Class 7 Mathematics Chapter 3: A Peek Beyond the Point
- 3 topics
- 75 MCQs
- 21 short answers
- 9 long answers
- 51 flashcards
- 15 games
- ~120 min read
- English
Tenths and hundredths
When a whole unit is too big to measure with, it is cut into ten equal parts called tenths, written after a decimal point. Cutting a tenth into ten again gives hundredths. Each place is one tenth of the place on its left.
What you will be able to do
- Explain why units smaller than one whole are needed.
- Write and read tenths and hundredths in figures.
- Place a decimal between two whole numbers on a number line.
- Say what each digit of a number like 2.36 counts.
- Distinguish a zero that adds nothing from one that moves the digits.
Key points
A measurement has to be a number, so "a little over 3" will not do.
One whole cut into ten equal parts gives tenths, and ten tenths rebuild the whole.
One tenth is written 0.1, in the first place after the decimal point.
One tenth cut into ten again gives hundredths, written 0.01.
Counting in tenths, 0.9 is followed by 1, because ten tenths regroup.
In 2.36 the 2 counts wholes, the 3 counts tenths and the 6 counts hundredths.
A zero after the digits adds nothing, so 0.40 and 0.4 are equal.
A zero before the digits moves them smaller, so 0.04 is ten times less than 0.4.
Each decimal place is one tenth of the place on its left.
Full explanation
A little over three metres
A carpenter is asked for a plank three metres long. The saw leaves it a little longer than that.
How much longer? "A little" is not a measurement. Nobody can order a second plank to match it, and nobody can check the work. A number is needed.
Whole units are sometimes too big
The trouble is that the metre is the smallest unit in use here. The plank is more than 3 and less than 4, and there is nothing in between to say it with.
So we make something in between.
Split the unit into ten equal parts
Take one whole and cut it into ten equal pieces. Each piece is one tenth of the whole.
Ten of those pieces put back together make the whole again. That is what makes them tenths and not just small bits.
Writing a tenth
One tenth is written 0.1. The dot is called the decimal point, and it marks where the whole units end.
To the left of the point sit the whole units. To the right sit the tenths. In 0.1 there are no wholes and one tenth.
Reading tenths aloud
0.3 is three tenths. 0.7 is seven tenths. Say the digits after the point by their place, not as an ordinary number.
0.7 is seven tenths, which is a little over a half. It is not seven of anything whole.
Tenths live between the whole numbers
On a number line, 0.1 to 0.9 fill the gap between 0 and 1. The gap is divided into ten equal steps, and each step is one tenth.
That is the picture to keep. A tenth is a step inside a gap that used to have nothing in it.
A whole and some tenths together
A plank 2 metres and 4 tenths long is written 2.4 metres. The 2 counts whole metres and the 4 counts tenths.
So 2.4 sits between 2 and 3, four steps along. And 1.4 sits between 1 and 2, four steps along the gap below it.
Ten tenths make one whole
Counting up in tenths from 0.7 gives 0.8, then 0.9, and the next step is not 0.10. It is 1.
Ten tenths are a whole, so they are written as one whole. This is the same regrouping that turns ten ones into a ten.
When a tenth is still too big
Our carpenter measures again. The plank is more than 3.6 metres and less than 3.7 metres.
Once again the measurement falls between two marks, and once again we need something smaller.
Split a tenth into ten
Take one tenth and cut it into ten equal pieces. Each piece is one hundredth of the original whole.
A hundred of them make the whole, because ten of them make a tenth, and ten tenths make the whole.
Writing a hundredth
One hundredth is written 0.01. The second place after the point is the hundredths place.
So the places run: wholes, then tenths, then hundredths. Each new place is one tenth the size of the place before it.
Reading hundredths aloud
0.07 is seven hundredths. There are no tenths at all, so a zero holds the tenths place open.
0.42 is four tenths and two hundredths. Reading it as forty-two hundredths is also right, because four tenths are forty hundredths.
Hundredths live between the tenths
Between 0.4 and 0.5 there is another gap, and it divides into ten steps of one hundredth each.
So 0.42 sits two small steps past 0.4. Every time we need a finer answer, we cut the gap we landed in.
Wholes, tenths and hundredths together
A plank 3 metres, 6 tenths and 4 hundredths long is written 3.64 metres. Reading right to left: 4 hundredths, 6 tenths, 3 wholes.
In the same way 2.36 is 2 wholes, 3 tenths and 6 hundredths, and 1.36 is one whole less than that.
Why 0.4 and 0.40 are the same
0.4 is four tenths. 0.40 is four tenths and no hundredths. Nothing was added, so the value did not change.
A zero on the end of a decimal adds no quantity. A zero before the digits is a different matter, because 0.04 is four hundredths.
What carries forward
A decimal is not a new kind of number. It is the same place-value idea carried past the units, each place a tenth of the one on its left.
Two slips are worth naming now. 0.7 is seven tenths, never seven hundredths. And 0.4 equals 0.40, while 0.04 is ten times smaller.
Concepts and terms
When a length falls between two whole units there is nothing to say it with. The whole is cut into ten equal parts so that the position can be named.
One of ten equal parts of a whole, written 0.1. Ten tenths make the whole again, which is what makes them tenths rather than small pieces.
