What is long division?

Long division is a method for splitting a big number into equal groups. Every problem has three parts:

  DIVIDEND  ÷  DIVISOR  =  QUOTIENT

  "the total   "number of   "how many
   to share"    groups"      each"
        

Think of 84 cookies shared between 3 friends. The dividend is 84, the divisor is 3, and the quotient — the answer you're after — is how many cookies each person gets.

The 4-step loop

Every digit in the dividend runs through the same four steps. Here's the mnemonic to remember them:

Step by step: 84 ÷ 3

We'll run DMSB twice — once for each digit in the dividend. Set up looks like this:

       _ _
     ────
  3 ) 8 4

Round 1 — the digit 8

       2
     ────
  3 ) 8 4
      6 ↓
      ───
      2 4

Round 2 — the number 24

       2 8
     ────
  3 ) 8 4
      6
      ───
      2 4
      2 4
      ───
        0

What about remainders?

Not every division comes out clean. Try 85 ÷ 3 — only the last digit changes:

       2 8
     ────
  3 ) 8 5
      6
      ───
      2 5
      2 4
      ───
        1    ← remainder!

Write it as 28 r 1.

The 7 things to remember

1. Dividend ÷ Divisor = Quotient
2. D — Divide: how many times does it fit?
3. M — Multiply: divisor × that number
4. S — Subtract: find what's left over
5. B — Bring down: pull the next digit
6. ↺ Repeat D → M → S → B until no digits remain
7. Remainder = what's left (always smaller than the divisor)

What you need

• Tyre levers 2 minimum — plastic, not metal
• Patch kit Sandpaper or roughing pad · vulcanising glue · patches
• Pump Check it fits your valve type — Presta or Schrader
• Bowl of water Optional — useful for finding slow punctures

Items 1–3 fit in a saddlebag and cost under £10. Worth keeping there permanently.

The 4-step sequence

Every puncture repair follows the same four steps — and the words themselves are the memory aid:

Run through OUT → FIND → FIX → IN in order, every time. Don't skip steps.

Step by step

OUT — Remove the tube

   tyre (outer rubber)
  ╔═══════════════════════════╗
  ║     inner tube            ║  ← this is what you're fixing
  ╚═══════════════════════════╝
  ═══════════════════════════    ← rim

FIND — Locate the hole

  ─────────────────●─────────────────
                   ↑
                 hole (feel for air or watch for bubbles)

FIX — Patch it correctly

  rough  →  glue  →  WAIT (dull/tacky)  →  patch  →  press 60s

  ░░░░░░░░  ░░░░░░░░  ▒▒▒▒▒▒▒▒  ▒▒▒▒▒▒▒▒  ████████
  ▓▓▓▓▓▓▓▓  ▓▓▓▓▓▓▓▓  ▓▓▓▓▓▓▓▓  ▓▓▓▓▓▓▓▓  ▓▓▓▓▓▓▓▓

IN — Refit everything

The 4 things to remember

1. OUT — release brake · remove wheel · lever bead · pull tube
2. FIND — inflate slightly · lips or water · mark the hole
3. FIX — rough · glue · wait for tacky · patch · press 60s
4. IN — check inside tyre · refit tube · bead on · inflate

What a ratio actually means

A ratio doesn't describe an amount. It describes a relationship — how two quantities compare to each other.

  6 : 2

  "for every 6 of one thing,
   there are 2 of the other"
        

As long as that relationship stays the same, you can write the ratio with any equivalent pair of numbers — just like 2/4 and 1/2 are the same fraction.

  6 : 2   →   same relationship as   →   3 : 1

  ██████ ██          ███ █
  6 red · 2 blue     3 red · 1 blue

  Same mix. Fewer pieces.
        

Why you can simplify

When you divide both sides by the same number, you scale the ratio down equally — the relationship doesn't change.

  6 : 2

  ÷2      ÷2
   ↓        ↓

  3 : 1
        

The number you divide by must go into both sides exactly — no remainders. The best number to use is the Highest Common Factor (HCF): the largest number that divides into both sides.

