Module 10 of 45Lesson 1 of 16 · Unit 5 · Repeating Forever
Number · Unit N05
Repeating Forever
Divide 1 by 3 and the calculator never finishes. Divide 1 by 8 and it stops after three digits. This unit shows you why a fraction only ever does one of those two things, how to write the endless ones down in full, and how to turn them straight back into fractions.
By the end of this unit you can
- Tell a terminating decimal from a recurring one, and say why there is no third kind
- Write a recurring decimal in dot notation, with the dots in the right places
- Turn a fraction into its decimal by dividing, and spot the cycle in the remainders
- Turn a recurring decimal back into a fraction with the ×10ⁿ subtraction method
- Match the power of 10 to the length of the repeating block
- Decide from the denominator alone whether a fraction will terminate
- Explain why √2 and π are not fractions at all
The one idea behind all of it
Everything in this unit comes from watching the remainders while you divide. There are only so many remainders available, so one of two things has to happen:
a remainder of 0 → it stops · a repeated remainder → it repeats
Nothing else can happen, because the remainders run out.
Dividing by 7 leaves only six possible non-zero remainders, so within six steps either the division ends or a remainder comes round again — and the moment it does, every later line is a copy of an earlier one.
How we write it · the notation convention
A dot above a digit means that digit repeats for ever. When a whole block repeats, put a dot on the first digit of the block and a dot on the last digit of the block, and nothing in between.
On paper and in the PDF version of this unit the same thing is shown with a bar over the repeating block. A dot pair and a bar mean exactly the same thing; use whichever your pen makes cleanly.
House rules. Give every final answer to 3 significant figures unless it is exact, and always give a fraction in its simplest form. Round, never truncate. Money is in Pula and is always written to 2 decimal places.
EXAMPLE
0.3 with 3 recurring means 0.333… · 0.27 with 27 recurring means 0.272727… · 0.16 with 6 recurring means 0.1666… · 0.142857 with 142857 recurring means 0.142857142857…
Notice that in 0.16 with 6 recurring only the 6 carries a dot — the 1 happens once and never comes back.
0.3 with 3 recurring means 0.333… · 0.27 with 27 recurring means 0.272727… · 0.16 with 6 recurring means 0.1666… · 0.142857 with 142857 recurring means 0.142857142857…
Notice that in 0.16 with 6 recurring only the 6 carries a dot — the 1 happens once and never comes back.
1
Does it stop, or does it repeat?
The only two things a fraction’s decimal can do
Turn any fraction into a decimal by dividing, and watch the remainders. If a remainder of 0 ever appears the division is finished and the decimal terminates. If it never appears, a remainder must eventually repeat — there are only finitely many to choose from — and from that moment the digits repeat too. That is a recurring decimal. There is no third possibility for a fraction.
Same starting number, two different endings. 1 ÷ 4 reaches a remainder of 0 and finishes; 1 ÷ 3 returns to remainder 1 and is trapped there for ever.
Worked example 1 · Show that 13 = 0.3 with 3 recurring
1
Set out 1 ÷ 3. Three does not go into 1, so start with 0 and a point
0.
2
Bring down a zero: 10 ÷ 3 = 3 remainder 1
digit 3, remainder 1
3
Bring down another zero: 10 ÷ 3 = 3 remainder 1 again
digit 3, remainder 1
4
The remainder is back to 1, which is exactly where the division started
the step is forced to repeat
5
So the digit 3 is produced for ever. One repeating digit, so one dot
13 = 0.3 with 3 recurring
The proof is not that we got bored of dividing. It is that the remainder came back to 1. Once a remainder repeats, every later line of the division is a copy of an earlier one, so the digits have no choice but to repeat.
Practice · Does it stop, or does it repeat? · 10 questions
1.1Write down 12 as a decimal.Fluency
Answer0.5
1.2Write down 14 as a decimal.Fluency
Answer0.25
1.3Write down 15 as a decimal.Fluency
Answer0.2
1.4Write down 18 as a decimal.Fluency
Answer0.125
1.5Write down 13 as a decimal, using dot notation.Applied
Answer0.3 with 3 recurring1 ÷ 3: the remainder is 1 at every step, so the digit 3 is produced over and over. The block is one digit long, so one dot sits on the 3.