The mark that shows where the whole units end. Digits to its left count wholes and digits to its right count tenths, hundredths and smaller parts.
One of ten equal parts of a tenth, written 0.01. A hundred of them make one whole, because ten make a tenth and ten tenths make the whole.
The one rule behind every decimal place. It gives tenths, then hundredths, and continues without end, so no new idea is needed for each new place.
Watch out
Common misconceptions
The tailor and the shelf
Scenario:
A tailor needs a strip of cloth for a shelf edge and measures the shelf with a metre stick. The length is more than 2 metres and less than 3, so she writes it as 2.4 metres, four tenths past the second mark. At the shop the assistant cuts carefully and finds the shelf edge is a shade longer than her strip. She measures again, more closely this time, and the length falls between 2.4 and 2.5 metres. She divides that gap into ten and reads off 2.36 for a second shelf and 2.43 for the first. The assistant asks why she did not simply say two and a half metres at the start.
Analysis:
Each measurement landed inside a gap, and each time the gap was divided into ten to name the position exactly.
Activity:
Measure the length of your desk with a ruler and write it to two places after the point. Then say which gap each digit came from.
Cut the gap twice
Title:
Cut the gap twice
Safety:
No safety concerns. Ruler, paper and pencil only.
What You Notice:
Every gap can be cut into ten again, and each cut adds one place after the point. The second drawing shows why 0.42 is only two small steps past 0.4 and nowhere near 0.5.
Did you know
Ten tenths and a hundred hundredths are both exactly one whole.
A trailing zero changes nothing, which is why a shop can price something at 0.50 rupees or 0.5 rupees and mean the same.
Every time a measurement lands inside a gap, that gap can be cut into ten again, and there is no last cut.
Memory tricks
Left of the point counts wholes, right of it counts parts.
A zero at the end is decoration; a zero in front does work.
Practice MCQs with answers (25)
- A 0.01
- B 1.0
- C 10
- D 0.1
Show answer
Why:
One whole cut into ten equal pieces gives 0.1.
- A 0.01
- B 0.1
- C 1.00
- D 100
Show answer
Why:
A tenth cut into ten again gives 0.01.
- A the full stop
- B the place marker
- C the unit sign
- D the decimal point
Show answer
Why:
It marks where the whole units end.
- A one hundred
- B five
- C ten
- D one thousand
Show answer
Why:
Ten equal pieces were cut from the whole.
- A ten
- B one hundred
- C fifty
- D one thousand
Show answer
Why:
Ten make a tenth, and ten tenths make the whole.
- A three tenths
- B three hundredths
- C thirty
- D three wholes
Show answer
Why:
The digit after the point sits in the tenths place.
- A seven tenths exactly
- B seventy altogether
- C seven hundredths
- D seven whole units
Show answer
Why:
The zero holds the tenths place, so the 7 is in hundredths.
- A four whole metres
- B four tenths of a metre
- C four hundredths of a metre
- D four centimetres
Show answer
Why:
The first place after the point is tenths.
- A 1
- B 0.10
- C 0.91
- D 1.1
Show answer
Why:
Ten tenths regroup into one whole.
- A nine equal steps
- B a hundred steps of one tenth
- C two equal halves
- D ten equal steps of one tenth
Show answer
Why:
The marks are 0.1 up to 0.9, so there are ten steps.
- A 2 and 3
- B 1 and 2
- C 4 and 5
- D 0 and 1
Show answer
Why:
It is two wholes and a bit more, four tenths along.
- A 2 and 3
- B 1 and 2
- C 0 and 1
- D 4 and 5
Show answer
Why:
One whole and four tenths.
- A two hundredths past 0.4
- B two tenths past 0.4
- C two wholes past 0.4
- D exactly halfway to 1
Show answer
Why:
The gap between 0.4 and 0.5 divides into hundredths.
- A Both true, and the reason explains the assertion
- B Both true, but the reason does not explain it
- C The assertion is true and the reason is false
- D The assertion is false and the reason is true
Show answer
Why:
The assertion is true and the reason is false: a zero BEFORE the digits does change the value, which is why 0.04 is not 0.4.
- A 0.40
- B 4.0
- C 0.44
- D 0.04
Show answer
Why:
A zero placed before the digits moves them one place smaller.
- A forty-two tenths
- B forty-two hundredths
- C four hundred and two
- D four and two tenths
Show answer
Why:
Four tenths are the same quantity as forty hundredths.
- A one tenth of its size
- B ten times its size
- C the same size
- D one hundredth of its size
Show answer
Why:
That one rule generates tenths, hundredths and all the rest.
- A planks cannot be measured accurately
- B the metre is far too small a unit here
- C a little over is not a measurement
- D the saw was faulty
Show answer
Why:
Nobody can match or check the work without a number.
- A six hundredths
- B six wholes
- C six tenths
- D six thousandths
Show answer
Why:
The first place after the point is tenths.
- A four tenths
- B four wholes
- C four thousandths
- D four hundredths
Show answer
Why:
The second place after the point is hundredths.
- A I and II only
- B II and III only
- C I and III only
- D I, II and III
Show answer
Why:
II and III are true. The first fails: the 6 stands one place further right, so it counts hundredths.