The 2-step method

No acronym needed — the steps say it themselves:

  For 6:2

  Factors of 6:  1  2  3  6
  Factors of 2:  1  2
                    ↑
              Highest shared = 2

  Divide both sides by 2  →  3:1
          

A ratio is fully simplified when the only shared factor is 1 — nothing bigger divides into both sides evenly.

Step by step: 6:2

Step 1 — FIND the HCF

Step 2 — DIVIDE both sides

The 5 things to remember

1. A ratio describes a relationship, not an amount
2. Dividing both sides equally keeps that relationship the same
3. FIND the HCF — the biggest number that divides into both sides exactly
4. DIVIDE both sides by the HCF
5. Done when the only shared factor is 1

Three ways particles can hang out

Everything is made of tiny particles. The same particles behave in three very different ways depending on how much energy they have.

   SOLID              LIQUID             GAS

   ● ● ● ● ●          ●  ●              ●         ●
   ● ● ● ● ●            ● ●  ●                ●
   ● ● ● ● ●          ●      ●           ●          ●
   ● ● ● ● ●             ● ●                  ●

   Locked in.         Sliding past.     Bouncing off the walls.

More energy means more movement. Heat adds energy. Cold takes it away.

The three states at a glance

The pattern: Solid → Liquid → Gas, particles gain energy, gain space, gain freedom. Reverse the direction and they slow down, pull close, lock in.

Remember it by asking why

The fastest way to lock this in is to answer three questions. Say each answer out loud — seriously.

Step by step: melting an ice cube

Watch what happens to the particles as you add heat.

  ── STAGE 1: Solid ice at −5°C ──────────────────
  ● ● ● ●     Particles locked in a pattern.
  ● ● ● ●     Vibrating, but not going anywhere.
  ● ● ● ●

         ↓  Add heat

  ── STAGE 2: Melting at 0°C ─────────────────────
  ● ●  ●     Particles gain energy.
   ●  ● ●    The pattern breaks. They start sliding.
  ●  ● ●

         ↓  Still adding heat

  ── STAGE 3: Liquid water ────────────────────────
   ●  ●  ●   Free to slide past each other.
   ● ●  ●    Still pulled close by weak attraction.
    ● ● ●

         ↓  Much more heat — up to 100°C

  ── STAGE 4: Steam ───────────────────────────────
    ●          Particles break free completely.
         ●     Bouncing around at high speed.
   ●      ●    Fill any space you put them in.

Where the heat goes

Here's what a thermometer reads as you heat water from frozen to steam:

  Temperature
  (°C)

  120 |                                      ╱
  100 |                          ─────────     ← boiling: stays at 100°C
   80 |                        ╱               until all water is gone
   60 |                      ╱
   40 |                    ╱
   20 |                  ╱
    0 |        ─────────   ← melting: stays at 0°C until all ice is gone
  −20 |      ╱
       ──────────────────────────────────── →  Time (heat applied)

Those flat lines answer the question from the opening. At those points, the energy isn't raising the temperature — it's breaking the bonds between particles. Pulling locked particles apart (solid → liquid) or freeing sliding ones entirely (liquid → gas) takes energy. A lot of it.

It takes 5× more energy to turn boiling water into steam than it took to heat that water from frozen to boiling.

The 6 things to remember

1. Same particles — three very different behaviours
2. Solid — locked in place, vibrating
3. Liquid — sliding past each other, still close
4. Gas — far apart, moving freely
5. Heat = energy = movement. Cold takes it away.
6. At a phase change, heat breaks bonds — not raises temperature. That hidden energy is called latent heat.

The answer

The person who saved for 10 years and stopped has more money.

  Alex    saved age 22–32  ·  10 years  ·  £12,000 paid in  →  ~£170,000 at 65
  Sam     saved age 32–65  ·  33 years  ·  £39,600 paid in  →  ~£150,000 at 65

  (£100/month · 7% average annual growth — typical long-term stock market return)

Alex saved for a third of the time, put in less than a third of the money, and ended up with more.