1.6Write down 16 as a decimal, using dot notation.Applied
Answer0.16 with 6 recurring1 ÷ 6 gives 0.1 with remainder 4. From then on the remainder is 4 every single time, so 6 repeats. The 1 is produced once and never comes back, so the dot goes on the 6 only.
1.7Write down 29 as a decimal, using dot notation.Applied
Answer0.2 with 2 recurring2 ÷ 9 = 0.2 remainder 2, and the remainder 2 returns immediately, so 2 repeats forever.
1.8Write down 512 as a decimal, using dot notation.Applied
Answer0.416 with 6 recurring5 ÷ 12 = 0.41 with remainder 8, then 80 ÷ 12 = 6 remainder 8 for ever. The 4 and the 1 happen once; only the 6 repeats.
1.9Determine which of 38, 56, 720 and 415 give terminating decimals, and write down the other two in dot notation.Challenge
Answer38 = 0.375 and 720 = 0.35 terminate; 56 = 0.83 with 3 recurring and 415 = 0.26 with 6 recurring recurSimplify first — all four are already in lowest terms. 8 = 2 × 2 × 2 and 20 = 2 × 2 × 5 contain only 2s and 5s, so those two stop. 6 = 2 × 3 and 15 = 3 × 5 both carry a 3, so those two repeat.
1.10Write down every remainder that can appear when you divide by 7, before the division ends, and hence determine the longest repeating block a seventh could possibly have.Challenge
Answerremainders 1, 2, 3, 4, 5, 6; longest possible block 6 digitsA remainder must be smaller than 7, and a remainder of 0 would end the division, so only 1, 2, 3, 4, 5 and 6 are available. Once a remainder repeats the digits repeat, so the block can never be longer than 6. 17 uses all six, so 6 really is reached.
2
Dot notation
Writing forever without writing forever
A dot above a digit means that digit repeats for ever. When a whole block repeats, put a dot on the first digit of the block and a dot on the last digit of the block, and nothing in between. Digits written before the dots happen once and never come back.
The repeating block is the unit of measurement here. Six digits, then the same six digits, then the same six again — which is why two dots can replace an infinite amount of writing.
Practice · Dot notation · 10 questions
2.1Write down 0.444… in dot notation.Fluency
Answer0.4 with 4 recurring
2.2Write down 0.818181… in dot notation.Fluency
Answer0.81 with 81 recurring
2.3Write down the first six decimal places of 0.2 with 2 recurring.Fluency
Answer0.222222
2.4Write down the first six decimal places of 0.45 with 45 recurring.Fluency
Answer0.454545
2.5Write down 0.5111… in dot notation.Applied
Answer0.51 with 1 recurringOnly the 1 repeats. The 5 is produced once, so it stays outside the dot.
2.6Write down the first eight decimal places of 0.16 with 6 recurring.Applied
Answer0.16666666The dot is on the 6 only, so the 1 is written once and the 6 fills every place after it.
2.7Write down 0.123123123… in dot notation.Applied
Answer0.123 with 123 recurringThe block is 123, three digits long. Dots go on the first digit and the last digit of the block, never on the middle ones.
2.8Determine which is larger, 0.4 with 4 recurring or 0.45.Applied
Answer0.450.4 with 4 recurring = 0.4444… and 0.45 = 0.4500…. Compare place by place: the tenths tie at 4, and in the hundredths 4 < 5. So 0.45 is larger.
2.9Determine which is larger, 0.16 with 6 recurring or 0.16 with 16 recurring.Challenge
Answer0.16 with 6 recurring0.16 with 6 recurring = 0.16666… and 0.16 with 16 recurring = 0.161616…. The tenths tie at 1 and the hundredths tie at 6; in the thousandths 6 > 1. Moving one dot changes the number.
2.10Write down the 100th decimal digit of 0.123 with 123 recurring.Challenge
Answer1The block 123 is 3 digits long, and 100 = 3 × 33 + 1, so the 100th digit is the 1st digit of a fresh block, which is 1.
3
Fraction to decimal, by dividing
Follow the remainders and the cycle appears
Set out the division as usual and keep adding zeros after the point. Write down each remainder as you go. The first remainder that appears twice is the start of the cycle, and everything from there onwards repeats in the same order for ever.
Left: the division, with the remainder written under the digit it produced. Right: the same six remainders drawn as a closed ring. Diagram not to scale.
