---
title: "Fractions and Powers"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 63
exercises: 10
source: https://one-course.com/books/math/1/en/chapter/63-fractions-and-powers
---

# Chapter 63 — Fractions and Powers

[Fractions](#def-g9-fractions-fraction) and [powers](#def-g9-fractions-power) are the everyday tools of computation: sharing quantities, comparing proportions, writing very large or very small numbers. This chapter consolidates the rules for computing with them — with every intermediate step written out — and introduces [scientific notation](#def-g9-fractions-scientific).

## 63.1 Computing with fractions

**Definition 63.1 (Fraction).**

A *fraction* $\dfrac ab$ (with $b \neq 0$) is the [quotient](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) of $a$ by $b$. Two fractions are equal when one is obtained from the other by multiplying (or dividing) [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) by the same nonzero number:

$$
\frac ab = \frac{a \times k}{b \times k} \qquad (k \neq 0).
$$

**Example 63.2.**

Simplify $\dfrac{42}{56}$ step by step:

$$
\frac{42}{56} = \frac{42 \div 2}{56 \div 2} = \frac{21}{28}
= \frac{21 \div 7}{28 \div 7} = \frac34 .
$$

(In [Chapter 64](https://one-course.com/books/math/1/en/chapter/64-arithmetic-divisors-and-prime-numbers#ch-g9-arith) the greatest common [divisor](https://one-course.com/books/math/1/en/chapter/32-division-and-multiples#def-g5-division-multiple) will do this in one step.)

**Method 63.3 (Adding and subtracting fractions).**

1. Put both [fractions](#def-g9-fractions-fraction) over a *common [denominator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def)* (a common [multiple](https://one-course.com/books/math/1/en/chapter/32-division-and-multiples#def-g5-division-multiple) of the two [denominators](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) );
2. add or subtract the [numerators](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) , keeping the [denominator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) ;
3. simplify the result if possible.

**Example 63.4.**

Compute $\dfrac56 + \dfrac34$. A common [denominator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) of $6$ and $4$ is $12$:

$$
\frac56 + \frac34
= \frac{5 \times 2}{6 \times 2} + \frac{3 \times 3}{4 \times 3}
= \frac{10}{12} + \frac{9}{12}
= \frac{19}{12}.
$$

Compute $2 - \dfrac37$. Write $2$ as a [fraction](#def-g9-fractions-fraction) over $7$:

$$
2 - \frac37 = \frac{14}{7} - \frac37 = \frac{11}{7}.
$$

![Why a common denominator: cutting both wholes into 12 equal boxes makes the fractions comparable and addable — together, 10 + 9 = 19 twelfths.](https://one-course.com/images/onecourse/chapters/math-1/g9-fractions/fig-5ea923ad981a.svg)

*Why a common [denominator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def): cutting both wholes into $12$ equal boxes makes the [fractions](#def-g9-fractions-fraction) comparable and addable — together, $10 + 9 = 19$ twelfths.*

**Method 63.5 (Multiplying and dividing fractions).**

1. To multiply, multiply [numerators](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) together and [denominators](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) together: $\dfrac ab \times \dfrac cd = \dfrac{ac}{bd}$ (simplify *before* multiplying whenever possible);
2. to divide, multiply by the [inverse](https://one-course.com/books/math/1/en/chapter/55-fractions-all-four-operations#def-g8-fractions-inverse) of the [divisor](https://one-course.com/books/math/1/en/chapter/32-division-and-multiples#def-g5-division-multiple) : $\dfrac ab \div \dfrac cd = \dfrac ab \times \dfrac dc$ .

