Decimals & Long Division — Print
Place value carries on past the point, which is why the old methods still work.
What the practice questions cover
These questions use 3 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Divisor — The polynomial you are dividing by.
- Remainder — What is left when the divisor no longer fits. Always lower degree than the divisor.
- Place value — What each digit is worth, depending on its column.
A bill for $47.85, split five ways.
You already know how to divide 47 by 5: it goes nine times with 2 left over.
That leftover $2.85 still has to be shared. It does not vanish because the whole dollars ran out.
Keep going past the point and you get $9.57 each. Nothing new happened — the columns just carried on.
A decimal point is not a barrier. Each column is a tenth of the one before it, all the way down, so every method you learned for whole numbers keeps working.
Lining up the point
To add or subtract decimals, line up the points, not the ends of the numbers.
4.5 + 12.35 needs the 4.5 written as 4.50 so the columns match.
Adding a zero on the end changes nothing, because there were already no hundredths there.
Ends do not line up, points do
Writing 4.5 under 12.35 flush right puts the 5 in the hundredths column. That turns 4.5 into 0.45 without you noticing.
Multiplying: count the places
Ignore the points, multiply as whole numbers, then count the decimal places in both original numbers.
0.3 × 0.4: 3 × 4 = 12, and there are two decimal places altogether, so the answer is 0.12.
It is smaller than either number, because multiplying by less than one shrinks things.
Estimate as a check
0.3 × 0.4 is roughly a third of a half, so well under one. An answer of 1.2 would fail that check immediately.
Dividing by a decimal
Divide by a whole number if you possibly can — so move the point in the divisor until it is whole.
Move it the same number of places in the dividend and the answer is unchanged, since you have scaled both by the same power of ten.
1.44 ÷ 0.6 becomes 14.4 ÷ 6 = 2.4.
Both or neither
Moving the point in only one number changes the answer by a factor of ten. Whatever you do to one, do to the other.
Long division that does not stop
When there is a remainder, add a decimal point and zeros to the dividend and carry on.
47 ÷ 5: nine, remainder two. Bring down a zero to make twenty tenths, which is four tenths each.
Some divisions stop; others repeat forever, like 1 ÷ 3 = 0.333…
The point goes straight up
Put the decimal point in the answer directly above the one in the dividend, before you start. Then the columns look after themselves.
Decimals, fractions and money
0.75 means 75100, which simplifies to 34. Reading the place value gives the fraction.
To go the other way, divide: 38 = 3 ÷ 8 = 0.375.
Money is just decimals with two places, which is why every rule here applies to a till receipt.
Rounding money
Bills are given to the nearest cent, so 9.57 each is $47.85 exactly. Not every split works out so neatly, and then somebody pays the extra penny.
Worked examples
Adding
Work out 4.5 + 12.35.
- Write 4.50 so both have two decimal places. The extra zero changes nothing.
- Line up the points and add column by column. Hundredths, then tenths, then ones.
- 4.50 + 12.35 = 16.85. Roughly 4 + 12 = 16, so this is sensible.
16.85
Dividing by a decimal
Work out 1.44 ÷ 0.6.
- Move the point one place right in 0.6 to make it 6. Dividing by a whole number is far easier.
- Do the same to 1.44, giving 14.4. Both scaled by ten, so the answer is unchanged.
- 14.4 ÷ 6 = 2.4. Check: 2.4 × 0.6 = 1.44.
2.4
Splitting a bill
Share $47.85 equally among 5 people.
- 47 ÷ 5 = 9 with 2 left over. Whole dollars first.
- The $2.85 left is 285 cents; 285 ÷ 5 = 57. Carry on past the point.
- So each person pays $9.57. Check: 9.57 × 5 = 47.85.
$9.57
Words to know
- Place value
- What each digit is worth, depending on its column.
- Dividend
- The number being divided — the one under the bar.
- Divisor
- The number you are dividing by.
- Quotient
- The answer to a division.
- Terminating decimal
- One that stops, like 0.75, rather than repeating forever.
Mistakes to watch for
Lining up the right-hand ends instead of the points. Writing 4.5 flush right under 12.35 puts the 5 in the hundredths column, silently turning it into 0.45. Line up the points and pad with zeros.
Losing a decimal place when multiplying. 0.3 × 0.4 is 0.12, not 1.2 — there are two decimal places in the question, so there must be two in the answer.
Moving the point in only one number when dividing. 1.44 ÷ 0.6 becomes 14.4 ÷ 6, not 1.44 ÷ 6. Both numbers must be scaled by the same amount.
Stopping at the remainder. 47 ÷ 5 is 9.4, not "9 remainder 2", when the question is about money or measurement. Add a point and zeros and keep going.
Assuming multiplying always makes things bigger. 0.3 × 0.4 = 0.12 is smaller than both. Multiplying by less than one shrinks.
The decimal point is only about four hundred years old. Before that people wrote awkward fractions, and long division was a specialist skill.
Different countries put it in different places: much of Europe writes 47,85 with a comma, which is why international forms often say which they mean.
A calculator storing 13 as 0.333333 is always slightly wrong, which is why bank software works in whole cents rather than decimals.
Problem 1
Work out 4.5 + 12.35.
Worked answer
16.85
4.50 + 12.35 = 16.85
Problem 2
Work out 0.7 + 0.35.
Worked answer
1.05
0.70+0.35 = 1.05
Problem 3
Work out 9.4 - 3.75.
Worked answer
5.65
9.40-3.75 = 5.65
Problem 4
Work out 0.3 × 0.4.
Worked answer
0.12
3 × 4 = 12
Two places: 0.12
Problem 5
Work out 1.2 × 5.
Worked answer
6
12 × 5 = 60
One place: 6.0
Problem 6
Work out 7.2 ÷ 4.
Worked answer
1.8
72 ÷ 4 = 18
So 1.8
Problem 7
Work out 1.44 ÷ 0.6.
Worked answer
2.4
14.4 ÷ 6 = 2.4
Problem 8
Work out 0.96 ÷ 0.4.
Worked answer
2.4
9.6 ÷ 4 = 2.4
Problem 9
Work out 0.25 × 0.8.
Worked answer
0.2
25 × 8 = 200
Three places: 0.200 = 0.2
Problem 10
Share $47.85 equally among 5 people. How much each?
Worked answer
9.57
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Problem 11
A $18.60 bill is split 3 ways. How much each?
Worked answer
6.2
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Problem 12
Write 0.75 as a fraction in lowest terms.
Worked answer
3/4
75100
Divide both by 25: 34
Problem 13
Write 38 as a decimal.
Worked answer
0.375
3 ÷ 8 = 0.375
Problem 14
Which is the best estimate for 0.48 × 19.7?
- About 10
- About 1
- About 100
- About 0.5
Worked answer
About 10
0.5 × 20 = 10
Problem 15
Work out 47 ÷ 5 as a decimal.
Worked answer
9.4
47 ÷ 5 = 9 remainder 2
20 ÷ 5 = 4 tenths
9.4
Problem 16
A ribbon costs $1.25 per metre. What do 3.2 metres cost?
Worked answer
4
125 × 32 = 4000
Three decimal places: 4.000