- A false, it means seventy
- B true
- C false, it means seven tenths
- D false, it means seven wholes
Show answer
Why:
The first place after the point is tenths.
- A true, it becomes larger
- B true, it becomes smaller
- C false, 0.40 is the same value
- D false, but only for whole numbers
Show answer
Why:
The extra zero adds no hundredths.
- A round it down to 3 metres flat
- B cut that gap into hundredths
- C call it a little over 3.6 metres
- D use a larger unit of measure instead
Show answer
Why:
Each time an answer lands inside a gap, we divide that gap again.
- A It shows the reader where the figure should be rounded
- B It marks the end of the number that is being written
- C It separates two whole numbers written side by side
- D The places on each side of it mean different things
Show answer
Why:
Without the point there is nothing to say which digit counts wholes and which counts tenths, so 2.36 and 236 would look alike.
Important short questions (7)
One tenth is 0.1 and one hundredth is 0.01.
Ten tenths make a whole, so they are regrouped and written as one whole. It is the same step as ten ones making a ten.
The 2 counts wholes, the 3 counts tenths and the 6 counts hundredths.
There are no tenths in this quantity, and the zero holds the tenths place open. Without it the 7 would fall into the tenths place instead.
Four tenths and two hundredths, or forty-two hundredths. Four tenths are the same quantity as forty hundredths.
The zero on the end of 0.40 adds no hundredths, so nothing changed. In 0.04 the zero sits before the 4 and pushes it a place smaller.
Each place is one tenth the size of the place on its left. That gives tenths, then hundredths, and continues without end.
Long answer questions (3)
The trouble is that the metre is the smallest unit available. The plank is more than 3 and less than 4, and there is nothing between them to say it with. So the whole metre is cut into ten equal parts, and each part is a tenth. Now the length can be given as 3.6 metres. Suppose the plank falls between 3.6 and 3.7. The same move is made again, and the tenth is cut into ten hundredths. The length can now be given as 3.64 metres. Each time a measurement lands inside a gap, that gap is divided into ten, and a new place appears to record it.
In 0.4 the 4 sits in the tenths place, so the value is four tenths. In 0.40 the 4 is still in the tenths place and the 0 sits in hundredths. That zero adds no hundredths at all, so nothing was added and the value is unchanged. The two are the same number written to different lengths. In 0.04 the position is different. The zero now comes first and holds the tenths place, which pushes the 4 into hundredths. Four hundredths is ten times smaller than four tenths. So a zero after the digits changes nothing, while a zero before them changes everything.
Both claims have the places the wrong way round. In 0.7 the 7 is in the first place after the point, which is tenths, so the value is seven tenths. That is a little over a half. In 0.07 a zero holds the tenths place, so the 7 has been pushed into the second place, which is hundredths. The pupil has read the digits without reading their positions. The single idea that prevents this is that each place is one tenth of the place on its left, counting outwards from the point. Once the order of the places is fixed, the value of a digit follows from where it sits rather than from what it looks like.
Flashcards (17)
Play this topic
- Match the decimal to its words
- How many places after the point?
- Fact or myth
- Put these lengths in order, smallest first
- Match each place to what it counts
Place value, and the units we measure in
The place-value table does not stop at the ones. Each place is one tenth of the place on its left, which gives tenths, hundredths and thousandths. Because the metric units are built on tens, converting between them is a walk along that table.
What you will be able to do
- Extend the place-value table past the decimal point to thousandths.
- Read a decimal place by place, naming what each digit counts.
- Tell a place-holding zero apart from one that adds nothing.
- Convert between metric units of length, mass and capacity.
- Explain why a number without its unit is not a measurement.
Key points
Each place is one tenth of the place on its left, before and after the point.
The places after the point run tenths, hundredths, thousandths.
One thousandth is written 0.001, and a thousand of them make one whole.
In 0.125 the digits count one tenth, two hundredths and five thousandths.
A zero inside a decimal holds a place open and cannot be removed.
A zero at the very end records none of that place and adds nothing.
A metre holds 100 centimetres, so 2.35 m is 235 cm.
A kilogram holds 1000 grams, so 0.25 kg is 250 g.
The unit is part of the answer, because each unit gives a different number.
Full explanation
The table does not stop at the units
In a whole number the places run ones, tens, hundreds, each ten times the one on its right.
Read that the other way and the rule is simpler: each place is one tenth of the place on its left. Nothing says it has to stop at the ones.
Carry the rule past the point
One tenth of a one is a tenth. One tenth of a tenth is a hundredth. One tenth of a hundredth is a thousandth.
So the table continues: ones, tenths, hundredths, thousandths. The decimal point simply marks where the ones end.
A third place
A thousandth is written 0.001. A thousand of them make one whole.
Three places after the point is as far as this chapter goes, but the rule does not run out. There is no last place.
Reading a number place by place
Take 0.125. The 1 is in tenths, the 2 in hundredths and the 5 in thousandths.
So 0.125 is one tenth, two hundredths and five thousandths. It can also be read as a hundred and twenty-five thousandths.
Zeros inside a decimal do real work
In 1.004 there is one whole, no tenths, no hundredths and four thousandths.
Those two zeros are not decoration. They hold the tenths and hundredths places open so that the 4 lands in thousandths.