**Example 63.6.**

$$
\frac{7}{15} \times \frac{25}{14}
= \frac{7 \times 25}{15 \times 14}
= \frac{7 \times 25}{15 \times 2 \times 7}
= \frac{25}{30} = \frac56,
$$

where we simplified by the common factor $7$, then by $5$. And a division:

$$
\frac{3}{4} \div \frac{9}{8}
= \frac34 \times \frac89
= \frac{3 \times 8}{4 \times 9}
= \frac{24}{36} = \frac23 .
$$

## 63.2 Powers

**Definition 63.7 (Power).**

For a real number $a$ and a positive integer $n$, the *power* $a^n$ is the [product](https://one-course.com/books/math/1/en/chapter/10-multiplication-first-steps#def-g2-mult-def) of $n$ factors equal to $a$:

$$
a^n = \underbrace{a \times a \times \dots \times a}_{n \text{ factors}} .
$$

By convention $a^0 = 1$ (for $a \neq 0$), and negative [exponents](https://one-course.com/books/math/1/en/chapter/56-powers#def-g8-powers-def) denote [inverses](https://one-course.com/books/math/1/en/chapter/55-fractions-all-four-operations#def-g8-fractions-inverse):

$$
a^{-n} = \frac{1}{a^n}.
$$

**Example 63.8.**

$2^4 = 2 \times 2 \times 2 \times 2 = 16$; $(-3)^2 = 9$ but $-3^2 = -(3^2) = -9$ (the [exponent](https://one-course.com/books/math/1/en/chapter/56-powers#def-g8-powers-def) binds before the minus sign); $10^{-3} = \frac{1}{1000} = 0.001$; $5^1 = 5$ and $5^0 = 1$.

**Theorem 63.9 (Rules of exponents).**

For all nonzero $a$, $b$ and all integers $m$, $n$:

$$
a^m \times a^n = a^{m+n},
\qquad
\frac{a^m}{a^n} = a^{m-n},
\qquad
\left(a^m\right)^n = a^{mn},
\qquad
(ab)^n = a^n b^n .
$$

**Proof.** For positive [exponents](https://one-course.com/books/math/1/en/chapter/56-powers#def-g8-powers-def), count the factors. For the first rule: $a^m \times a^n$ is $m$ factors $a$ followed by $n$ factors $a$, in total $m + n$ factors. For the third: $\left(a^m\right)^n$ repeats $n$ times a block of $m$ factors, giving $mn$ factors. The other two rules are similar; the extension to [zero](https://one-course.com/books/math/1/en/chapter/1-counting-to-20#def-g1-counting-zero) and negative [exponents](https://one-course.com/books/math/1/en/chapter/56-powers#def-g8-powers-def) is checked from $a^{-n} = \frac{1}{a^n}$ (and makes the convention $a^0 = 1$ the only consistent choice, since $a^n \times a^0$ must be $a^{n+0} = a^n$). ∎

**Example 63.10.**

Simplify step by step:

$$
\frac{3^7 \times 3^{-2}}{3^4}
= \frac{3^{7 + (-2)}}{3^4}
= \frac{3^5}{3^4}
= 3^{5-4} = 3^1 = 3 .
$$

And with two letters: $(2a^3)^2 \times a = 4 a^6 \times a = 4a^7$.

## 63.3 Powers of ten and scientific notation

**Proposition 63.11 (Powers of ten).**

For a positive integer $n$: $10^n$ is written “$1$ followed by $n$ [zeros](https://one-course.com/books/math/1/en/chapter/1-counting-to-20#def-g1-counting-zero)”, and $10^{-n} = 0.0\dots01$ with the $1$ in the $n$-th decimal place. Multiplying a [decimal number](https://one-course.com/books/math/1/en/chapter/38-decimal-numbers#def-g6-decimals-places) by $10^n$ (resp. $10^{-n}$) shifts its [decimal point](https://one-course.com/books/math/1/en/chapter/25-tenths-and-hundredths#def-g4-decimals-point) $n$ places to the right (resp. left).

