) that have one, and say which [fraction](#def-g6-fractions-def) does not: $\frac12$; $\frac34$; $\frac13$; $\frac{7}{10}$; $\frac{9}{4}$.

**Solution of Exercise 39.9.**

$\frac12 = 0.5$; $\frac34 = 0.75$; $\frac{7}{10} = 0.7$; $\frac94 = 2.25$. The [odd](https://one-course.com/books/math/1/en/chapter/14-numbers-up-to-10-000#def-g3-numbers-evenodd) one out is $\frac13 = 0.333\dots$, whose decimal writing never ends: it is not a [decimal number](https://one-course.com/books/math/1/en/chapter/38-decimal-numbers#def-g6-decimals-places).

**Exercise 39.10 ★★.**

A class has $28$ students. Three [quarters](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of them come to school by bus, and the rest walk. How many students walk? Solve step by step, and check that your two group sizes add up to $28$.

**Solution of Exercise 39.10.**

Bus users: $\frac34$ of $28$: $28 \div 4 = 7$, then $7 \times 3 = 21$ students. Walkers: $28 - 21 = 7$ students (that is the remaining [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half): $28 \div 4 = 7$). Check: $21 + 7 = 28$.

**Exercise 39.11 ★★.**

Leo spent $\frac23$ of his savings on a game that cost $18$ euros. How much money did Leo have before? (One third of his savings is [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of $18$ … think about why, or find the size of one part first.)

**Solution of Exercise 39.11.**

Two thirds of the savings are $18$ euros, so *one* third is $18 \div 2 = 9$ euros, and the whole savings (three thirds) were $9 \times 3 = 27$ euros. Check: $\frac23$ of $27$ is $18$.

**Exercise 39.12 ★★★.**

A tank is filled to $\frac58$ of its [capacity](https://one-course.com/books/math/1/en/chapter/12-money-and-measures#def-g2-measure-units). After using $10$ liters, it is [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) full. What is the [capacity](https://one-course.com/books/math/1/en/chapter/12-money-and-measures#def-g2-measure-units) of the tank? (Which [fraction](#def-g6-fractions-def) of the tank do the $10$ liters represent?)

**Solution of Exercise 39.12.**

The used water went from $\frac58$ to $\frac48$ (a [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) is four eighths) of the tank: the $10$ liters represent $\frac58 - \frac48 = \frac18$ of the [capacity](https://one-course.com/books/math/1/en/chapter/12-money-and-measures#def-g2-measure-units). So the full tank holds $10 \times 8 = 80$ liters. Check: $\frac58$ of $80$ is $50$, minus $10$ leaves $40$, which is indeed [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of $80$.

## 39.5 Problem: The chocolate bar you can never finish

**Problem 39.1.**

Weekend problem — halves of halves: the sum $\frac12 + \frac14 + \frac18 + \dots$ creeps up to $1$ without ever touching it

Take a chocolate bar. Eat [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of it. Then eat [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of *what is left*. Then again [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of what is left, and so on. Two things seem to be true at once: you never finish the bar (something always remains), and yet you eat almost all of it. This problem turns both feelings into exact statements about [fractions](#def-g6-fractions-def) — and meets, on the way, a runner from ancient Greece and a sharing puzzle solved entirely with a knife that only cuts things in [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half).

**Part I — Halves of halves.** The bar is a rectangle of $16$ equal squares.

1. First bite: you eat [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) the bar. How many squares is that? Write the bite as a [fraction](#def-g6-fractions-def) of the bar in two ways, with [denominator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) $2$ and with [denominator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) $16$ ( [Proposition 39.6](#prop-g6-fractions-equal) ).
2. Second bite: [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of what is left. How many squares? Show, with the picture or with equivalent [fractions](#def-g6-fractions-def) , that this bite is $\frac14$ of the whole bar: *[half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of a [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) is a [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half)* .
3. Third and fourth bites, same rule: give each one in squares and as a [fraction](#def-g6-fractions-def) of the bar. What happens to the [denominator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) from one bite to the next?
4. After the four bites, how many squares have been eaten in all? What [fraction](#def-g6-fractions-def) of the bar is that, and what [fraction](#def-g6-fractions-def) is left?
5. Question 4 says, written in sixteenths: $$\frac{8}{16} + \frac{4}{16} + \frac{2}{16} + \frac{1}{16}  = \frac{15}{16},  \qquad\text{that is}\qquad  \frac12 + \frac14 + \frac18 + \frac{1}{16}  = \frac{15}{16} .$$ Check each of the four rewritings ($\frac12 = \frac{8}{16}$, and so on).

