---
title: "Fractions: Comparing and Adding"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 47
exercises: 12
source: https://one-course.com/books/math/1/en/chapter/47-fractions-comparing-and-adding
---

# Chapter 47 — Fractions: Comparing and Adding

Grade 6 introduced [fractions](https://one-course.com/books/math/1/en/chapter/39-fractions-first-steps#def-g6-fractions-def) as shares and as exact [quotients](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) ([Chapter 39](https://one-course.com/books/math/1/en/chapter/39-fractions-first-steps#ch-g6-fractions)). This chapter learns to *compute* with them: recognizing equal [fractions](https://one-course.com/books/math/1/en/chapter/39-fractions-first-steps#def-g6-fractions-def), comparing, adding and subtracting — for now with friendly [denominators](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def); the general case comes in [Chapter 63](https://one-course.com/books/math/1/en/chapter/63-fractions-and-powers#ch-g9-fractions).

## 47.1 Equal fractions

**Theorem 47.1 (Equal fractions).**

A [fraction](https://one-course.com/books/math/1/en/chapter/39-fractions-first-steps#def-g6-fractions-def) is unchanged 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 multiplied, or divided, by the same nonzero number:

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

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

**Example 47.2.**

$\dfrac{3}{4} = \dfrac{3 \times 5}{4 \times 5} = \dfrac{15}{20}$, and in the other direction $\dfrac{42}{30} = \dfrac{42 \div 6}{30 \div 6}
= \dfrac{7}{5}$ (simplified by $6$).

To test whether $\dfrac{8}{12}$ and $\dfrac{10}{15}$ are equal, simplify both: $\dfrac{8}{12} = \dfrac{2}{3}$ and $\dfrac{10}{15} = \dfrac{2}{3}$ — yes, equal.

## 47.2 Comparing fractions

**Method 47.3 (Comparing two fractions).**

1. *Same [denominator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def)* : compare the [numerators](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) : $\frac{5}{9} < \frac{7}{9}$ .
2. *One [denominator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) is a [multiple](https://one-course.com/books/math/1/en/chapter/32-division-and-multiples#def-g5-division-multiple) of the other* : rewrite the coarser [fraction](https://one-course.com/books/math/1/en/chapter/39-fractions-first-steps#def-g6-fractions-def) , then compare: $\frac{5}{6}$ vs $\frac{7}{12}$ : $\frac56 = \frac{10}{12} >  \frac{7}{12}$ .
3. *Compare with $1$ or $\frac12$* when possible: $\frac{9}{8} > 1 > \frac{7}{9}$ settles $\frac98$ vs $\frac79$ instantly.

![Comparing 5/6 and 7/12 on twin bars: cutting the sixths in half shows 5/6 = 10/12, more than 7/12.](https://one-course.com/images/onecourse/chapters/math-1/g7-fractions/fig-342a23a205ce.svg)

*Comparing $\frac56$ and $\frac{7}{12}$ on twin bars: cutting the sixths in [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) shows $\frac56 = \frac{10}{12}$, more than $\frac{7}{12}$.*

## 47.3 Adding and subtracting

**Theorem 47.4 (Same denominator).**

[Fractions](https://one-course.com/books/math/1/en/chapter/39-fractions-first-steps#def-g6-fractions-def) with the same [denominator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) are added (or subtracted) by adding (or subtracting) the [numerators](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def):

$$
\frac{a}{d} + \frac{b}{d} = \frac{a + b}{d},
\qquad
\frac{a}{d} - \frac{b}{d} = \frac{a - b}{d} .
$$

**Why.** $a$ parts of size $\frac1d$ plus $b$ parts of the same size make $a + b$ parts of that size. ∎

**Method 47.5 (Different denominators).**

When one [denominator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) is a [multiple](https://one-course.com/books/math/1/en/chapter/32-division-and-multiples#def-g5-division-multiple) of the other:

1. rewrite the [fraction](https://one-course.com/books/math/1/en/chapter/39-fractions-first-steps#def-g6-fractions-def) with the smaller [denominator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) so both have the larger one ( [Theorem 47.1](#thm-g7-fractions-equal) );
2. add or subtract the [numerators](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) ;
3. simplify the result if possible.