A longer example
5.062 is five wholes, no tenths, six hundredths and two thousandths.
Say the places aloud as you read the digits. The digit tells you how many; the place tells you how big each one is.
Wholes can be long too
23.45 has two tens and three ones before the point, then four tenths and five hundredths after it.
208.7 has two hundreds, no tens, eight ones and seven tenths. The zero before the point works exactly like the zeros after it.
One number, two zeros, two jobs
In 45.07 the zero sits in tenths, so the 7 is seven hundredths.
Compare that with 45.70, where the zero sits in hundredths and adds nothing. Same digits, and only the positions differ.
Why measurement needs this
A metre is a whole unit. A centimetre is a hundredth of it, and a millimetre is a thousandth of it.
The units of length were built on tens, which is why a decimal describes them so neatly. The place-value table and the measuring tape match each other.
From metres to centimetres
There are 100 centimetres in a metre. So a length of 2.35 metres is 2 whole metres and 35 hundredths of a metre.
Thirty-five hundredths of a metre is 35 centimetres, so 2.35 m is 235 cm. Multiplying by 100 moves every digit two places.
From centimetres to millimetres
There are 10 millimetres in a centimetre. A length of 4.7 cm is 4 whole centimetres and 7 tenths.
Seven tenths of a centimetre is 7 millimetres, so 4.7 cm is 47 mm. This time the digits move one place, because the step is ten and not a hundred.
Mass works the same way
A gram is a thousandth of a kilogram, so there are 1000 grams in a kilogram.
A packet of 0.25 kg is twenty-five hundredths of a kilogram. Twenty-five hundredths of 1000 grams is 250 grams.
And so does capacity
A millilitre is a thousandth of a litre, so a litre holds 1000 millilitres.
A bottle of 1.5 litres holds one whole litre and five tenths. Five tenths of 1000 millilitres is 500, so the bottle holds 1500 millilitres.
Longer distances
A kilometre is a thousand metres. So 3.08 km is 3 whole kilometres and eight hundredths of one.
Eight hundredths of 1000 metres is 80 metres, giving 3080 metres in all. The zero in the tenths place is doing the work here.
The unit decides the number
The same distance is 3.08 km, or 3080 m, or 308000 cm. The quantity never changed; only the unit it is counted in did.
This is why a measurement without its unit is not a measurement. The number alone cannot say how much.
What carries forward
One rule generates the whole table, forwards and backwards: each place is a tenth of the place on its left.
Because the metric units are built on tens, converting between them is the same act as moving along that table. Read the places, not just the digits.
Concepts and terms
The places of a whole number carry on past the ones without any new idea. Each one is a tenth of the place on its left, which produces every decimal place.
One of ten equal parts of a hundredth, written 0.001. A thousand of them make one whole, which is why the third place after the point carries that name.
A zero written between the point and a digit keeps a place open, as in 1.004. Removing it would move every digit after it into a larger place.
A zero written at the end records that there is none of that place. It adds no quantity, so 45.70 and 45.7 are the same number.
Metric units step by tens, so changing unit moves every digit a fixed number of places. Ten moves one place, a hundred two, a thousand three.
Watch out
Common misconceptions
Three labels, one shelf
Scenario:
A shopkeeper is stocking a shelf and three labels are already printed. A rice packet says 0.25 kg, a bottle of oil says 1.5 L and a roll of tape says 4.7 cm wide. A customer asks for all three in the smaller units, because the recipe she is following is written in grams and millilitres, and her ruler is marked in millimetres. The shopkeeper works through them one at a time. He then notices a fourth label reading 3.08 km on a coil of wire, and a note beside it in another hand claiming the coil holds 380 metres of wire.
Analysis:
Every conversion is one move along the place-value table, and the number of places moved is set by the step between the units.
Activity:
Collect three labelled packets from a kitchen and rewrite each quantity in the smaller unit, saying how many places the digits moved and why.
Build the table and walk along it
Title:
Build the table and walk along it
Safety:
No safety concerns. Paper, ruler and kitchen packets only.
What You Notice:
Converting a unit is not a new operation. It is the same digits walking along the table, and the step between the units says how many columns they walk.
Did you know
A millimetre is a thousandth of a metre, so a metre stick carries a thousand of the smallest marks on it.
The same coil of wire can be labelled 3.08 km, 3080 m or 308000 cm, and none of those labels is more correct than the others.
Because the units step by tens, no metric conversion ever needs anything harder than moving the digits along.
Memory tricks
A zero works only when a digit follows it.
Ten moves one place, a hundred two, a thousand three.
Practice MCQs with answers (25)
- A hundredths
- B tenths
- C ten-thousandths
- D thousandths
Show answer
Why:
Each place is one tenth of the one before it.
- A 0.01
- B 0.1
- C 0.001
- D 1000
Show answer
Why:
A thousand of them make one whole.
- A one hundred
- B one thousand
- C ten
- D ten thousand
Show answer
Why:
Ten make a hundredth and a hundred make a tenth.
- A one whole
- B one tenth
- C one hundredth
- D one thousandth
Show answer
Why:
The 1 stands to the left of the point, so it counts wholes. The places after the point begin with the tenths.