Practice · Fraction to decimal, by dividing · 10 questions
3.1Divide to write 13 as a decimal.Fluency
Answer0.3 with 3 recurring
3.2Divide to write 19 as a decimal.Fluency
Answer0.1 with 1 recurring
3.3Divide to write 23 as a decimal.Fluency
Answer0.6 with 6 recurring
3.4Divide to write 111 as a decimal.Fluency
Answer0.09 with 09 recurring
3.5Divide to write 17 as a decimal.Applied
Answer0.142857 with 142857 recurringRemainders in order: 1, 3, 2, 6, 4, 5, then 1 again. The digits they produce are 1, 4, 2, 8, 5, 7. The moment remainder 1 returns, the whole run starts over.
3.6Divide to write 311 as a decimal.Applied
Answer0.27 with 27 recurring30 ÷ 11 = 2 remainder 8, then 80 ÷ 11 = 7 remainder 3, and remainder 3 is where we started, so the pair 27 repeats.
3.7Divide to write 56 as a decimal.Applied
Answer0.83 with 3 recurring50 ÷ 6 = 8 remainder 2, then 20 ÷ 6 = 3 remainder 2 for ever. Only the 3 repeats.
3.8Divide to write 712 as a decimal.Applied
Answer0.583 with 3 recurring70 ÷ 12 = 5 r 10, 100 ÷ 12 = 8 r 4, 40 ÷ 12 = 3 r 4, and the remainder 4 now sticks. The 5 and the 8 happen once each.
3.9Divide to write 27 as a decimal, and describe how its repeating block compares with the block for 17.Challenge
Answer0.285714 with 285714 recurring; the same six digits in the same cyclic order, entered at a different point17 gives 142857 and 27 gives 285714. Reading 142857 round in a circle and starting at the 2 gives 285714 — the cycle of remainders is the same ring, joined in a different place.
3.10The remainders for 1 ÷ 7 run 1, 3, 2, 6, 4, 5 and then back to 1. Use that cycle to write 47 as a decimal without dividing again.Challenge
Answer0.571428 with 571428 recurringJoin the ring at remainder 4. From 4 the remainders continue 4, 5, 1, 3, 2, 6 and the digits they produce are 5, 7, 1, 4, 2, 8. So 47 = 0.571428 with 571428 recurring, the same six digits once more.
4
Recurring to fraction · one repeating digit
The ×10 subtraction method, with the scaffold given
Give the number a name, x. Multiply by the power of 10 that shifts the decimal point past exactly one block. The two numbers then have identical tails, so subtracting wipes the tail out completely and leaves whole numbers. Solve, then simplify.
The scaffold · use it every time
1. Give the number a name: let x = the recurring decimal.
2. Count the digits in the repeating block. Call it k.
3. Multiply both sides by 10ᵏ — that is 10, 100, 1000 and so on.
4. Subtract the smaller line from the larger. The tails are identical, so they cancel exactly.
5. Solve for x, simplify the fraction, then divide it back out to check.
Worked example 2 · Write 0.7 with 7 recurring as a fraction
1
Give the number a name
x = 0.7 with 7 recurring = 0.777…
2
The block is 1 digit long, so multiply by 10¹ = 10
10x = 7.777…
3
Line the two up and subtract. The tails are identical, so they cancel exactly
10x − x = 7.777… − 0.777…
4
The left side is 9x and the right side is a whole number
9x = 7
5
Divide by 9, then check nothing simplifies
x = 79
6
Check by dividing back: 7 ÷ 9
0.7 with 7 recurring ✓
The subtraction is doing all the work. Both tails are the same forever, so they vanish exactly — nothing is rounded, nothing is approximated, and the answer is exact.
Practice · Recurring to fraction · one repeating digit · 10 questions
4.1Write 0.5 with 5 recurring as a fraction in its simplest form.Fluency
Answer59
4.2Write 0.8 with 8 recurring as a fraction in its simplest form.Fluency
Answer89
4.3Write 0.1 with 1 recurring as a fraction in its simplest form.Fluency
Answer19
4.4Write 0.6 with 6 recurring as a fraction in its simplest form.Fluency
Answer23
4.5Write 0.7 with 7 recurring as a fraction in its simplest form.Applied
Answer79Let x = 0.7 with 7 recurring. The block is 1 digit long, so multiply by 10: 10x = 7.7 with 7 recurring. Subtract: 10x − x = 7.777… − 0.777… = 7, so 9x = 7 and x = 79. 7 and 9 share no factor, so it is already simplest.