**Example 47.6.**

Compute $\dfrac{3}{4} + \dfrac{5}{8}$, step by step:

1. $8$ is a [multiple](https://one-course.com/books/math/1/en/chapter/32-division-and-multiples#def-g5-division-multiple) of $4$ : rewrite $\dfrac34 = \dfrac{3 \times 2}{4 \times 2} = \dfrac{6}{8}$ ;
2. add: $\dfrac{6}{8} + \dfrac{5}{8} = \dfrac{11}{8}$ ;
3. nothing to simplify: the result is $\dfrac{11}{8}$ , i.e. $1 + \dfrac38$ .

Another: $2 - \dfrac{4}{5} = \dfrac{10}{5} - \dfrac{4}{5} =
\dfrac{6}{5}$ (write the whole number over the [denominator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) $5$ first).

**Example 47.7 (A classic trap).**

$\dfrac12 + \dfrac13$ is *not* $\dfrac{2}{5}$! Adding [numerators](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) and [denominators](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) separately is wrong — the answer $\frac25$ would be smaller than $\frac12$, absurd when *adding* something positive to $\frac12$. (The correct [sum](https://one-course.com/books/math/1/en/chapter/2-addition-first-steps#def-g1-addition-def), $\frac56$, needs a common [denominator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) $6$; the general method is in [Chapter 63](https://one-course.com/books/math/1/en/chapter/63-fractions-and-powers#ch-g9-fractions).)

## 47.4 Multiplying a fraction by a number

**Proposition 47.8 (Fraction times a number).**

For a number $k$:

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

Taking $\frac ab$ *of* a quantity means multiplying by $\frac ab$.

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

**Example 47.9.**

$6 \times \dfrac{2}{3} = \dfrac{12}{3} = 4$: six times two thirds is four wholes. And “$\frac34$ of $60$ euros” is $\frac34 \times 60 = \frac{180}{4} = 45$ euros — the same answer as dividing by $4$ then multiplying by $3$ ([Method 39.8](https://one-course.com/books/math/1/en/chapter/39-fractions-first-steps#met-g6-fractions-ofquantity)).

## 47.5 Exercises

**Exercise 47.1 ★.**

Complete: $\dfrac{2}{5} = \dfrac{?}{20}$; $\dfrac{18}{24} = \dfrac{3}{?}$; $\dfrac{7}{3} = \dfrac{28}{?}$; $\dfrac{?}{9} = \dfrac{20}{36}$.

**Solution of Exercise 47.1.**

$\dfrac{2}{5} = \dfrac{8}{20}$; $\dfrac{18}{24} = \dfrac{3}{4}$; $\dfrac{7}{3} = \dfrac{28}{12}$; $\dfrac{5}{9} = \dfrac{20}{36}$.

**Exercise 47.2 ★.**

Simplify as much as possible: $\dfrac{12}{16}$; $\dfrac{30}{45}$; $\dfrac{27}{36}$; $\dfrac{48}{12}$.

**Solution of Exercise 47.2.**

$\dfrac{12}{16} = \dfrac34$; $\dfrac{30}{45} = \dfrac23$; $\dfrac{27}{36} = \dfrac34$; $\dfrac{48}{12} = 4$.

**Exercise 47.3 ★.**

Are $\dfrac{15}{35}$ and $\dfrac{21}{49}$ equal? And $\dfrac{16}{28}$ and $\dfrac{20}{36}$? Justify by simplifying.

**Solution of Exercise 47.3.**

$\dfrac{15}{35} = \dfrac37$ and $\dfrac{21}{49} = \dfrac37$: equal.

$\dfrac{16}{28} = \dfrac47$ and $\dfrac{20}{36} = \dfrac59$: to compare, $\frac47 = \frac{36}{63}$ and $\frac59 = \frac{35}{63}$ — not equal.