**Part II — The staircase towards $1$.** To keep biting, imagine a finer bar of $64$ squares.

6. List the sizes of the first six bites, in squares ( $32$ , then …), and check that after the sixth bite exactly one square is left. What [fraction](#def-g6-fractions-def) of the bar remains?
7. After each bite, what [fraction](#def-g6-fractions-def) of the bar remains? Write the list ( $\frac12$ , $\frac14$ , …) down to the sixth bite, and describe the pattern. If an imaginary knife could halve [even](https://one-course.com/books/math/1/en/chapter/14-numbers-up-to-10-000#def-g3-numbers-evenodd) the last square, what would remain after a seventh bite?
8. Explain why the eaten total can *never* reach $1$ , no matter how many bites are taken. (Compare with the twenty nines of [Problem 38.1](https://one-course.com/books/math/1/en/chapter/38-decimal-numbers#pb-g6-decimals-1) : same story?)
9. Yet the total passes any target below $1$ . A friend challenges you to eat more than $\frac{99}{100}$ of the bar. Which is smaller, $\frac{1}{128}$ or $\frac{1}{100}$ — and after which bite is the challenge won?
10. On a number line from $0$ to $1$ graduated in sixteenths, place the eaten totals after each of the first four bites: $\frac12$ , $\frac34$ , $\frac78$ , $\frac{15}{16}$ . Where does each new total land, compared with the previous one and with $1$ ?

**Part III — Zeno’s runner, and a knife that only halves.**

11. Twenty-four centuries ago, the Greek philosopher Zeno told this story: to reach a wall, a runner must first cover [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) the distance, then [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of what remains, then [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of what remains again … “so the runner never reaches the wall!” Using questions 8 and 9, say what is *right* in Zeno’s story, and where the trap is. (Does the runner really need forever to make all those steps?)
12. The sharing puzzle: share $15$ identical chocolate bars fairly among $16$ children, using only cuts into halves (of a bar, of a half-bar, of a quarter-bar …). Describe how to cut: how many bars are cut into halves, how many into [quarters](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) , into eighths, into sixteenths, so that every child receives one piece of each size.
13. Check your sharing: write each child’s share as a [sum](https://one-course.com/books/math/1/en/chapter/2-addition-first-steps#def-g1-addition-def) of [fractions](#def-g6-fractions-def) , compute it in sixteenths, and verify that the $16$ shares together use up exactly the $15$ bars.
14. In the class next door, $7$ bars are shared fairly among $8$ children. Which child receives more chocolate — one of the $16$ , or one of the $8$ ? (Write both shares with [denominator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) $16$ , [Exercise 39.8](#exo-g6-fractions-8) .)
15. The finale: a friend claims that with *infinitely* many bites one would eat *exactly* the whole bar. Draw a square and shade the bites inside it: [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) the square, then a [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) , then an eighth … What do you observe about the unshaded corner? State carefully the two true facts of this whole problem: what remains after every finite number of bites, and how small that [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) becomes. (What an “infinite [sum](https://one-course.com/books/math/1/en/chapter/2-addition-first-steps#def-g1-addition-def) ” exactly means is a story for the High School volume.)

**Solution of Problem 39.1.**

**1.** [Half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of $16$ squares is $8$ squares. As a [fraction](#def-g6-fractions-def): $\frac12$ of the bar, and since each [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) is $8$ squares out of $16$, also $\frac{8}{16}$ — the same [fraction](#def-g6-fractions-def) written two ways ([Proposition 39.6](#prop-g6-fractions-equal)).

**2.** [Half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of the $8$ remaining squares is $4$ squares, which is $\frac{4}{16}$ of the bar. Simplifying by $4$: $\frac{4}{16} = \frac14$. On the picture: cutting the remaining [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) in two produces two [quarters](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of the original bar — [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of a [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) is a [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half).

**3.** Third bite: [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of $4$ squares is $2$ squares, i.e. $\frac{2}{16} = \frac18$ of the bar. Fourth bite: $1$ square, i.e. $\frac{1}{16}$. From one bite to the next, the [denominator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) *doubles*: $2$, $4$, $8$, $16$.