**Definition 63.12 (Scientific notation).**

The *scientific notation* of a positive [decimal number](https://one-course.com/books/math/1/en/chapter/38-decimal-numbers#def-g6-decimals-places) is the unique way of writing it as

$$
a \times 10^n,
\qquad\text{with } 1 \leq a < 10 \text{ and } n \text{ an integer.}
$$

**Example 63.13.**

The Earth–Sun distance is about $149\,600\,000$ km $= 1.496 \times
10^8$ km. The size of a water molecule is about $0.000\,000\,000\,28$ m $= 2.8 \times 10^{-10}$ m. To compute with such numbers, group the decimal parts and the [powers](#def-g9-fractions-power) of ten:

$$
(3 \times 10^5) \times (2.5 \times 10^{-2})
= (3 \times 2.5) \times 10^{5 + (-2)}
= 7.5 \times 10^{3}.
$$

**Method 63.14 (Putting a number in scientific notation).**

1. Move the [decimal point](https://one-course.com/books/math/1/en/chapter/25-tenths-and-hundredths#def-g4-decimals-point) just after the first nonzero digit; this gives the factor $a$ with $1 \leq a < 10$ ;
2. count the number of places the point moved: that count is $n$ , positive if the original number was $\geq 10$ , negative if it was $< 1$ ;
3. check: $a \times 10^n$ must reproduce the original number.

## 63.4 Exercises

**Exercise 63.1 ★.**

Compute and give the result as a fully simplified [fraction](#def-g9-fractions-fraction):

$$
\frac13 + \frac15, \qquad
\frac72 - \frac53, \qquad
\frac{5}{12} + \frac38, \qquad
3 - \frac45 .
$$

**Solution of Exercise 63.1.**

$\dfrac13 + \dfrac15 = \dfrac{5}{15} + \dfrac{3}{15} = \dfrac{8}{15}$.

$\dfrac72 - \dfrac53 = \dfrac{21}{6} - \dfrac{10}{6} = \dfrac{11}{6}$.

$\dfrac{5}{12} + \dfrac38 = \dfrac{10}{24} + \dfrac{9}{24} =
\dfrac{19}{24}$.

$3 - \dfrac45 = \dfrac{15}{5} - \dfrac45 = \dfrac{11}{5}$.

**Exercise 63.2 ★.**

Compute and simplify:

$$
\frac59 \times \frac{27}{35}, \qquad
\frac{8}{15} \div \frac{4}{5}, \qquad
\frac23 \times \frac34 \times \frac45 .
$$

**Solution of Exercise 63.2.**

$\dfrac59 \times \dfrac{27}{35} = \dfrac{5 \times 27}{9 \times 35}
= \dfrac{27}{9} \times \dfrac{5}{35} = 3 \times \dfrac17 = \dfrac37$ (simplify by $9$ and by $5$).

$\dfrac{8}{15} \div \dfrac45 = \dfrac{8}{15} \times \dfrac54
= \dfrac{8 \times 5}{15 \times 4} = \dfrac{40}{60} = \dfrac23$.

$\dfrac23 \times \dfrac34 \times \dfrac45 = \dfrac{2 \times 3 \times 4}
{3 \times 4 \times 5} = \dfrac25$ (the $3$s and $4$s cancel).

**Exercise 63.3 ★.**

Compute:

$$
2^5, \qquad (-2)^4, \qquad -2^4, \qquad 4^{-2}, \qquad
\left(\tfrac23\right)^3, \qquad 7^0 .
$$

**Solution of Exercise 63.3.**

$2^5 = 32$; $(-2)^4 = 16$; $-2^4 = -16$; $4^{-2} = \frac{1}{16}$; $\left(\frac23\right)^3 = \frac{8}{27}$; $7^0 = 1$.

**Exercise 63.4 ★.**

Write as a single [power](#def-g9-fractions-power):

$$
5^3 \times 5^6, \qquad
\frac{7^9}{7^5}, \qquad
\left(2^3\right)^4, \qquad
\frac{3^2 \times 3^7}{3^5}, \qquad
2^5 \times 4 .
$$

**Solution of Exercise 63.4.**

$5^3 \times 5^6 = 5^9$; $\dfrac{7^9}{7^5} = 7^4$; $\left(2^3\right)^4 = 2^{12}$; $\dfrac{3^2 \times 3^7}{3^5} = 3^{9-5} = 3^4$; $2^5 \times 4 = 2^5 \times 2^2 = 2^7$.