4.6Write 0.4 with 4 recurring as a fraction in its simplest form.Applied
Answer49x = 0.4 with 4 recurring, 10x = 4.4 with 4 recurring, subtract to get 9x = 4, so x = 49.
4.7Write 0.3 with 3 recurring as a fraction in its simplest form.Applied
Answer13x = 0.3 with 3 recurring, 10x = 3.3 with 3 recurring, subtract to get 9x = 3, so x = 39. Divide top and bottom by 3: x = 13. Always finish by simplifying.
4.8Write 0.9 with 9 recurring as a fraction in its simplest form.Applied
Answer1x = 0.9 with 9 recurring, 10x = 9.9 with 9 recurring, subtract to get 9x = 9, so x = 1. This is not a rounding error and not a trick: 0.999… and 1 are two names for one number.
4.9Write 0.16 with 6 recurring as a fraction in its simplest form.Challenge
Answer16One digit sits outside the block, so shift past it first. x = 0.16 with 6 recurring, 10x = 1.6 with 6 recurring and 100x = 16.6 with 6 recurring. Subtract the pair whose tails match: 100x − 10x = 16.666… − 1.666… = 15, so 90x = 15 and x = 1590 = 16.
4.10Write 0.583 with 3 recurring as a fraction in its simplest form.Challenge
Answer712Two digits sit outside the block. x = 0.583 with 3 recurring, 100x = 58.3 with 3 recurring and 1000x = 583.3 with 3 recurring. Subtract: 900x = 583.333… − 58.333… = 525, so x = 525900 = 712.
5
Recurring to fraction · longer blocks
Match the power of 10 to the length of the block
The power of 10 is decided by the block, not by the number of digits you can see. A block of 1 digit needs ×10, a block of 2 digits needs ×100, a block of 3 needs ×1000. If digits sit outside the block, shift past those first and subtract the two lines whose tails match.
Match the multiplier to the block
- block of 1 digit → ×10
- block of 2 digits → ×100
- block of 3 digits → ×1000
- block of k digits → ×10ᵏ, and the subtraction leaves (10ᵏ − 1)x, a string of k nines
Digits outside the block
- Shift past them first, then multiply by 10ᵏ again
- Subtract the two lines whose tails match, not always x itself
- 0.16 with 6 recurring: use 100x − 10x, not 100x − x
- The bottom of the fraction ends up as nines followed by zeros
Worked example 3 · Write 0.27 with 27 recurring as a fraction
1
Give the number a name
x = 0.27 with 27 recurring = 0.272727…
2
The block is 2 digits long, so multiply by 10² = 100
100x = 27.272727…
3
Subtract — the tails match, so they cancel
100x − x = 27
4
The left side is 99x
99x = 27
5
Divide by 99
x = 2799
6
Simplify: 27 and 99 both divide by 9
x = 311
7
Check by dividing back: 3 ÷ 11
0.27 with 27 recurring ✓
2799 was correct but not finished. In this topic more marks are lost on the last line than on the first — always simplify, then divide back to check.
Practice · Recurring to fraction · longer blocks · 10 questions
5.1Write 0.12 with 12 recurring as a fraction in its simplest form.Fluency
Answer433
5.2Write 0.45 with 45 recurring as a fraction in its simplest form.Fluency
Answer511
5.3Write 0.81 with 81 recurring as a fraction in its simplest form.Fluency
Answer911
5.4Write 0.09 with 09 recurring as a fraction in its simplest form.Fluency
Answer111
5.5Write 0.27 with 27 recurring as a fraction in its simplest form.Applied
Answer311The block is 2 digits long, so multiply by 100, not 10. x = 0.27 with 27 recurring, 100x = 27.27 with 27 recurring. Subtract: 99x = 27, so x = 2799. Divide top and bottom by 9: x = 311.
5.6Write 0.36 with 36 recurring as a fraction in its simplest form.Applied
Answer411x = 0.36 with 36 recurring, 100x = 36.36 with 36 recurring, subtract to get 99x = 36, so x = 3699 = 411.
5.7Write 0.123 with 123 recurring as a fraction in its simplest form.Applied
Answer41333The block is 3 digits long, so multiply by 1000. x = 0.123 with 123 recurring, 1000x = 123.123 with 123 recurring, subtract to get 999x = 123, so x = 123999. Both divide by 3: x = 41333.