- A one hundred
- B ten
- C one thousand
- D fifty
Show answer
Why:
So a centimetre is a hundredth of a metre.
- A one hundred
- B one thousand
- C ten
- D ten thousand
Show answer
Why:
So a gram is a thousandth of a kilogram.
- A five hundredths
- B five tenths
- C five thousandths
- D five wholes
Show answer
Why:
It sits in the third place after the point.
- A two thousandths
- B two tenths
- C two wholes
- D two hundredths
Show answer
Why:
It sits in the second place after the point.
- A four thousandths
- B four hundredths
- C four tenths
- D four wholes
Show answer
Why:
Two zeros hold the tenths and hundredths places open.
- A I and II only
- B II and III only
- C I and III only
- D I, II and III
Show answer
Why:
I and III are true. The second fails: 1.4 is not 1.004, because a zero inside a decimal cannot be dropped.
- A seven tenths
- B seven thousandths
- C seven wholes
- D seven hundredths
Show answer
Why:
The zero takes the tenths place, pushing the 7 along one.
- A 45.07
- B 45.007
- C 45.70
- D 4.57
Show answer
Why:
A zero in the last place adds no hundredths.
- A 5 wholes, 6 hundredths and 2 thousandths
- B 5 wholes, 6 tenths and 2 hundredths exactly
- C 5 whole ones and 62 tenths of one altogether
- D 50 wholes, 6 tenths and 2 hundredths exactly
Show answer
Why:
A zero holds the tenths place.
- A no tenths
- B no tens
- C no hundreds
- D no ones
Show answer
Why:
It sits between the hundreds and the ones.
- A 23.5 cm
- B 2350 cm
- C 235 cm
- D 2.35 cm
Show answer
Why:
Two metres is 200 cm and 35 hundredths of a metre is 35 cm.
- A 470 mm
- B 47 mm
- C 4.7 mm
- D 0.47 mm
Show answer
Why:
A centimetre holds 10 mm, so seven tenths of one is 7 mm.
- A 250 g
- B 25 g
- C 2500 g
- D 0.25 g
Show answer
Why:
A kilogram holds 1000 g, and twenty-five hundredths of that is 250.
- A 1500 mL
- B 150 mL
- C 15 mL
- D 15000 mL
Show answer
Why:
Five tenths of 1000 mL is 500, added to one whole litre.
- A 308 m
- B 30800 m
- C 3080 m
- D 3800 m
Show answer
Why:
Eight hundredths of 1000 m is 80 m, added to 3000 m.
- A make the number read a little more easily aloud
- B show the figure was rounded
- C mark the end of the whole number part of it
- D show there are no tenths of a kilometre
Show answer
Why:
Without it the 8 would mean eight tenths, which is 800 metres.
- A Both true, and the reason explains the assertion
- B Both true, but the reason does not explain it
- C The assertion is true and the reason is false
- D The assertion is false and the reason is true
Show answer
Why:
The reason is true and the assertion is false: the quantity is one and the same, and only the number changes with the unit.
- A the units are built on twelves
- B decimals were invented for them
- C both use a point
- D the units are built on tens
Show answer
Why:
Moving between units is moving along the place-value table.
- A true, zeros never matter
- B false, the 4 would move to tenths
- C true, but only in measurement
- D false, the number would be longer
Show answer
Why:
They hold the tenths and hundredths places open.
- A true, if the number is exact
- B false, the unit decides the number
- C true, for lengths only
- D false, unless it is a whole number
Show answer
Why:
The same quantity has a different number in every unit.
- A one place
- B three places
- C no places
- D two places
Show answer
Why:
A hundred is two steps along the table, so 2.35 m becomes 235 cm.
Important short questions (7)
Tenths, then hundredths, then thousandths. Each is one tenth of the place before it.
One tenth, two hundredths and five thousandths.
They hold the tenths and hundredths places open, because this quantity has none of either. Without them the 4 would land in the tenths place.
A metre holds 100 centimetres. Two whole metres is 200 cm and 35 hundredths of a metre is 35 cm, giving 235 cm.
A kilogram holds 1000 grams. Twenty-five hundredths of 1000 grams is 250 grams.
In 45.07 the zero holds the tenths place, so the 7 means seven hundredths. In 45.70 the zero sits last and adds nothing at all.
The same quantity takes a different number in every unit. One distance is 3.08 km, 3080 m or 308000 cm, so the unit is part of the answer.
Long answer questions (3)
In a whole number the places run ones, tens, hundreds, each ten times the place on its right. Read the other way, each place is one tenth of the place on its left. Nothing in that rule says it must stop at the ones. Carrying it on gives tenths, then hundredths, then thousandths, and the decimal point simply marks where the ones end. Applying it to 0.125, the 1 sits in tenths, the 2 in hundredths and the 5 in thousandths. So the number is one tenth, two hundredths and five thousandths, which can also be read as a hundred and twenty-five thousandths.
Both conversions rest on the metric units being built on tens. A gram is a thousandth of a kilogram, so a kilogram holds 1000 grams. The packet is twenty-five hundredths of a kilogram, and twenty-five hundredths of 1000 grams is 250 grams. A millilitre is a thousandth of a litre, so a litre holds 1000 millilitres. The bottle is one whole litre and five tenths. Five tenths of 1000 millilitres is 500, so the bottle holds 1500 millilitres in all. In each case the digits moved along the place-value table rather than changing.