**Exercise 63.5 ★.**

Write in [scientific notation](#def-g9-fractions-scientific): $45\,000$; $0.0072$; $302.5$; $0.000\,000\,9$; twelve million.

**Solution of Exercise 63.5.**

$45\,000 = 4.5 \times 10^4$; $0.0072 = 7.2 \times 10^{-3}$; $302.5 = 3.025 \times 10^2$; $0.000\,000\,9 = 9 \times 10^{-7}$; twelve million $= 1.2 \times 10^7$.

**Exercise 63.6 ★★.**

Compute, giving the result in [scientific notation](#def-g9-fractions-scientific):

$$
(2 \times 10^7) \times (4.5 \times 10^{-3}),
\qquad
\frac{9 \times 10^5}{3 \times 10^{-2}},
\qquad
(5 \times 10^3)^2 .
$$

**Solution of Exercise 63.6.**

$(2 \times 10^7) \times (4.5 \times 10^{-3}) = 9 \times 10^{4}$.

$\dfrac{9 \times 10^5}{3 \times 10^{-2}} = 3 \times 10^{5 - (-2)}
= 3 \times 10^7$.

$(5 \times 10^3)^2 = 25 \times 10^6 = 2.5 \times 10^7$.

**Exercise 63.7 ★★.**

A tank is $\frac35$ full. After adding $30$ liters it is $\frac34$ 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? (Express the added [fraction](#def-g9-fractions-fraction) of the tank first.)

**Solution of Exercise 63.7.**

The added water represents $\dfrac34 - \dfrac35 = \dfrac{15}{20} - \dfrac{12}{20} = \dfrac{3}{20}$ of the tank. So $\frac{3}{20}$ of the [capacity](https://one-course.com/books/math/1/en/chapter/12-money-and-measures#def-g2-measure-units) is $30$ liters: $\frac{1}{20}$ is $10$ liters, and the [capacity](https://one-course.com/books/math/1/en/chapter/12-money-and-measures#def-g2-measure-units) is $200$ liters.

**Exercise 63.8 ★★.**

Alice eats $\frac14$ of a cake, then Ben eats $\frac13$ of what remains, then Carla eats $\frac12$ of what still remains. What [fraction](#def-g9-fractions-fraction) of the cake is left at the end? (Compute the [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) after each step.)

**Solution of Exercise 63.8.**

After Alice: $1 - \frac14 = \frac34$ of the cake remains. Ben eats $\frac13 \times \frac34 = \frac14$, leaving $\frac34 - \frac14 = \frac12$. Carla eats [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of that, $\frac14$, leaving $\frac14$ of the cake.

**Exercise 63.9 ★★.**

Light travels $3 \times 10^8$ meters per second. The Sun is about $1.5 \times 10^{11}$ m away. How long does sunlight take to reach us? Give the answer in seconds ([scientific notation](#def-g9-fractions-scientific) not needed), then in minutes and seconds.

**Solution of Exercise 63.9.**

Time $= \dfrac{\text{distance}}{\text{speed}} =
\dfrac{1.5 \times 10^{11}}{3 \times 10^8} = 0.5 \times 10^3 = 500$ seconds, i.e. $8$ minutes and $20$ seconds.

**Exercise 63.10 ★★★.**

Which is larger, $2^{30}$ or $3^{20}$? (Hint: write both as [powers](#def-g9-fractions-power) with [exponent](https://one-course.com/books/math/1/en/chapter/56-powers#def-g8-powers-def) $10$ using $\left(a^m\right)^n = a^{mn}$, and compare $2^3$ with $3^2$.)

**Solution of Exercise 63.10.**

Write both numbers as [powers](#def-g9-fractions-power) with [exponent](https://one-course.com/books/math/1/en/chapter/56-powers#def-g8-powers-def) $10$:

$$
2^{30} = \left(2^3\right)^{10} = 8^{10},
\qquad
3^{20} = \left(3^2\right)^{10} = 9^{10}.
$$

Since $8 < 9$, multiplying ten copies of each: $8^{10} < 9^{10}$, so $2^{30} < 3^{20}$.

