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CBSE · Class 7 · Mathematics

Chapter 3: A Peek Beyond the Point (Decimals)

5 concepts · 40 practice questions with answers · Free to read

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Tenths, hundredths and thousandths

From daily life

A pencil is 14 cm and 7 mm long. 7 mm is 7 tenths of a centimetre, so the length is 14.7 cm. A decimal point lets us write parts smaller than 1.

Simple definition

Cut 1 into 10 equal parts: each part is a tenth = 1/10 = 0.1. Cut a tenth into 10: a hundredth = 1/100 = 0.01. Again: a thousandth = 1/1000 = 0.001. A decimal has a whole part, a decimal point and a fraction part.

37/100 = 0.3737

In the figure: A square cut into 100 equal boxes; 37 are shaded: 37/100 = 0.37 (3 full columns = 3 tenths, plus 7 hundredths).

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Tenths, hundredths and thousandths

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Rule 1: Fraction to decimal — The number of zeros in the denominator (10, 100, 1000) is the number of digits after the point.

  • Easy: 7/10 = 0.7.
  • Medium: 23/100 = 0.23.
  • Exam-type: 8/1000 = 0.008 (two zeros fill the gap).

Rule 2: Decimal to fraction — Write the digits over 10, 100 or 1000, then simplify if you can.

  • Easy: 0.9 = 9/10.
  • Medium: 0.45 = 45/100 = 9/20.
  • Exam-type: 0.375 = 375/1000 = 3/8.

Rule 3: Read a decimal — Read the whole part as a number, say "point", then read each digit after the point one by one.

  • Easy: 0.5 = zero point five.
  • Medium: 3.07 = three point zero seven.
  • Exam-type: 0.274 = zero point two seven four (not "twenty-seven four").
Remember

One digit after the point means tenths, two digits hundredths, three digits thousandths.

Common mistake

Writing 7/100 as 0.7. It is 0.07, because 100 has two zeros.

Shortcut

Count the zeros in the denominator and give that many places after the point. Memory clue: "Zeros below = digits beyond the point."

Spot the difference

PartTenthHundredthThousandth
Fraction1/101/1001/1000
Decimal0.10.010.001
Digits after point123

Practice questions

Easy
  1. 1Write 6/10 as a decimal.
  2. 2Write 0.9 as a fraction.
  3. 3Write 47/100 as a decimal.
Medium
  1. 4Write 8/1000 as a decimal.
  2. 5Write 0.45 as a fraction in lowest terms.
  3. 6How do you read 3.07?
Challenging
  1. 7Write 125/1000 as a decimal and as a simple fraction.
  2. 8Write 7/100 + 3/10 as a decimal.
Show answers (try them yourself first)
  1. 10.6
  2. 29/10
  3. 30.47
  4. 40.008
  5. 59/20
  6. 6three point zero seven
  7. 70.125 = 1/8
  8. 80.37 (0.30 + 0.07)

Decimal place value and expanded form

From daily life

A petrol pump bill shows 24.375 litres. Every digit has its own place: the digits after the point are tenths, hundredths and thousandths.

Simple definition

Places to the left of the point are ones, tens, hundreds. To the right they are tenths (1/10), hundredths (1/100), thousandths (1/1000). Every place is 10 times the place on its right. The expanded form adds the value of every digit.

2T4O31/1071/10051/1000whole partfraction part24.375 = 20 + 4 + 3/10 + 7/100 + 5/1000

In the figure: 24.375: the 3 is 3 tenths, the 7 is 7 hundredths and the 5 is 5 thousandths.

Rule 1: Value of a digit — Value = digit × value of its place.

  • Easy: In 5.24 the 2 stands for 2 tenths = 0.2.
  • Medium: In 7.386 the 8 stands for 8 hundredths = 0.08.
  • Exam-type: In 120.509 the 9 stands for 9 thousandths = 0.009.

Rule 2: Expanded form — Write each digit with its place value and add; a zero digit contributes nothing.

  • Easy: 3.6 = 3 + 6/10.
  • Medium: 7.05 = 7 + 0/10 + 5/100 = 7 + 5/100.
  • Exam-type: 0.274 = 2/10 + 7/100 + 4/1000.

Rule 3: Ten times, ten times again — Multiplying by 10 moves the point one place right; dividing by 10 moves it one place left.

  • Easy: 0.3 is 10 times 0.03.
  • Medium: 0.06 × 10 = 0.6.
  • Exam-type: 4.5 × 100 = 450.
Remember

The point only separates the whole part from the fraction part; digits after it are read one by one.

