What Lives in the Space Between 4 and 5
Learn why we need pieces smaller than one, what each digit after the point is worth, how to switch between fractions and decimals, and how to write money and lengths.
Why do we need numbers smaller than one?
Because whole numbers run out. Between 4 and 5 there is nothing at all in whole numbers, yet a real measurement often lands there — a stick 4 and a half centimetres long, or a price of four and a half rupees. Decimals give names to those in-between amounts by splitting each unit into ten equal parts, then splitting again.
This page covers everything in the CBSE Class 7 Mathematics chapter's first half: tenths, hundredths and thousandths, decimal place value and expanded form, converting between fractions and decimals, and writing money and measures in decimal form.
This page covers everything in the CBSE Class 7 Mathematics chapter's first half: tenths, hundredths and thousandths, decimal place value and expanded form, converting between fractions and decimals, and writing money and measures in decimal form.
What are tenths, hundredths and thousandths?
Split one whole into ten equal parts and each part is a tenth, written or 0.1. Split a tenth into ten again and each piece is a hundredth, or 0.01. Split once more and you have a thousandth, or 0.001.
So each step to the right is one-tenth the size of the step before it — the same pattern as whole numbers, continued past the point.
Three tenths is . Seven hundredths is . Note the zero in 0.07: it holds the tenths place empty, and leaving it out would give 0.7, which is ten times larger.
For example, a shopkeeper's weighing scale showing 0.250 kg is showing 250 thousandths of a kilogram — that is, 250 grams.
The zero directly after the point is the one students drop most often, and it changes the value every time.
So each step to the right is one-tenth the size of the step before it — the same pattern as whole numbers, continued past the point.
Three tenths is . Seven hundredths is . Note the zero in 0.07: it holds the tenths place empty, and leaving it out would give 0.7, which is ten times larger.
For example, a shopkeeper's weighing scale showing 0.250 kg is showing 250 thousandths of a kilogram — that is, 250 grams.
The zero directly after the point is the one students drop most often, and it changes the value every time.
How do you write a decimal in expanded form?
Give every digit its place value and add them.
Take 4.375. The 4 is in the ones place, the 3 in tenths, the 7 in hundredths and the 5 in thousandths:
You can also write it with decimals instead of fractions:
Another: , with nothing in the tenths place.
Reading it aloud helps: 4.375 is "four point three seven five", said digit by digit after the point — not "four point three hundred seventy-five", which suggests a whole number and hides the place values.
The expanded form is what makes comparing and adding decimals straightforward later, because it shows exactly which places line up with which.
Take 4.375. The 4 is in the ones place, the 3 in tenths, the 7 in hundredths and the 5 in thousandths:
You can also write it with decimals instead of fractions:
Another: , with nothing in the tenths place.
Reading it aloud helps: 4.375 is "four point three seven five", said digit by digit after the point — not "four point three hundred seventy-five", which suggests a whole number and hides the place values.
The expanded form is what makes comparing and adding decimals straightforward later, because it shows exactly which places line up with which.
How do you convert between fractions and decimals?
When the denominator is 10, 100 or 1000 the conversion is immediate: the number of zeros tells you how many decimal places to use.
Going the other way, write the digits after the point over the matching power of ten and simplify:
For other fractions, first make the denominator 10, 100 or 1000 if you can:
Those three are worth knowing on sight, since they appear constantly.
If the denominator cannot be turned into a power of ten, simply divide the numerator by the denominator — , which is exactly the number from the previous section.
Going the other way, write the digits after the point over the matching power of ten and simplify:
For other fractions, first make the denominator 10, 100 or 1000 if you can:
Those three are worth knowing on sight, since they appear constantly.
If the denominator cannot be turned into a power of ten, simply divide the numerator by the denominator — , which is exactly the number from the previous section.
How do you write money and lengths as decimals?
Both work the same way, because both use units built in tens and hundreds.