## 63.5 Problem: The secret code of repeating decimals

**Problem 63.1.**

Weekend problem — every fraction has a decimal writing that stops or repeats, every repeating decimal is a fraction, and $0.999\ldots$ equals exactly $1$

Divide $3$ by $11$ and the digits fall into a chant: $0.272727\ldots$ forever. Divide $1$ by $4$ and the digits stop dead: $0.25$. This problem proves that these are the only two possible behaviors for a [fraction](#def-g9-fractions-fraction) — and then reverses the machine: given any repeating decimal, it rebuilds the [fraction](#def-g9-fractions-fraction) hiding behind it. On the way it settles, once and for all, the most argued-about equality in school mathematics.

**Part I — From [fraction](#def-g9-fractions-fraction) to decimal.**

1. Compute by long division the decimal writings of $\frac14$ , $\frac13$ , $\frac56$ and $\frac{3}{11}$ . Which stop, which repeat, and with what repeating block?
2. Explain why the decimal writing of *any* [fraction](#def-g9-fractions-fraction) $\frac ab$ must either stop or eventually repeat. (In the long division by $b$ , which values can the successive [remainders](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) take? What happens the moment a [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) shows up for the second time?)
3. Push the division of $1$ by $7$ far enough to find the repeating block. How long is it, and how does that length compare with the bound promised by question 2?
4. Some [fractions](#def-g9-fractions-fraction) stop because their [denominator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) can be blown up to a [power](#def-g9-fractions-power) of ten: $\frac14 =  \frac{25}{100}$ , $\frac{3}{8} = \frac{375}{1000}$ . Explain the criterion, and why no whole-number multiplier can ever turn $3$ , $6$ or $11$ into $10$ , $100$ , $1000, \dots$ (what digit would the [product](https://one-course.com/books/math/1/en/chapter/10-multiplication-first-steps#def-g2-mult-def) end in?).
5. Without dividing, sort $\frac{9}{40}$ , $\frac{5}{12}$ and $\frac{33}{50}$ into “stops” and “repeats”, then give the decimal writing of the two that stop.

**Part II — From repeating decimal back to the [fraction](#def-g9-fractions-fraction).**

6. The shift trick. Let $x = 0.7777\ldots$ Compute $10x$ , then $10x - x$ , and deduce $x$ as a [fraction](#def-g9-fractions-fraction) . Check by dividing.
7. Let $x = 0.727272\ldots$ Which [power](#def-g9-fractions-power) of ten shifts $x$ by exactly one repeating block? Use it to write $x$ as a [fraction](#def-g9-fractions-fraction) in lowest terms.
8. A mixed case: $x = 0.58333\ldots$ (the $3$ alone repeats). Compute $1000x - 100x$ and deduce $x$ as a [fraction](#def-g9-fractions-fraction) in lowest terms.
9. The famous one: apply the shift trick to $x = 0.9999\ldots$ What [fraction](#def-g9-fractions-fraction) — what *number* — do you find? Reconcile the result with your intuition, remembering the phantom neighbour of [Problem 38.1](https://one-course.com/books/math/1/en/chapter/38-decimal-numbers#pb-g6-decimals-1) and the shrinking [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) of [Problem 39.1](https://one-course.com/books/math/1/en/chapter/39-fractions-first-steps#pb-g6-fractions-1) : is $0.999\ldots$ “just below” $1$ , or another name for $1$ ?
10. Convert $2.454545\ldots$ and $0.123123123\ldots$ into [fractions](#def-g9-fractions-fraction) in lowest terms.