The pupil has read the 8 as eight tenths of a kilometre. Eight tenths of 1000 metres would indeed be 800 metres, and the pupil then seems to have mislaid a zero as well. In fact the 8 sits in the hundredths place, because the zero occupies the tenths place in front of it. Eight hundredths of 1000 metres is 80 metres. Added to the three whole kilometres, which are 3000 metres, the distance is 3080 metres. The zero was holding the tenths place open, and ignoring it made the answer ten times too large before the second slip made it smaller again.
Flashcards (17)
Play this topic
- Match the digit to what it counts
- How many places do the digits move?
- Fact or myth
- Order these steps for converting 3.08 km to metres
- Match each unit to its size
Comparing decimals, and adding and subtracting them
A longer decimal is not a larger one, because its extra digits sit in smaller places. Comparison runs place by place from the left and stops at the first difference. Addition and subtraction need the points in one column, with zeros padding the shorter number.
What you will be able to do
- Compare two decimals place by place and justify the result.
- Explain why more digits does not mean a larger decimal.
- Locate a decimal between two neighbours on a number line.
- Add and subtract decimals by lining up the points and padding.
- Diagnose a total produced by right-aligning the digits.
Key points
A whole number gains digits in bigger places; a decimal gains them in smaller ones.
Compare place by place from the left, and the first difference decides.
So 0.8 is larger than 0.75, because eight tenths beats seven.
Padding with end zeros changes no value and makes places line up.
A comparison has three outcomes, because 2.4 and 2.40 are equal.
Locating a decimal means cutting the gap it falls in into ten again.
For addition and subtraction the decimal points go in one column.
A carry out of the tenths lands in the ones, because ten tenths make a whole.
Right-aligning the digits instead of the points turns 9.45 into 9.41.
Full explanation
Longer does not mean larger
Which is more, 0.8 or 0.75? Many people answer 0.75, because it has more digits.
It is the smaller of the two. With whole numbers more digits does mean more, and that habit is exactly what goes wrong here.
Why the habit fails
In a whole number, extra digits are added on the LEFT, in bigger places. In a decimal, extra digits are added on the RIGHT, in smaller places.
So a long decimal is long because it is measured finely, not because it is large.
Compare place by place from the left
Start at the biggest place and work right. The first place where the two differ decides the answer, and nothing after it can change that.
0.8 has eight tenths. 0.75 has seven tenths. Eight tenths beats seven, so 0.8 is larger, and the 5 never gets a say.
Padding makes it obvious
Write 0.8 as 0.80. Now both numbers show two places, and the question is whether 80 hundredths beats 75 hundredths.
The padding changed nothing, because a zero on the end adds nothing. It only made the comparison easier to see.
A second example
Which is more, 1.09 or 1.9? Both have one whole, so move to the tenths.
1.09 has no tenths and 1.9 has nine. So 1.9 is far larger, even though 1.09 has more digits written.
A third, where the tenths tie
Compare 0.6 and 0.55. Six tenths against five tenths settles it at once, so 0.6 is larger.
Pad to 0.60 and 0.55 if it helps. Sixty hundredths against fifty-five hundredths says the same thing.
Equal numbers can look different
2.4 and 2.40 are the same number. Padding one of them to match the other does not make it bigger.
So comparison has three outcomes, not two. One may be larger, or smaller, or they may be equal while looking different.
Finding a decimal on the line
To place 0.75, find the gap between 0.7 and 0.8 and divide it into ten hundredths.
0.75 sits five of those steps along, exactly halfway across the gap. Locating a decimal is the same cutting that made it.
Lining up for addition
To add 5.6 and 3.85, write them one above the other with the points in a column.
Padding 5.6 to 5.60 keeps the columns honest. Now tenths sit under tenths and hundredths under hundredths.
Working the addition
5.60 plus 3.85. Five hundredths and no hundredths give five hundredths. Six tenths and eight tenths give fourteen tenths.
Fourteen tenths is one whole and four tenths, so the whole is carried. Five and three and the carried one make nine. The answer is 9.45.
Why the carry works the same way
Ten tenths make a whole, just as ten ones make a ten. So a carry out of the tenths column lands in the ones column.
Nothing new was needed. The carrying rule you already know simply applies one column further right.
A second addition
Add 12.5 and 0.75. Pad the first to 12.50 so both show hundredths.
Zero and five hundredths give five hundredths. Five tenths and seven tenths give twelve tenths, which is one whole and two tenths. Carrying gives 13.25.
Lining up for subtraction
Take 4.65 from 7.2. Pad 7.2 to 7.20 so that both numbers show hundredths.
Without that padding there is nothing above the 5, and the column cannot be worked at all.
Working the subtraction
7.20 take away 4.65. Zero hundredths cannot give five, so borrow a tenth, which is ten hundredths.
Now ten take five leaves five hundredths. One tenth remains above the 6, so borrow again: eleven tenths take six leaves five tenths. Six take four leaves two. The answer is 2.55.
A subtraction where the shorter number is below
Take 3.7 from 9.04. Pad the 3.7 to 3.70, because it is the number of places that must match, not the size.