Common mistake

Writing 7.05 as 7.5. The zero holds the tenths place; without it 5 would become 5 tenths.

Shortcut

Draw a small place-value table and put one digit in each box. Memory clue: Names ending in "ths" (tenths, hundredths, thousandths) live after the point.

Spot the difference

PointLeft of the pointRight of the point
Placesones, tens, hundredstenths, hundredths, thousandths
Values1, 10, 1001/10, 1/100, 1/1000
Going outward×10 each step÷10 each step

Practice questions

Easy
  1. 1In 8.35, which place does the digit 3 occupy?
  2. 2Write 3.6 in expanded form.
  3. 3What is the value of 8 in 7.386?
Medium
  1. 4Write 0.274 in expanded form.
  2. 50.3 is 10 times which number?
  3. 6Find 4.5 × 100.
Challenging
  1. 7What is the value of 9 in 120.509?
  2. 8Write 7.05 in expanded form. Why can the 0 not be dropped?
Show answers (try them yourself first)
  1. 1tenths place
  2. 23 + 6/10
  3. 30.08
  4. 42/10 + 7/100 + 4/1000
  5. 50.03
  6. 6450
  7. 70.009
  8. 87 + 5/100; the 0 keeps 5 in the hundredths place (without it 7.5 = 7 + 5/10)

Units of measurement and decimals

From daily life

A tailor measures cloth as 2 m 35 cm, which is 2.35 m. A parcel of 3 kg 250 g weighs 3.250 kg. Decimals help us change units.

Simple definition

1 cm = 10 mm, 1 m = 100 cm, 1 km = 1000 m, 1 kg = 1000 g, 1 L = 1000 mL, ₹1 = 100 paise. To go to a bigger unit, divide (point moves left); to go to a smaller unit, multiply (point moves right).

012345675.6 cm = 56 mm1 cm = 10 mm

In the figure: The mark is at 56 small steps (mm) = 5 cm and 6 mm = 5.6 cm.

Rule 1: Small unit to big unit — Divide by 10, 100 or 1000 (move the point left).

  • Easy: 56 mm = 56 ÷ 10 = 5.6 cm.
  • Medium: 235 cm = 235 ÷ 100 = 2.35 m.
  • Exam-type: 4,750 m = 4,750 ÷ 1000 = 4.75 km.

Rule 2: Big unit to small unit — Multiply by 10, 100 or 1000 (move the point right).

  • Easy: 3.5 cm = 35 mm.
  • Medium: 2.35 kg = 2,350 g.
  • Exam-type: 0.75 km = 750 m.

Rule 3: Mixed units — Change the small unit to the big unit using the right number of places, then add the parts.

  • Easy: 2 m 35 cm = 2.35 m.
  • Medium: 3 kg 250 g = 3.250 kg = 3.25 kg.
  • Exam-type: ₹45 and 7 paise = ₹45.07.
Remember

A misplaced point can be dangerous: 0.5 mg of a medicine is 10 times 0.05 mg.

Common mistake

Writing 3 kg 50 g as 3.50 kg. Grams need 3 places: 3.050 kg.

Shortcut

Zeros to remember: cm-mm 1, m-cm 2, km-m 3, kg-g 3, ₹-paise 2. Memory clue: "1-2-3": cm→mm is 1 zero, m→cm is 2, km→m is 3.

Spot the difference

PointSmall → bigBig → small
Operationdividemultiply
Point movesleftright
Example56 mm = 5.6 cm5.6 cm = 56 mm

Practice questions

Easy
  1. 1Convert 56 mm to cm.
  2. 2Convert 2.35 m to cm.
  3. 3Write 2 m 35 cm in metres.
Medium
  1. 4Convert 4,750 m to km.
  2. 5Convert 3.5 kg to g.
  3. 6Write 3 kg 50 g in kg.
Challenging
  1. 7Convert 0.75 km to m.
  2. 8A doctor writes 0.05 mg, a nurse gives 0.5 mg. How many times the dose is given?
Show answers (try them yourself first)
  1. 15.6 cm
  2. 2235 cm
  3. 32.35 m
  4. 44.75 km
  5. 53,500 g
  6. 63.05 kg (3.050)
  7. 7750 m
  8. 810 times (0.5 ÷ 0.05 = 10)

Locating, comparing and ordering decimals

From daily life

Three runners finish a race in 12.4, 12.04 and 12.40 seconds. To say who was fastest we must compare decimals correctly.

Simple definition

Equivalent decimals: adding zeros at the end does not change the value: 0.2 = 0.20 = 0.200 (but 0.02 is different). To compare: first compare the whole parts, then tenths, then hundredths, then thousandths; the first difference decides. On a number line, cut the gap between two whole numbers into 10 equal steps to mark tenths.