5.8Write 0.142857 with 142857 recurring as a fraction in its simplest form.Applied
Answer17The block is 6 digits long, so multiply by 1 000 000. That gives 999999x = 142857, so x = 142857999999. Since 142857 × 7 = 999999, the fraction is exactly 17.
5.9Write 0.163 with 63 recurring as a fraction in its simplest form.Challenge
Answer955One digit is outside the block and the block is 2 digits long. x = 0.163 with 63 recurring, 10x = 1.63 with 63 recurring and 1000x = 163.63 with 63 recurring. Subtract the matching tails: 990x = 163.6363… − 1.6363… = 162, so x = 162990 = 955.
5.10Write 2.72 with 72 recurring as a fraction in its simplest form.Challenge
Answer3011The whole-number part comes along for the ride. x = 2.72 with 72 recurring, 100x = 272.72 with 72 recurring, subtract to get 99x = 270, so x = 27099 = 3011. Check: 3011 = 2.727272… ✓
6
Which denominators terminate
Simplify, then look at the prime factors
Write the fraction in its simplest form first, then split the denominator into prime factors. If the only primes there are 2s and 5s, the decimal terminates. If any other prime survives, the decimal recurs. The reason is that our decimals are built on tenths, and 10 = 2 × 5.
Break the denominator into primes and the answer is immediate. Every 2 and every 5 can be cleared into a power of 10; a 3 never can.
Worked example 4 · Determine which denominators terminate — a mini-investigation
| n | 1n as a decimal | Stops? | Prime factors of n |
|---|---|---|---|
| 2 | 0.5 | yes | 2 |
| 3 | 0.3 with 3 recurring | no | 3 |
| 4 | 0.25 | yes | 2 × 2 |
| 5 | 0.2 | yes | 5 |
| 6 | 0.16 with 6 recurring | no | 2 × 3 |
| 7 | 0.142857 with 142857 recurring | no | 7 |
| 8 | 0.125 | yes | 2 × 2 × 2 |
| 9 | 0.1 with 1 recurring | no | 3 × 3 |
| 10 | 0.1 | yes | 2 × 5 |
| 11 | 0.09 with 09 recurring | no | 11 |
| 12 | 0.083 with 3 recurring | no | 2 × 2 × 3 |
The pattern has nothing to do with how big the denominator is. It is about which primes are hiding inside it. Our decimals are built on tenths, and 10 = 2 × 5, so a denominator made only of 2s and 5s can always be turned into a power of 10 — and then it stops. Any other prime never clears, and the division runs for ever.
Practice · Which denominators terminate · 10 questions
6.1Determine whether 18 terminates.Fluency
AnswerYes — 8 = 2 × 2 × 2
6.2Determine whether 16 terminates.Fluency
AnswerNo — 6 = 2 × 3
6.3Determine whether 120 terminates.Fluency
AnswerYes — 20 = 2 × 2 × 5
6.4Determine whether 115 terminates.Fluency
AnswerNo — 15 = 3 × 5
6.5Determine whether 740 terminates, and write down the decimal.Applied
AnswerYes, 0.17540 = 2 × 2 × 2 × 5 — only 2s and 5s, so it stops. 7 ÷ 40 = 0.175.
6.6Determine whether 512 terminates, and write down the decimal.Applied
AnswerNo, 0.416 with 6 recurring12 = 2 × 2 × 3. The 3 cannot be cleared, so the decimal must repeat: 0.416 with 6 recurring.
6.7Determine whether 615 terminates, and write down the decimal.Applied
AnswerYes, 0.4Simplify first: 615 = 25. The denominator is now 5, so it stops: 25 = 0.4. Testing 15 before simplifying would have given the wrong answer.
6.8Determine whether 924 terminates, and write down the decimal.Applied
AnswerYes, 0.375924 simplifies to 38, and 8 = 2 × 2 × 2, so it stops: 0.375. Unsimplified, 24 = 2 × 2 × 2 × 3 looks as though it should repeat.
6.9Determine every whole number n from 2 to 20 for which 1n terminates.Challenge
Answer2, 4, 5, 8, 10, 16, 20These are exactly the numbers built from 2s and 5s only: 2, 4 = 2², 5, 8 = 2³, 10 = 2 × 5, 16 = 2⁴, 20 = 2² × 5. Every other n from 2 to 20 carries a prime other than 2 or 5.