**Part III — The dictionary completed.**

11. Questions 6–10 suggest a dictionary: a block of $k$ digits repeating from the [decimal point](https://one-course.com/books/math/1/en/chapter/25-tenths-and-hundredths#def-g4-decimals-point) equals that block over $k$ nines ( $0.727272\ldots = \frac{72}{99}$ ). Use the dictionary to predict $0.037037\ldots$ as a [fraction](#def-g9-fractions-fraction) , simplify — and marvel: which surprisingly simple [fraction](#def-g9-fractions-fraction) chants “ $037$ ”? Check it by division.
12. Combine the dictionary with a shift: write $0.0272727\ldots$ as a [fraction](#def-g9-fractions-fraction) in lowest terms.
13. The number $0.101001000100001\ldots$ (a $1$ , then one $0$ , then a $1$ , then two $0$ s, then a $1$ , then three $0$ s, and so on) never stops. Is it a repeating decimal? What does Part I then say about it — can it be a [fraction](#def-g9-fractions-fraction) ? (You have just met your first provably *irrational* number; the most famous one is caught in [Chapter 65](https://one-course.com/books/math/1/en/chapter/65-square-roots#ch-g9-sqrt) .)
14. Roughly how many digits does $2^{30}$ have? Use $2^{10} = 1\,024 \approx 1.024 \times 10^3$ and the [exponent](https://one-course.com/books/math/1/en/chapter/56-powers#def-g8-powers-def) rules ( [Theorem 63.9](#thm-g9-fractions-rules) ) to write $2^{30}$ in (approximate) [scientific notation](#def-g9-fractions-scientific) , and conclude. (Compare [Exercise 63.10](#exo-g9-fractions-10) .)
15. Finale: write $0.20262026\ldots$ as a [fraction](#def-g9-fractions-fraction) , and state the two-way moral of the whole problem: which decimal writings are [fractions](#def-g9-fractions-fraction) , and which [fractions](#def-g9-fractions-fraction) have which decimal writings?

**Solution of Problem 63.1.**

**1.** $\frac14 = 0.25$ (stops); $\frac13 = 0.333\ldots$ (block $3$); $\frac56 = 0.8333\ldots$ (block $3$ after the $8$); $\frac{3}{11} = 0.272727\ldots$ (block $27$).

**2.** At each step of the division by $b$, the [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) is one of the $b$ values $0, 1, \dots, b - 1$ ([Theorem 37.7](https://one-course.com/books/math/1/en/chapter/37-whole-numbers#thm-g6-wholes-division)). If a [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $0$ appears, the division stops. Otherwise, after at most $b$ steps some [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) must appear a second time — there are more steps than possible [remainders](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) — and from that moment the division repeats exactly the sequence of digits it produced after the first appearance: the decimal writing cycles forever. Stop or repeat: no third behavior exists.

**3.** $\frac17 = 0.142857\,142857\ldots$: the block $142857$ has length $6$. Question 2 promised a repetition within at most $7$ steps — here the [remainders](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) $1, 3, 2, 6, 4, 5$ all appear before the $1$ returns: the longest possible chant for a division by $7$.

**4.** $\frac ab$ stops exactly when some $\frac{a \times k}{b \times k}$ has [denominator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) $10$, $100$, $1000, \dots$ — that is, when the [denominator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) can be multiplied up to a [power](#def-g9-fractions-power) of ten, as $4 \times 25 = 100$ or $8 \times 125 = 1\,000$. For $3$ and $6$ this is hopeless: a [power](#def-g9-fractions-power) of ten has digit [sum](https://one-course.com/books/math/1/en/chapter/2-addition-first-steps#def-g1-addition-def) $1$, and every [multiple](https://one-course.com/books/math/1/en/chapter/32-division-and-multiples#def-g5-division-multiple) of $3$ has a digit [sum](https://one-course.com/books/math/1/en/chapter/2-addition-first-steps#def-g1-addition-def) that is a [multiple](https://one-course.com/books/math/1/en/chapter/32-division-and-multiples#def-g5-division-multiple) of $3$ — so no [multiple](https://one-course.com/books/math/1/en/chapter/32-division-and-multiples#def-g5-division-multiple) of $3$ (hence none of $6$) is ever $10$, $100$, $1\,000, \dots$ For $11$: dividing $10$, $100$, $1\,000$, $10\,000$ by $11$ always leaves a [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) ($10$, $1$, $10$, $1, \dots$), never $0$. [Denominators](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) that reach a [power](#def-g9-fractions-power) of ten stop; all others repeat.