**Exercise 47.4 ★.**

Compare (write the reasoning, not just the answer):

$$
\frac{7}{11} \text{ and } \frac{9}{11}; \qquad
\frac{5}{6} \text{ and } \frac{11}{18}; \qquad
\frac{13}{12} \text{ and } \frac{19}{20}.
$$

**Solution of Exercise 47.4.**

$\dfrac{7}{11} < \dfrac{9}{11}$ (same [denominator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def)).

$\dfrac56 = \dfrac{15}{18} > \dfrac{11}{18}$.

$\dfrac{13}{12} > 1$ while $\dfrac{19}{20} < 1$, so $\dfrac{13}{12} > \dfrac{19}{20}$.

**Exercise 47.5 ★.**

Compute and simplify if possible:

$$
\frac{5}{9} + \frac{2}{9}, \qquad
\frac{11}{7} - \frac{4}{7}, \qquad
\frac{5}{8} + \frac{7}{8} .
$$

**Solution of Exercise 47.5.**

$\dfrac59 + \dfrac29 = \dfrac79$; $\dfrac{11}{7} - \dfrac47 = \dfrac77 = 1$; $\dfrac58 + \dfrac78 = \dfrac{12}{8} = \dfrac32$.

**Exercise 47.6 ★.**

Compute using [Method 47.5](#met-g7-fractions-add):

$$
\frac{2}{3} + \frac{5}{6}, \qquad
\frac{7}{10} - \frac{2}{5}, \qquad
\frac{3}{4} + \frac{5}{12} .
$$

**Solution of Exercise 47.6.**

$\dfrac23 + \dfrac56 = \dfrac46 + \dfrac56 = \dfrac96 = \dfrac32$.

$\dfrac{7}{10} - \dfrac25 = \dfrac{7}{10} - \dfrac{4}{10} =
\dfrac{3}{10}$.

$\dfrac34 + \dfrac{5}{12} = \dfrac{9}{12} + \dfrac{5}{12} =
\dfrac{14}{12} = \dfrac76$.

**Exercise 47.7 ★.**

Compute:

$$
3 - \frac{5}{4}, \qquad
1 + \frac{3}{8}, \qquad
2 - \frac{7}{6} .
$$

**Solution of Exercise 47.7.**

$3 - \dfrac54 = \dfrac{12}{4} - \dfrac54 = \dfrac74$; $1 + \dfrac38 = \dfrac{11}{8}$; $2 - \dfrac76 = \dfrac{12}{6} - \dfrac76 = \dfrac56$.

**Exercise 47.8 ★.**

Compute:

$$
5 \times \frac{3}{10}, \qquad
\frac{7}{8} \times 4, \qquad
\frac{2}{3} \text{ of } 45 .
$$

**Solution of Exercise 47.8.**

$5 \times \dfrac{3}{10} = \dfrac{15}{10} = \dfrac32$; $\dfrac78 \times 4 = \dfrac{28}{8} = \dfrac72$; $\dfrac23$ of $45$ is $\dfrac{2 \times 45}{3} = \dfrac{90}{3} = 30$.

**Exercise 47.9 ★★.**

Marc ate $\frac14$ of a pizza and Julie $\frac38$ of the same pizza. What [fraction](https://one-course.com/books/math/1/en/chapter/39-fractions-first-steps#def-g6-fractions-def) of the pizza did they eat together? What [fraction](https://one-course.com/books/math/1/en/chapter/39-fractions-first-steps#def-g6-fractions-def) is left?

**Solution of Exercise 47.9.**

Together: $\dfrac14 + \dfrac38 = \dfrac28 + \dfrac38 = \dfrac58$ of the pizza. Left: $1 - \dfrac58 = \dfrac38$.

**Exercise 47.10 ★★.**

A bottle contains $\frac34$ L of juice. Lea pours out $\frac16$ L twice. How much juice remains? (Common [denominator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def): $12$.)