6.10Determine the smallest whole number n greater than 1 for which 1n stops after exactly three decimal places, and the smallest for which it stops after exactly four.Challenge
Answer8 and 1612 = 0.5 and 15 = 0.2 stop after 1 place; 14 = 0.25 after 2; 18 = 0.125 after 3; 110 = 0.1 after 1. The next terminating denominator is 16, and 116 = 0.0625 stops after 4. For a denominator 2ᵃ × 5ᵇ the number of places is the larger of a and b.
The routine
Four moves, whichever direction you are travelling in.
1
Divide, and write the remainders downA remainder of 0 ends it — the decimal terminates. A remainder you have seen before starts the cycle.
2
Mark the block with dotsOne dot for a single repeating digit; a dot on the first and last digit of a longer block. Digits produced before the cycle stay outside.
3
Going back? Name it x and match the power of 10Block of k digits needs ×10ᵏ. Digits outside the block need a shift past those first, so subtract the two lines whose tails match.
✓
Subtract, solve, simplify, then divide backThe tails cancel exactly. Finish by simplifying the fraction, then check by dividing it out again — you should land back on the decimal you started with.
The mistakes that cost the most marks
1Multiplying by the wrong power of 10
Wrong0.27 with 27 recurring: x = 0.272727…, so 10x = 2.727272… and 10x − x = 2.454545…
Right0.27 with 27 recurring: the block is 2 digits, so 100x = 27.272727… and 100x − x = 27
Multiplying by 10 shifted the point past only half a block, so the two tails did not match and the subtraction left 9x = 2.45 with 45 recurring — still recurring, and no further on. Count the digits in the block first: 1 digit needs ×10, 2 digits need ×100, 3 digits need ×1000.
2Putting the dot over the wrong digit
Wrong16 = 0.16 with 16 recurring
Right16 = 0.16 with 6 recurring
0.16 with 16 recurring means 0.161616…, which is 1699, not 16. The 1 is produced once, before the cycle of remainders begins, so it sits outside the dots. Only digits that come back round again get a dot.
3Assuming every decimal is rational
Wrong“Every decimal can be written as a fraction, so √2 must be one too.”
Right√2 = 1.41421356… and π = 3.14159265… never stop and never repeat, so neither is a fraction.
Terminating and recurring decimals are exactly the fractions — the rational numbers. A decimal that neither stops nor settles into a repeating block, such as 0.101001000100001… where the run of zeros grows every time, is irrational. Recurring means rational; endless-and-never-repeating means irrational.
Tasks to submit
Criterion B · Investigating patterns
How long before it comes round again?
Every fraction 1n either terminates or repeats. When it repeats, the number of digits in the repeating block is called the period. 13 = 0.3 with 3 recurring has period 1. 111 = 0.09 with 09 recurring has period 2. 17 = 0.142857 with 142857 recurring has period 6.
The period is not random and it is not obvious. Your job is to find out what decides it.
Part 1 · structured
| n | 1n as a decimal | Terminates? | Period |
|---|---|---|---|
| 2 | 0.5 | yes | 0 |
| 3 | 0.3 with 3 recurring | no | 1 |
| 4 | |||
| 5 | |||
| 6 | |||
| 7 | |||
| 8 | |||
| 9 | |||
| 10 | |||
| 11 | |||
| 12 | |||
| 13 |
- Complete the table for every n from 2 to 13. Write each decimal in dot notation, say whether it terminates, and record the period — the number of digits in the repeating block. Use a period of 0 for a decimal that terminates.
- Predict the period of 114 and the period of 115 before you divide. Write both predictions down, then divide and check whether you were right.
- Identify the patterns in your table. Which values of n terminate? Which give a period of exactly 1? Which give a period of exactly 2? And what do you notice when you compare each period with the value of n itself?
- Determine a general rule that predicts the period for any n you are handed. Two ideas worth testing: the period is 1 exactly when n divides 9, and the period is 2 exactly when n divides 99 but not 9. If those hold, what would a period of 3 mean? State your rule so that it works for any n, not only the ones in your table.
- Verify your rule on n = 37 and n = 41, neither of which is in your table. Predict the period from the rule first, then divide and check that the two agree.
- Justify why the period of 1n can never be as large as n. Your explanation should talk about the remainders in the division, not about the digits.