**5.** $\frac{9}{40}$: $40 \times 25 = 1\,000$, stops: $\frac{225}{1000} = 0.225$. $\frac{33}{50}$: $50 \times 2 = 100$, stops: $\frac{66}{100} = 0.66$. $\frac{5}{12}$: $12$ is a [multiple](https://one-course.com/books/math/1/en/chapter/32-division-and-multiples#def-g5-division-multiple) of $3$, repeats ($0.41666\ldots$).

**6.** $10x = 7.777\ldots$, so $10x - x = 7$ (the tails cancel perfectly), $9x = 7$ and $x = \frac79$. Division check: $7 \div 9 = 0.777\ldots$

**7.** The block has two digits, so shift by $100$: $100x - x = 72.7272\ldots - 0.7272\ldots = 72$, hence $99x = 72$ and $x = \frac{72}{99} = \frac{8}{11}$.

**8.** $1000x = 583.333\ldots$ and $100x = 58.333\ldots$; subtracting, $900x = 525$, so $x = \frac{525}{900} = \frac{7}{12}$ (divide by $75$). Indeed $\frac{7}{12} = 0.58333\ldots$

**9.** $10x - x = 9.999\ldots - 0.999\ldots = 9$, so $9x = 9$ and $x = 1$. Not “just below” $1$: $0.999\ldots$ *is* the number $1$, written in a second costume. Any number strictly below $1$ leaves a gap, and [Problem 38.1](https://one-course.com/books/math/1/en/chapter/38-decimal-numbers#pb-g6-decimals-1) showed a gap always contains further numbers — while nothing fits between $0.999\ldots$ and $1$. The endless nines are the chocolate [remainder](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-remainder) of [Problem 39.1](https://one-course.com/books/math/1/en/chapter/39-fractions-first-steps#pb-g6-fractions-1) shrunk below every positive amount: nothing is left.

**10.** $2.4545\ldots = 2 + \frac{45}{99} = 2 +
\frac{5}{11} = \frac{27}{11}$. And $0.123123\ldots = \frac{123}{999} = \frac{41}{333}$ (divide by $3$).

**11.** Dictionary: $0.037037\ldots = \frac{37}{999}$. Since $999 = 27 \times 37$, this simplifies to $\frac{1}{27}$: the [fraction](#def-g9-fractions-fraction) $\frac{1}{27}$ chants “$037$”. Division check: $1 \div 27 = 0.037037\ldots$

**12.** $0.0272727\ldots = \frac{1}{10} \times
0.272727\ldots = \frac{1}{10} \times \frac{27}{99}
= \frac{27}{990} = \frac{3}{110}$ (divide by $9$).

**13.** The blocks of [zeros](https://one-course.com/books/math/1/en/chapter/1-counting-to-20#def-g1-counting-zero) grow without end, so no fixed block can repeat forever: it is *not* a repeating decimal (and it never stops). By Part I, every [fraction](#def-g9-fractions-fraction) stops or repeats — so this number is no [fraction](#def-g9-fractions-fraction) at all: it is irrational, built to order. The superstar of irrationality, $\sqrt2$, is unmasked in [Chapter 65](https://one-course.com/books/math/1/en/chapter/65-square-roots#ch-g9-sqrt).

**14.** $2^{30} = \left(2^{10}\right)^3 \approx
(1.024)^3 \times 10^9 \approx 1.07 \times 10^9$: a number slightly above $10^9$, hence with $10$ digits. (Exactly: $2^{30} = 1\,073\,741\,824$.)

**15.** $0.20262026\ldots = \frac{2026}{9999}$, already in lowest terms ($9999 = 3 \times 3 \times 11 \times 101$ shares no factor with $2026 = 2 \times 1013$). The completed dictionary: *the [fractions](#def-g9-fractions-fraction) are exactly the decimals that stop or repeat* — [denominators](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) of tens’ friends stop, all others chant a block; and conversely every chant, [even](https://one-course.com/books/math/1/en/chapter/14-numbers-up-to-10-000#def-g3-numbers-evenodd) your graduation year, is a [fraction](#def-g9-fractions-fraction) in disguise.