Lesson Guide
Explanation
Everything in this lesson comes back to place value. In 347.562, each digit's value depends on its column, and each column is worth ten times the one to its right. The decimal point isn't a wall — it's just a marker showing where the ones column sits. Tenths, hundredths and thousandths continue the same ten-times pattern.
Your Turn
What is the 6 actually worth in 347.562? And what happens to every digit if you multiply the number by 10? Check: 6 hundredths = 0.06; every digit slides one column to the left.
Watch Out
Think 'the digits slide' rather than 'the point moves' — it is the more truthful picture, and it will pay off when we divide by decimals shortly.
Explanation
To add or subtract decimals, line up the decimal points so that tenths sit under tenths and hundredths under hundredths. 12.7 + 3.48: write 12.70 — that trailing zero costs nothing and keeps the columns honest — then add normally: 16.18. Subtraction works exactly the same way.
Your Turn
Set out 8.1 − 2.36 with the points lined up, adding a zero where you need one. Check: 8.10 − 2.36 = 5.74.
Watch Out
The error to avoid: lining up the right-hand ends like whole numbers, which treats 12.7 + 3.48 as 127 + 348. Writing the extra zero (12.70) is the vaccine.
Explanation
To multiply decimals, ignore the points and multiply the whole numbers: 4 × 12 = 48. Then count the decimal places in the question — one in 0.4, two in 0.12, three in total — and put that many places back into the answer: 0.048. Always sanity-check: multiplying by a number less than 1 makes things smaller, so an answer bigger than 0.12 would be a red flag.
Your Turn
Predict before computing: will 0.4 × 0.12 be bigger or smaller than 0.12? Then work out 0.7 × 0.3 and 0.05 × 0.8. Check: smaller; 0.21; 0.04.
Watch Out
Primary school taught you 'multiplying makes bigger' — that stops being true once numbers go below 1. Estimate first, compute second, and the habit protects you.
Explanation
Never divide by a decimal directly — shift it away. 5.6 ÷ 0.07: multiply both numbers by 100, and the question becomes 560 ÷ 7 = 80. This is legal because a division is really a fraction, and multiplying top and bottom by the same amount gives an equivalent fraction. Only the divisor decides how far to shift — just enough to make it whole.
Your Turn
Write 5.6 ÷ 0.07 as the fraction 5.6/0.07 and scale it yourself. Then try 4.8 ÷ 0.6 and 0.084 ÷ 0.007. Check: 48 ÷ 6 = 8, and 84 ÷ 7 = 12.
Watch Out
Shifting only the divisor (0.07 → 7 but leaving 5.6 alone) gives an answer 100× too small. It's both numbers or neither — the fraction picture makes that obvious.
Explanation
Rounding answers the question: which neat value is this closest to? 3.276, to two decimal places, sits between 3.27 and 3.28 — and it's past the halfway mark, so it rounds up to 3.28. The procedure: keep the digits you're asked for, look at the single next digit, and round up if it's 5 or more. You never look further than that one digit.
Your Turn
Round 17.386 to 1 d.p., 0.6349 to 2 d.p., and 2.397 to 2 d.p. Check: 17.4; 0.63 (the 4 says stay); 2.40 — the round-up carries, and you must write the zero.
Watch Out
Two snags: chaining roundings (0.6349 → 0.635 → 0.64 is wrong — one look, one round) and dropping the final zero of 2.40, which silently changes the precision you are claiming.
Explanation
Significant figures measure how precisely a number is known, and the game is deciding which digits carry information. Start counting from the first non-zero digit. Zeros trapped between non-zero digits count — in 407 the zero matters. Leading zeros never count — in 0.00407 they only position the number. And a zero written at the end after the decimal point does count: writing 0.040 says you measured to that extra digit on purpose.
Your Turn
How many significant figures in 5300, 0.082, 30.06 and 0.0500? Check: 2, 2, 4, 3. If 5300 or 0.0500 surprised you, re-read the zero rules above — those two carry the whole idea.
Watch Out
Trailing zeros are the battleground: in 5300 they are ambiguous placeholders (treat as 2 s.f.); in 0.0500 they are deliberate precision. Scientific notation fixes this properly — it comes in Chapter 6.
Explanation
Rounding to significant figures uses the same up-or-down rule — only the starting point changes. 34682 to 2 s.f.: the first two figures are 3 and 4, the next digit is 6, so round up to 35000 — the zeros stay to keep the number the right size. 0.039 62: the first two significant figures are 3 and 9, the next digit 6 rounds 39 up to 40, giving 0.040 — and that final zero must stay, because it IS the second significant figure.
Your Turn
Round 0.020 496 to 2 s.f., and 986 to 2 s.f. Check: 0.020, and 990 — not 99! Then head to the Exercise tab, then the Quiz.
Watch Out
Rounding must never change the SIZE of a number, only its precision. 986 → 99 is the give-away error — keep the place-holder zero: 990.
📌 Key Points — this lesson in a nutshell
- Adding or subtracting: line up the decimal points.
- Multiplying: ignore the points, multiply, then count total decimal places back in.
- Dividing: shift BOTH numbers until the divisor is whole.
- Rounding: look at the next digit only — 5 or more rounds up.
- Significant figures start at the first non-zero digit; leading zeros never count.
Step 1: Introduction
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