For money, 100 paise make 1 rupee, so paise are hundredths of a rupee:
So Rs 2.50 means two rupees and fifty paise, and 5 paise is Rs 0.05 — the zero in the tenths place again mattering.
For length, 10 mm make 1 cm, so millimetres are tenths of a centimetre:
And since 100 cm make a metre, 3 m 40 cm is 3.40 m.
For example, a shop bill reading Rs 47.75 is forty-seven rupees and seventy-five paise, and a tailor measuring 1.65 m is measuring 1 m 65 cm.
The unit must always be stated. Writing "2.50" without saying rupees leaves the number meaningless, and in measurement questions the unit usually carries a mark of its own.
For money, 100 paise make 1 rupee, so paise are hundredths of a rupee:
So Rs 2.50 means two rupees and fifty paise, and 5 paise is Rs 0.05 — the zero in the tenths place again mattering.
For length, 10 mm make 1 cm, so millimetres are tenths of a centimetre:
And since 100 cm make a metre, 3 m 40 cm is 3.40 m.
For example, a shop bill reading Rs 47.75 is forty-seven rupees and seventy-five paise, and a tailor measuring 1.65 m is measuring 1 m 65 cm.
The unit must always be stated. Writing "2.50" without saying rupees leaves the number meaningless, and in measurement questions the unit usually carries a mark of its own.
Exam tip
The mistake most students make with the zero after the point
Students write 0.7 when they mean seven hundredths, losing a factor of ten.
Count the places deliberately: tenths first, hundredths second, thousandths third. Seven hundredths needs the hundredths place filled, so the tenths place must be held by a zero — 0.07.
The same care applies when converting money. Five paise is five hundredths of a rupee, so Rs 0.05, not Rs 0.5, which would be fifty paise — a tenfold error in real money.
A quick check: the number of digits after the point must equal the number of zeros in the fraction's denominator. has two zeros, so its decimal has two places.
Count the places deliberately: tenths first, hundredths second, thousandths third. Seven hundredths needs the hundredths place filled, so the tenths place must be held by a zero — 0.07.
The same care applies when converting money. Five paise is five hundredths of a rupee, so Rs 0.05, not Rs 0.5, which would be fifty paise — a tenfold error in real money.
A quick check: the number of digits after the point must equal the number of zeros in the fraction's denominator. has two zeros, so its decimal has two places.
Did you know
Why does each place get ten times smaller after the point?
Because the whole number system is built on tens, and the decimal point does not interrupt that — it only marks where the whole units end.
Moving left, each place is ten times bigger: ones, tens, hundreds. Moving right, each place is ten times smaller: tenths, hundredths, thousandths. It is one continuous pattern, and the point is just a signpost in the middle of it.
Moving left, each place is ten times bigger: ones, tens, hundreds. Moving right, each place is ten times smaller: tenths, hundredths, thousandths. It is one continuous pattern, and the point is just a signpost in the middle of it.
Key takeaways
Decimals and place value: quick revision
- Decimals name amounts between whole numbers by splitting each unit into ten, then ten again: tenths 0.1, hundredths 0.01, thousandths 0.001.
- A zero immediately after the point holds the tenths place, so 0.07 and 0.7 differ by a factor of ten.
- Expanded form gives every digit its place value, as .
- Denominators of 10, 100 or 1000 convert directly; , and are worth knowing on sight.
- 100 paise make a rupee and 10 mm make a centimetre, so 250 paise is Rs 2.50 and 7 cm 5 mm is 7.5 cm — always with the unit stated.
You will remember all of this far better after answering five questions on it than after reading it twice.
- A zero immediately after the point holds the tenths place, so 0.07 and 0.7 differ by a factor of ten.
- Expanded form gives every digit its place value, as .
- Denominators of 10, 100 or 1000 convert directly; , and are worth knowing on sight.
- 100 paise make a rupee and 10 mm make a centimetre, so 250 paise is Rs 2.50 and 7 cm 5 mm is 7.5 cm — always with the unit stated.
You will remember all of this far better after answering five questions on it than after reading it twice.