**Solution of Exercise 47.10.**

Poured out: $2 \times \dfrac16 = \dfrac26 = \dfrac13$ L. Remaining:

$$
\frac34 - \frac13 = \frac{9}{12} - \frac{4}{12} = \frac{5}{12}
\text{ L}.
$$

**Exercise 47.11 ★★.**

Explain, with the argument of [Example 47.7](#ex-g7-fractions-trap), why $\frac35 + \frac12$ cannot be $\frac{4}{7}$, without computing the correct [sum](https://one-course.com/books/math/1/en/chapter/2-addition-first-steps#def-g1-addition-def).

**Solution of Exercise 47.11.**

Adding the positive quantity $\frac12$ to $\frac35$ must give a result *larger* than $\frac35$. But $\frac47 < \frac35$ (indeed $\frac47 = \frac{20}{35}$ and $\frac35 = \frac{21}{35}$): the claimed [sum](https://one-course.com/books/math/1/en/chapter/2-addition-first-steps#def-g1-addition-def) is smaller than one of its terms — impossible.

**Exercise 47.12 ★★★.**

Compute the [sum](https://one-course.com/books/math/1/en/chapter/2-addition-first-steps#def-g1-addition-def)

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

step by step (left to right). Observe the pattern of the partial [sums](https://one-course.com/books/math/1/en/chapter/2-addition-first-steps#def-g1-addition-def): what number do they approach as more and more terms $\frac{1}{32}, \frac{1}{64}, \dots$ are added?

**Solution of Exercise 47.12.**

Step by step:

$$
\frac12 + \frac14 = \frac34, \qquad
\frac34 + \frac18 = \frac68 + \frac18 = \frac78, \qquad
\frac78 + \frac1{16} = \frac{14}{16} + \frac1{16} = \frac{15}{16}.
$$

The partial [sums](https://one-course.com/books/math/1/en/chapter/2-addition-first-steps#def-g1-addition-def) $\frac12, \frac34, \frac78, \frac{15}{16}, \dots$ are each time half-way closer to $1$: at every step, the missing part is cut in [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half). The [sums](https://one-course.com/books/math/1/en/chapter/2-addition-first-steps#def-g1-addition-def) approach $1$ without ever reaching it.

## 47.6 Problem: The fractions hidden in a bar of music

**Problem 47.1.**

Weekend problem — note values are fractions that must add up exactly, rhythm counting hides a famous sequence, and parity rules the Balkan beat

Music is written in [fractions](https://one-course.com/books/math/1/en/chapter/39-fractions-first-steps#def-g6-fractions-def). A *whole note* lasts $1$ measure (in the common “four-four” time); a *[half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) note* lasts $\frac12$; then come the *[quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) note* $\frac14$, the *eighth note* $\frac18$ and the *sixteenth note* $\frac{1}{16}$. One inviolable law: *the durations inside a measure must add up to exactly the measure’s total* — $1$ in four-four time. Every question below is this chapter’s arithmetic ([Theorem 47.4](#thm-g7-fractions-add), [Method 47.5](#met-g7-fractions-add)) set to music; on the way you will meet a counting sequence discovered by Indian scholars a thousand years ago.

**Part I — Notes that must add up.**

1. Check that each of these measures is legal in four-four time: (a) [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) $+$ [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) $+$ [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) ; (b) [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) $+$ eighth $+$ eighth $+$ [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) .
2. A copyist wrote: [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) $+$ [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) $+$ eighth. What [fraction](https://one-course.com/books/math/1/en/chapter/39-fractions-first-steps#def-g6-fractions-def) of the measure is written, and which single note completes it?
3. How many sixteenth notes fill one whole measure? How many sixteenths make one eighth note?
4. A *dot* after a note adds [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) the note’s value: a dotted [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) is $\frac12 + \frac14$ , a dotted [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) is $\frac14 + \frac18$ . Compute both values.
5. A waltz is written in three-four time: each measure must total $\frac34$ . Check that a single dotted [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) fills a waltz measure, and write down two other legal waltz measures using only [quarters](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) and eighths.