Four hundredths take nothing leaves four. Zero tenths cannot give seven, so borrow a whole, which is ten tenths. Ten take seven leaves three. Eight take three leaves five. The answer is 5.34.
What carries forward
Comparison goes place by place from the left, and the first difference decides it. Length of writing is never the test.
Addition and subtraction need the points in a column, and padding with zeros is what makes the columns line up without changing any value.
Concepts and terms
Whole numbers take extra digits on the left, in bigger places, while decimals take them on the right, in smaller ones. Length records fineness, not size.
Work from the biggest place towards the smallest. The first place where the two numbers differ settles the answer, and nothing after it can change that.
Writing end zeros so that both numbers show the same places. It records none of that place, so it adds nothing, and it makes the columns line up.
In an addition or subtraction the decimal points sit in one column, which puts tenths under tenths. Aligning the last digits instead adds unlike places together.
Ten tenths make one whole and one tenth is ten hundredths, so the familiar carry and borrow work unchanged, simply one column further right.
Watch out
Common misconceptions
The weighing and the bill
Scenario:
A shopkeeper weighs 5.6 kilograms of rice and 3.85 kilograms of flour for one customer. He writes the total as 9.41 kilograms, having set the two numbers out one under the other with their last digits in line. The customer, who has brought her own ribbon to tie the sacks, also compares two lengths on the counter: one piece measures 0.8 metres and another measures 0.75 metres. She says the second is the longer of the two because it has more figures. The shopkeeper then takes 4.65 kilograms back out of a 7.2 kilogram bag and is unsure how to begin.
Analysis:
Both the shopkeeper and the customer are applying whole-number habits: right-aligning the digits, and treating length of writing as size.
Activity:
Weigh or measure three things at home, write each to two decimal places, then put them in order and find the total.
Pad, line up, and check
Title:
Pad, line up, and check
Safety:
No safety concerns. Paper, ruler and pencil only.
What You Notice:
The wrong setting-out gives 9.41 and the right one gives 9.45. The gap is four hundredths, and it appears the moment tenths are added to hundredths. Lining up the points is the whole of the method.
Did you know
A comparison of decimals has three possible answers, because two numbers can look different and still be equal.
Right-aligning 5.6 and 3.85 gives 9.41 instead of 9.45, and the gap of four hundredths is exactly the misplaced tenths.
Locating a decimal on a number line uses the very same cutting that created the decimal places in the first place.
Memory tricks
Points in a column, zeros on the end.
First difference from the left decides.
Practice MCQs with answers (25)
- A 0.75
- B they are equal
- C it cannot be decided
- D 0.8
Show answer
Why:
Eight tenths beats seven tenths, so the 5 never matters.
- A 1.09
- B they are equal
- C 1.9
- D it cannot be decided
Show answer
Why:
The wholes tie; then nine tenths beats no tenths.
- A 0.55
- B they are equal
- C it cannot be decided
- D 0.6
Show answer
Why:
Six tenths beats five tenths.
- A they are equal
- B 2.40 is larger
- C 2.4 is larger
- D it cannot be decided
Show answer
Why:
The trailing zero adds no hundredths.
- A place by place from the left
- B place by place from the right
- C by counting the digits
- D by adding them first
Show answer
Why:
The first place where they differ settles it.
- A is ignored
- B decides the answer
- C must be the tenths
- D is checked last
Show answer
Why:
No digit after that place can change the outcome.
- A its value, making it larger
- B its value, making it smaller
- C the place of the 8
- D nothing about its value
Show answer
Why:
A zero at the end records no hundredths.
- A Both true, and the reason explains the assertion
- B Both true, but the reason does not explain it
- C The assertion is true and the reason is false
- D The assertion is false and the reason is true
Show answer
Why:
Both are true and the second explains the first. 0.75 has more digits than 0.8, and 0.8 padded to 0.80 shows which is larger.
- A halfway between 0.5 and 1
- B halfway between 0.7 and 0.8
- C just below 0.7
- D just above 0.8
Show answer
Why:
That gap cuts into ten hundredths, and 0.75 is five steps along.
- A line up the decimal points
- B line up the last digits
- C remove the points
- D make both whole
Show answer
Why:
Then tenths sit under tenths and hundredths under hundredths.
- A 9.41
- B 9.45
- C 8.45
- D 9.5
Show answer
Why:
Pad to 5.60; fourteen tenths carry one whole across.
- A 13.25
- B 12.25
- C 13.15
- D 20.0
Show answer
Why:
Pad to 12.50; twelve tenths carry one whole across.
- A 2.55
- B 2.65
- C 3.55
- D 2.45
Show answer
Why:
Pad to 7.20, then borrow a tenth as ten hundredths.
- A 5.97
- B 6.34
- C 5.34
- D 5.44
Show answer
Why:
Pad the 3.7 to 3.70, then borrow a whole as ten tenths.
- A 5.06
- B 5.65
- C 5.60
- D 56.0
Show answer
Why:
A zero in the hundredths place keeps the columns honest.
- A the hundredths column
- B the ones column
- C the tens column
- D no column at all
Show answer
Why:
Ten tenths make one whole.
- A one hundredth
- B a hundred hundredths
- C ten tenths
- D ten hundredths
Show answer
Why:
A tenth is made of ten hundredths.