Why it works

Money invested doesn't just grow — it grows on its growth.

In year one, £1,000 at 7% becomes £1,070. In year two, you earn 7% on the £1,070 — not the original £1,000. You're earning interest on interest.

  Year 1:   £1,000  →  £1,070    (+£70)
  Year 5:   £1,000  →  £1,403    (+£403)
  Year 10:  £1,000  →  £1,967    (+£967)
  Year 20:  £1,000  →  £3,870    (+£2,870)
  Year 30:  £1,000  →  £7,612    (+£6,612)
  Year 40:  £1,000  →  £14,974   (+£13,974)

The number isn't just getting bigger — the rate of getting bigger is getting bigger.

Alex's contributions had 33 years to compound after Alex stopped. That is what Sam's later contributions — no matter how consistent — could never catch up to.

The number to remember: 72

The Rule of 72 gives you an instant feel for compounding without a calculator.

  72 ÷ 7% growth  =  money doubles every ~10 years

  Age 22 → 32:   £1 → £2     (1st double)
  Age 32 → 42:   £2 → £4     (2nd double)
  Age 42 → 52:   £4 → £8     (3rd double)
  Age 52 → 62:   £8 → £16    (4th double)

  Sam's same £1, invested at age 42:
  Age 42 → 52:   £1 → £2     (1st double)
  Age 52 → 62:   £2 → £4     (2nd double)

  Same £1. Invested 20 years later. Worth 4× less at retirement.

So what about pensions?

A pension is just a wrapper around an investment. In the UK, the government adds tax relief on top — for every £80 you pay in, £100 goes into the pot (basic rate taxpayer). Many employers match contributions too: free money that only lands if you're enrolled.

The maths above isn't about pensions specifically — it works for any long-held investment. But pensions are where most people encounter this choice, and where the decision most often gets deferred.

The thing nobody explains at 22 isn't the rule. It's what the rule does: every year you delay doesn't cost you one year of saving. It costs you one doubling period on everything you would have saved that year.

The good news

The same logic that made the past so valuable makes now valuable.

If you're 35, you still have three doubling periods ahead of you. Money invested today still has time to become 8× itself. That's not as good as starting at 25 — but it's dramatically better than starting at 45.

The Rule of 72 cuts both ways: it shows you what you lost, but it also shows you exactly what you still have.

The 5 things to remember

1. Early beats more — 10 years young outperforms 33 years late
2. Compounding = growth on growth (not just growth on original)
3. Rule of 72: money doubles every (72 ÷ interest rate) years
4. At 7%, £1 at age 22 is worth 4× more at 65 than £1 at age 42
5. The best time to start was then. The second best time is now.

What darning does

Darning recreates the structure of the fabric across a hole by weaving new thread through itself — not patching over the top. Done right, the repair stretches with the merino and stays invisible under a layer.

Two techniques exist. This guide covers weave darning, which works for holes up to roughly 1cm. For tiny worn patches with no actual hole, see the Swiss darning note at the end.

What you need

• Darning needle Sharp, with a fine eye
• Thread Fine wool or embroidery thread, close in colour to the garment
• Darning mushroom Or a light bulb, tennis ball, or smooth stone
• Scissors

The mushroom matters. Merino is stretchy — working flat causes the repair to pucker and pull when worn.

The 4-step sequence

Every darn follows the same four steps — and the words describe exactly what you do:

Run through MOUNT → WARP → WEAVE → SEAL in order, every time.