**Part II — The copyist’s workshop.**

6. A four-four measure contains: dotted [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) $+$ eighth $+$ [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) . What is missing?
7. Which lasts longer, a dotted eighth or a [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) note? Compare via sixteenths.
8. A *tie* glues durations together into one long sound: a [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) note tied to an eighth lasts how long? How many sixteenths is that?
9. The notes of a measure add up to $\frac{11}{16}$ ; the [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) is silence (a *rest* ). How long is the rest?
10. A drummer fills one [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) note ( $\frac{4}{16}$ of the measure) using only eighths ( $\frac{2}{16}$ ) and sixteenths ( $\frac{1}{16}$ ), order mattering: for instance eighth–sixteenth–sixteenth, or sixteenth–eighth–sixteenth. Check that both examples are legal, then list *all* the possible rhythms. How many are there?

**Part III — A thousand-year-old sequence, and an [odd](https://one-course.com/books/math/1/en/chapter/14-numbers-up-to-10-000#def-g3-numbers-evenodd) measure.**

11. Let us count the drummer’s rhythms for longer notes. Every rhythm ends either with a sixteenth or with an eighth; explain why this gives the rule: (rhythms filling $n$ sixteenths) $=$ (rhythms filling $n - 1$ ) $+$ (rhythms filling $n - 2$ ). Starting from $1$ rhythm for one sixteenth and $2$ for two, build the table up to $n = 8$ : how many rhythms fill a [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) note?
12. The numbers you found — $1, 2, 3, 5, 8, 13, 21, 34$ — were published by the Indian scholar Hemachandra around the year 1150, counting exactly such rhythms, two generations before Fibonacci wrote them down in Italy. State their defining pattern in one sentence, and check it on the last three entries of your table.
13. “Swing” style replaces two equal eighths by a long–short pair: $\frac16$ then $\frac{1}{12}$ . Verify that the swap is legal (same total duration), using the common [denominator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) $12$ .
14. A *triplet* squeezes three equal notes into one [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) note. How long is each note (check your answer by multiplying it by $3$ , [Proposition 47.8](#prop-g7-fractions-mult) )? Same question for three equal notes filling a [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) note.
15. Balkan dance music uses seven-eight time: each measure totals $\frac78$ . Verify that dotted [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) $+$ [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) $+$ [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) fills it. Then explain why *no* combination of [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) notes and [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) notes alone, however many, can ever fill a seven-eight measure. (Count in eighths and think about even and [odd](https://one-course.com/books/math/1/en/chapter/14-numbers-up-to-10-000#def-g3-numbers-evenodd) .)

**Solution of Problem 47.1.**

**1.** (a) $\frac12 + \frac14 + \frac14 = \frac24 + \frac14
+ \frac14 = \frac44 = 1$: legal. (b) $\frac14 + \frac18 +
\frac18 + \frac12 = \frac28 + \frac18 + \frac18 + \frac48 =
\frac88 = 1$: legal.

**2.** $\frac12 + \frac14 + \frac18 = \frac48 + \frac28 +
\frac18 = \frac78$: one eighth note is missing.

**3.** A measure is $1 = \frac{16}{16}$: sixteen sixteenth notes. An eighth is $\frac18 = \frac{2}{16}$: two sixteenths.

**4.** Dotted [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half): $\frac12 + \frac14 = \frac34$. Dotted [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half): $\frac14 + \frac18 = \frac28 + \frac18 = \frac38$.

**5.** The dotted [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) is worth $\frac34$ (question 4): exactly one waltz measure. Other fillings, for instance: [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) $+$ [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) $+$ [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) ($\frac14 + \frac14 + \frac14 =
\frac34$), or [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) $+$ eighth $+$ eighth $+$ [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) ($\frac14 + \frac18 + \frac18 + \frac14 = \frac{2 + 1 + 1 +
2}{8} = \frac68 = \frac34$).