**4.** Eaten: $8 + 4 + 2 + 1 = 15$ squares out of $16$, that is $\frac{15}{16}$ of the bar. Left: $1$ square, $\frac{1}{16}$.

**5.** $\frac12 = \frac{8}{16}$ (multiply top and bottom by $8$), $\frac14 = \frac{4}{16}$ (by $4$), $\frac18 = \frac{2}{16}$ (by $2$), and $\frac{1}{16}$ stays. Counting sixteenths: $8 + 4 + 2 + 1 = 15$ of them, so the total is $\frac{15}{16}$.

**6.** Bites: $32$, $16$, $8$, $4$, $2$, $1$ squares. Total eaten: $32 + 16 + 8 + 4 + 2 + 1 = 63$ squares, so $1$ square remains: $\frac{1}{64}$ of the bar.

**7.** After each bite there remains

$$
\frac12,\quad \frac14,\quad \frac18,\quad \frac{1}{16},\quad
\frac{1}{32},\quad \frac{1}{64} :
$$

the [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) halves each time — its [denominator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) doubles. A seventh, imaginary bite would leave $\frac{1}{128}$.

**8.** Every bite eats only *[half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half)* of what remains, so the other [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of the [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) is still there: after any number of bites, something is always left. The eaten total is therefore always below $1$ — exactly like the twenty nines of [Problem 38.1](https://one-course.com/books/math/1/en/chapter/38-decimal-numbers#pb-g6-decimals-1), which come ever closer to $3$ without reaching it.

**9.** Both [fractions](#def-g6-fractions-def) have [numerator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) $1$, and cutting into $128$ parts makes smaller parts than cutting into $100$: so $\frac{1}{128} < \frac{1}{100}$. After the seventh bite the [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) is $\frac{1}{128}$, so the eaten part exceeds $1 - \frac{1}{100} = \frac{99}{100}$: the challenge is won at bite seven.

**10.** In sixteenths, the totals are $\frac{8}{16}$, $\frac{12}{16}$, $\frac{14}{16}$, $\frac{15}{16}$. Each new total lands exactly *halfway* between the previous total and $1$: the staircase keeps halving its remaining distance to $1$, without ever stepping on it.

**11.** What is right: at every stage of Zeno’s description, some distance indeed remains (question 8) — the list of stages never ends. The trap: the stages become extremely short, in distance *and* in running time; the runner does not spend equal time on each stage. Describing the run in infinitely many shrinking pieces does not make the run itself endless: the runner reaches the wall, and question 9 shows the description passing any point short of it.

**12.** Cut $8$ bars into halves: $16$ half-pieces. Cut $4$ bars into [quarters](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half): $16$ quarter-pieces. Cut $2$ bars into eighths: $16$ eighth-pieces. Cut the last bar into sixteenths: $16$ small pieces. That uses $8 + 4 + 2 + 1 = 15$ bars, and each child receives one piece of each size.

**13.** Each child’s share is

$$
\frac12 + \frac14 + \frac18 + \frac{1}{16} = \frac{15}{16}
$$

of a bar (question 5). Sixteen such shares make $16 \times \frac{15}{16} = 15$ bars ([Theorem 39.4](#thm-g6-fractions-quotient)): exactly what was cut, with nothing left over — a fair sharing of $15$ bars among $16$ children, by halving alone.

**14.** The neighbours receive $\frac78$ of a bar each. With [denominator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) $16$: $\frac78 = \frac{14}{16}$, while our children receive $\frac{15}{16}$. So a child of the $16$ receives more: $\frac{15}{16} > \frac{14}{16}$, by one sixteenth of a bar.

**15.** In the square picture, each shaded bite fills [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of the still-unshaded corner, and the unshaded corner keeps shrinking: after each bite it is [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) as large as before, and it never disappears. The two true facts: (i) after every finite number of bites, a [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) is left — the total eaten is always strictly below $1$; (ii) that [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) becomes smaller than any [fraction](#def-g6-fractions-def) one cares to name ($\frac{1}{100}$, $\frac{1}{1000}$, …), if one bites long enough. Giving the words “the infinite [sum](https://one-course.com/books/math/1/en/chapter/2-addition-first-steps#def-g1-addition-def) equals $1$” their exact meaning is the job of *limits*, in the High School volume.