Step by step

MOUNT — Fabric on the mushroom

       hole
        │
  ~~~~~~○~~~~~~   ← fabric, taut over mushroom
  ──────────────  ← mushroom surface beneath

WARP — Parallel rows across

  ─ ─ ─ ─ ─ ─ ─ ─   ← rows start and end beyond the hole
  ─ ─ ─ ┌───┐─ ─
  ─ ─ ─ │   │─ ─   ← hole
  ─ ─ ─ └───┘─ ─
  ─ ─ ─ ─ ─ ─ ─ ─

WEAVE — Through the warp

  │ │ │ │ │ │   ← warp threads
  ─────────────  ← weft row 1 (over / under)
  ─────────────  ← weft row 2 (under / over)
  ─────────────  ← weft row 3 (over / under)

SEAL — Secure the ends

Swiss darning — for tiny holes and worn patches

If the hole is very small, or the fabric is just thin and worn rather than open, Swiss darning (duplicate stitch) is better. It recreates the knit structure rather than weaving across it.

        ┌───┐
  ──────┘   └──────  ← the V of a knit stitch
        ↑
  1. Bring needle up through base of the V
  2. Pass needle under both legs of the stitch above
  3. Return needle down through the same base point
  4. Move to the next V and repeat

Work row by row. Used over a worn patch, it thickens and reinforces without adding visible bulk.

The 4 things to remember

1. MOUNT — hole centred on mushroom · fabric taut · needle unknotted
2. WARP — parallel rows across · 5mm beyond hole each side · rows close
3. WEAVE — perpendicular · alternate over/under each row · light tension
4. SEAL — weave tail through reverse · no knot · trim close

The most common mistake is working flat. Always use the mushroom.

The foot has three zones

A shoe lace distributes tension across the whole foot from a single pull. Change the pattern, and you change where the tension goes.

  ┌─────────────────────────────────┐
  │                                 │
  │   HEEL        ← slipping?       │
  │                                 │
  │   INSTEP      ← numb or tight?  │
  │                                 │
  │   TOE BOX     ← cramped?        │
  │                                 │
  └─────────────────────────────────┘

Find your zone. That tells you which technique to use.

Three techniques, three zones

Match the symptom to the technique, then scroll to that section.

LOCK the heel. WINDOW the instep. SKIP the toe.

LOCK — for heel slippage and heel blisters

The heel is lifting inside the shoe. You're getting blisters at the back, or your foot is sliding forward on descents.

  ○ ──loop── ○   ← top eyelets: thread UP through same side
  │           │     only — creates a loop
  ○ ───────── ○   ← second from top: last normal cross

        loop      loop
  ○ ─┐       ┌─ ○
     │       │
  ○  └───────┘  ○   ← laces cross THROUGH opposite loops
                      Pull firm. Tie as normal.

The loop captures the lace and grips the heel collar. It's mechanical, not just tight — pull as hard as you like above and the heel stays locked.

WINDOW — for instep pressure and numbness

The top of your foot goes numb after an hour. Or there's a ridge of pressure across the instep — often from a high arch, or a shoe that fits narrow across the top.

  ○ ────────── ○   ← resume normal crossing above here
  ║            ║
  ║  (window)  ║   ← no cross at the pressure point:
  ║            ║     both laces run straight up same side
  ○ ────────── ○   ← last normal cross below pressure

SKIP — for toe box tightness and wide forefoot

Your toes feel cramped. Or the ball of the foot is being squeezed at its widest point — common with bunions, Morton's neuroma, or simply a wide foot in a standard-width shoe.

Option A — skip the first eyelet pair (toe relief)

  ○ ────────── ○   eyelet 6
  ...normal crossing...
  ○ ────────── ○   eyelet 2 ← start here
  ○            ○   eyelet 1 ← skipped entirely

Don't thread through eyelets 1 at all. Start from eyelet 2 and lace up normally. This leaves the toe box without any lace pressure.

Option B — skip at the widest point (forefoot relief)

  ○ ────────── ○   ← resumes crossing above
  ○ ══════════ ○   ← straight across (no cross) at widest point
  ○ ────────── ○   ← normal crossing below

Where the foot is widest (often eyelets 3–4), run the lace straight across rather than crossing. This stops the lacing pulling inward at that point.

The 3 things to remember

1. LOCK — heel slipping or blistering · loop at top eyelets · cross through opposite loop
2. WINDOW — top pressure or numbness · skip the cross at the pressure point · run straight up
3. SKIP — toe box tight or forefoot cramped · skip eyelet 1, or run straight across at widest point

Your two feet are different. Lace them differently.