Part 2 · open
Now choose your own line of enquiry. You decide which values of n to test, how many you need, how to set your results out and what would count as convincing evidence. Take one of these, or propose your own:
- Which values of n give a period of exactly 1, exactly 2, or exactly 3? What do the numbers 9, 99 and 999 have to do with your answer?
- What happens for values of n that contain a 2 or a 5 as well, such as 6, 12, 14, 15 and 28? Does the repeating block start straight after the decimal point, or later? How much later?
- Compare the decimals for 17, 27, 37, 47, 57 and 67. Is it the same block every time? Try the same experiment with elevenths and with thirteenths.
- For prime values of n, compare the period with n − 1. Which primes reach the largest period they possibly could, and how common are they?
- If you know the period of 1a and the period of 1b, can you predict the period of 1a × b? Test it on pairs such as 3 and 11, 7 and 11, 11 and 13.
Whichever you choose, take it through the same ladder: predict, identify the pattern, determine a general rule, test and verify it on values you have not already used, and justify why it must hold.
Submit: your tables, your rule and your reasoning, as a scan or a typed document. Marked out of 8 against Criterion B. Completing the table earns the lower bands; the general rule, the verification and the justification are what reach levels 5–8, and Part 2 must show your own choice of method.
Remember, you should communicate in an organized and coherent manner.
Criterion D · Applying maths in real-life contexts
Splitting the bill in thebe
Six friends finish a meal in Gaborone and ask for one bill so they can split it equally. Here is the slip they were handed. It contains more information than you need.
Till slip · Kgale View Grill, Gaborone
- 2 × Seswaa and pap @ P89.00 = P178.00
- 2 × Grilled chicken quarter and chips @ P76.50 = P153.00
- 1 × Beef burger and chips @ P82.00 = P82.00
- 1 × Vegetable pasta @ P74.00 = P74.00
- 6 × Soft drink 300 ml @ P18.00 = P108.00
- 2 × Malva pudding (to share) @ P75.00 = P150.00
- TOTAL DUE P745.00
- Table 9 · served by Kealeboga · 19:42, Saturday
- Prices include VAT at 14%. VAT content of this bill: P91.49
- Loyalty points earned on this visit: 74
- Kitchen closes 21:30 · takeaway boxes P6.50 each (none taken)
- The group travelled by kombi, P5.50 each way per person, paid separately
- Payment method: cash
Botswana money is written in pula and thebe, and 100 thebe make one pula, so every amount actually paid has exactly two decimal places. The exact share does not.
- i · Identify the relevant elements. Identify the figures on the slip that you need in order to split the bill equally, and describe why each of the other figures is not needed for that job.
- ii · Select appropriate strategies. Describe, in order and before you calculate anything, the steps you will take. Say which division you will do and what you will do with the answer if it does not stop.
- iii · Apply them to reach a solution. Calculate the exact share for each of the six friends and write it in dot notation. Calculate what each person actually pays, to the nearest thebe, and calculate the total the six of them hand over. Then a seventh friend arrives and offers to pay an equal share of the same bill: calculate the exact seventh-share in dot notation and what each of the seven would actually pay.
- iv · Accuracy of the solution. Describe how accurate your answers are. The exact share never stops, but money is paid in whole thebe, so something has to be given up. Describe how much each person is over- or under-paying, and how much the group is over or under in total. Describe what changes if everybody rounds down instead of rounding to the nearest thebe.
- v · Does it make sense in this situation. Describe whether your answer works in real life. Somebody has to absorb the leftover thebe — describe a fair way for the group to settle it. Describe what happens if the smallest coin the group has between them is 5 thebe, and describe whether an equal split is even the sensible thing to do when two people ordered a P150 dessert that everybody shared.
Submit: your working plus short written answers for strands i, ii, iv and v. Marked out of 8 against Criterion D. Strands iv and v carry as much weight as the arithmetic — an exact share written to eight decimal places will lose marks if you cannot say what happens to the leftover thebe.
When you are ready
The mastery quiz is 10 questions drawn at random from a bank of 25. You need 70% to pass and you have three attempts. Work through the practice questions above first — especially Concept 5, where the power of 10 has to match the block, and Concept 6, where the denominator gives the answer away.
Printable version
Download Unit 5 as a PDF — the same lesson, laid out for printing and for working offline.
Download Unit 5 as a PDF — the same lesson, laid out for printing and for working offline.