- A the answer would be too large
- B there is nothing above the 5
- C 7.2 is the smaller number
- D the points do not line up
Show answer
Why:
That column cannot be worked without a hundredths digit.
- A 9.04, to 9.040
- B neither of them
- C both of them
- D 3.7, to 3.70
Show answer
Why:
It is the number of places that must match, not the size.
- A I and II only
- B II and III only
- C I and III only
- D I, II and III
Show answer
Why:
All three are true. Aligning the points and padding 5.6 to 5.60 is exactly what gives 9.45 rather than 9.41.
- A larger or smaller only
- B equal only when the digits match
- C larger, smaller or equal
- D none of these
Show answer
Why:
2.4 and 2.40 look different and are equal.
- A Each gap is cut into ten again, exactly as the places were made
- B The number line has to be redrawn for every decimal that is placed
- C A decimal can only be placed once both its neighbours are known
- D Cutting a gap into ten is the only way to compare two decimals
Show answer
Why:
Tenths came from cutting the whole into ten, and hundredths from cutting a tenth into ten. Finding 0.42 does the same two cuts.
- A the 0.75 m ribbon
- B the 0.8 m ribbon
- C neither, they are equal
- D impossible to tell
Show answer
Why:
Eighty hundredths of a metre against seventy-five.
- A false, it is larger
- B false, it is smaller
- C true, the measure is finer
- D true only for lengths
Show answer
Why:
Extra places record smaller parts, not greater size.
- A true, it makes it larger
- B true, it makes it smaller
- C false, padding only adds empty places
- D false, but only beyond the point
Show answer
Why:
A zero at the end records none of that place.
Important short questions (7)
0.8 is larger. Eight tenths beats seven tenths, so the 5 in the hundredths place never gets a say.
Work place by place from the left. The first place where they differ decides the answer.
A whole number gains digits on the left, in bigger places. A decimal gains them on the right, in smaller places, so extra digits only record finer parts.
Pad 5.6 to 5.60 and line up the points. Then 5.60 plus 3.85 gives 9.45.
Pad 7.2 to 7.20 and line up the points. Then 7.20 take 4.65 leaves 2.55.
Because ten tenths make one whole. It is the same regrouping as ten ones making a ten, applied one column further right.
The 3.7 is padded to 3.70. It is the number of decimal places that has to match, not the size of the number.
Long answer questions (3)
The pupil has carried a habit over from whole numbers. There, extra digits are added on the left, in bigger places, so a longer number really is larger. In a decimal the extra digits are added on the right, in smaller places, so length records how finely something was measured rather than how much there is. Comparing properly means working place by place from the left. Both numbers have no wholes, so we move to the tenths. 0.8 has eight tenths and 0.75 has seven, so 0.8 is the larger and the 5 never matters. Padding 0.8 to 0.80 shows the same thing, because eighty hundredths beats seventy-five.
First pad 7.2 to 7.20 and write the numbers with their points in one column. Without that padding there is no digit above the 5, and the hundredths column cannot be worked at all. Now take the columns from the right. Zero hundredths cannot give five, so borrow one tenth, which is ten hundredths, and ten take five leaves five hundredths. The tenths column now holds one, which cannot give six, so borrow one whole as ten tenths. Eleven take six leaves five tenths. In the ones, six take four leaves two. The answer is 2.55.
The total should be 9.45 kg. The shopper has lined up the last digits written instead of the decimal points, so the six tenths of the rice sat under the eight hundredths of the flour. Digits in different places were then added together, which is why the hundredths came out as one instead of five. Setting the sum out correctly means padding 5.6 to 5.60 so that both numbers show hundredths, then placing the points in a single column. Five hundredths come straight down. Six tenths and eight tenths give fourteen tenths, which is one whole and four tenths, so one is carried into the ones. That gives 9.45 kg.
Flashcards (17)
Play this topic
- Match each sum to its answer
- Which is the larger of each pair?
- Fact or myth
- Order these steps for taking 4.65 from 7.2
- Match each idea to what it means
Class 7 Mathematics Chapter 3 — frequently asked questions
One tenth is 0.1 and one hundredth is 0.01.
Ten tenths make a whole, so they are regrouped and written as one whole. It is the same step as ten ones making a ten.
The 2 counts wholes, the 3 counts tenths and the 6 counts hundredths.
Tenths, then hundredths, then thousandths. Each is one tenth of the place before it.
One tenth, two hundredths and five thousandths.
They hold the tenths and hundredths places open, because this quantity has none of either. Without them the 4 would land in the tenths place.
0.8 is larger. Eight tenths beats seven tenths, so the 5 in the hundredths place never gets a say.
Work place by place from the left. The first place where they differ decides the answer.
A whole number gains digits on the left, in bigger places. A decimal gains them on the right, in smaller places, so extra digits only record finer parts.
All Class 7 Mathematics chapters
All Mathematics chapters · Every Class 7 subject · The whole NCERT library
Study this chapter properly, free
Everything above is on this page to read. In the Veda app the same chapter becomes spaced-repetition flashcards, timed tests that mark themselves, Veda Bites you can swipe, and games — with your progress tracked topic by topic.
Get Veda free