**6.** $\frac38 + \frac18 + \frac14 = \frac38 + \frac18 +
\frac28 = \frac68 = \frac34$: one [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) note ($\frac14$) is missing.

**7.** Dotted eighth: $\frac18 + \frac{1}{16} =
\frac{2}{16} + \frac{1}{16} = \frac{3}{16}$. [Quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half): $\frac{4}{16}$. The [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) is longer.

**8.** $\frac12 + \frac18 = \frac48 + \frac18 = \frac58$, that is $\frac{10}{16}$: ten sixteenths.

**9.** $1 - \frac{11}{16} = \frac{16}{16} - \frac{11}{16}
= \frac{5}{16}$ of the measure is silence.

**10.** Both examples total $\frac{2 + 1 + 1}{16} =
\frac{4}{16}$: legal. All rhythms filling four sixteenths with pieces of $2$ and $1$:

$$
2{+}2, \quad 2{+}1{+}1, \quad 1{+}2{+}1, \quad 1{+}1{+}2, \quad
1{+}1{+}1{+}1 :
$$

five rhythms.

**11.** A rhythm filling $n$ sixteenths ends either with a sixteenth — and what precedes fills $n - 1$ — or with an eighth — and what precedes fills $n - 2$. Every rhythm is counted exactly once this way, so $\text{ways}(n) = \text{ways}(n-1) + \text{ways}(n-2)$. Table:

| $n$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
| ways | $1$ | $2$ | $3$ | $5$ | $8$ | $13$ | $21$ | $34$ |

A [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) note ($8$ sixteenths) can be drummed in $34$ ways. ($n = 4$ recovers the five rhythms of question 10.)

**12.** Each number is the [sum](https://one-course.com/books/math/1/en/chapter/2-addition-first-steps#def-g1-addition-def) of the two before it: $3 + 5 = 8$, $5 + 8 = 13$, $8 + 13 = 21$, $13 + 21 = 34$ — the rule question 11 proved, a thousand years old and now known as the Fibonacci rule.

**13.** $\frac16 + \frac{1}{12} = \frac{2}{12} +
\frac{1}{12} = \frac{3}{12} = \frac14$: the same duration as two straight eighths ($\frac18 + \frac18 = \frac14$). The swing swap is legal.

**14.** Each triplet note lasts $\frac{1}{12}$: check, $3 \times \frac{1}{12} = \frac{3}{12} = \frac14$ ([Proposition 47.8](#prop-g7-fractions-mult)). For a [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) note: each lasts $\frac16$, since $3 \times \frac16 = \frac36 = \frac12$.

**15.** Dotted [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) $+$ [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) $+$ [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half): in eighths, $3 + 2 + 2 = 7$ eighths $= \frac78$: legal. Halves and [quarters](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) are worth $4$ and $2$ eighths — both *[even](https://one-course.com/books/math/1/en/chapter/14-numbers-up-to-10-000#def-g3-numbers-evenodd)* numbers. Any [sum](https://one-course.com/books/math/1/en/chapter/2-addition-first-steps#def-g1-addition-def) of [even numbers](https://one-course.com/books/math/1/en/chapter/14-numbers-up-to-10-000#def-g3-numbers-evenodd) is [even](https://one-course.com/books/math/1/en/chapter/14-numbers-up-to-10-000#def-g3-numbers-evenodd), but a seven-eight measure needs $7$ eighths, an [odd number](https://one-course.com/books/math/1/en/chapter/14-numbers-up-to-10-000#def-g3-numbers-evenodd): impossible. To fill an [odd](https://one-course.com/books/math/1/en/chapter/14-numbers-up-to-10-000#def-g3-numbers-evenodd) measure, something [odd](https://one-course.com/books/math/1/en/chapter/14-numbers-up-to-10-000#def-g3-numbers-evenodd) — an eighth or a dotted note — must appear.