88.598.38.750.11 tenth = 1 small step

In the figure: Between 8 and 9 there are 10 small steps of 0.1. 8.3 is 3 steps after 8 and 8.75 is halfway between 8.7 and 8.8.

Rule 1: Equivalent decimals — Zeros at the end can be added or dropped; zeros right after the point cannot.

  • Easy: 0.5 = 0.50.
  • Medium: 3.40 = 3.4.
  • Exam-type: 0.2 = 0.20 but 0.02 is ten times smaller.

Rule 2: Compare two decimals — Make the number of decimal places equal by adding zeros, then compare like whole numbers.

  • Easy: 6.2 > 5.9 (6 > 5).
  • Medium: 4.35 > 4.29 (tenths 3 > 2).
  • Exam-type: 0.7 > 0.68 because 0.70 > 0.68.

Rule 3: Order decimals — Write all with the same number of places, order them as whole numbers, then remove the extra zeros.

  • Easy: 0.6, 0.06, 0.66 → 0.06 < 0.6 < 0.66.
  • Medium: 1.23, 1.3, 1.203 → 1.230, 1.300, 1.203 → 1.203 < 1.23 < 1.3.
  • Exam-type: Descending: 5.5 > 5.055 > 5.05 > 5.005.
Remember

More digits does not mean a bigger decimal: 0.4 is bigger than 0.35.

Common mistake

Thinking 0.35 > 0.4 because 35 > 4. Write 0.4 as 0.40 and compare 40 with 35.

Shortcut

Pad with zeros to the same length, then compare as whole numbers. Memory clue: "Pad with zeros, then race the numbers."

Spot the difference

PointWhole numbersDecimals
More digitsbigger numbernot always bigger
Comparecount digits firstequalise places, then compare

Practice questions

Easy
  1. 1Which is greater: 0.7 or 0.07?
  2. 2Write 3.40 in its simplest form.
  3. 3Compare 6.2 and 5.9.
Medium
  1. 4Compare 0.35 and 0.4.
  2. 5Arrange in ascending order: 0.6, 0.06, 0.66.
  3. 6Write a decimal that lies between 8.3 and 8.4.
Challenging
  1. 7Arrange in descending order: 5.05, 5.5, 5.055, 5.005.
  2. 8How many hundredths lie strictly between 2.5 and 2.6?
Show answers (try them yourself first)
  1. 10.7
  2. 23.4
  3. 36.2 > 5.9
  4. 40.35 < 0.4 (0.35 < 0.40)
  5. 50.06 < 0.6 < 0.66
  6. 68.35 (or 8.31, 8.32 … 8.39)
  7. 75.5 > 5.055 > 5.05 > 5.005
  8. 89 (2.51 to 2.59)

Adding and subtracting decimals

From daily life

A shopkeeper sells 2.75 kg of rice and 1.5 kg of wheat. Total weight: 4.25 kg. We add decimals just like whole numbers, but with the points in a line.

Simple definition

Write the numbers one under the other so that the decimal points are in a line; fill empty places with zeros; add or subtract as for whole numbers; put the point straight down in the answer.

2 . 7 5+ 1 . 5 04 . 2 5points in a line↑

In the figure: 2.75 + 1.50 = 4.25. The extra zero in 1.50 makes both numbers have two decimal places.

Rule 1: Add decimals — Align the points, add from the right, carry as usual.

  • Easy: 0.4 + 0.3 = 0.7.
  • Medium: 2.75 + 1.5 = 2.75 + 1.50 = 4.25.
  • Exam-type: 12.08 + 5.9 + 0.375 = 12.080 + 5.900 + 0.375 = 18.355.

Rule 2: Subtract decimals — Align the points, fill zeros, borrow as usual. A whole number like 10 is 10.000.

  • Easy: 0.9 – 0.4 = 0.5.
  • Medium: 5.2 – 1.75 = 5.20 – 1.75 = 3.45.
  • Exam-type: 10 – 3.062 = 10.000 – 3.062 = 6.938.

Rule 3: Word problems — Decide whether to add or subtract, keep the units the same, then calculate.

  • Easy: ₹12.50 + ₹7.25 = ₹19.75.
  • Medium: A 5 m ribbon: 5 – 2.35 = 2.65 m is left.
  • Exam-type: Bags of 1.25, 2.4 and 0.875 kg: total 4.525 kg.
Remember

Check a subtraction by adding the answer back: 3.45 + 1.75 = 5.20.

Common mistake

Not lining up the points: 2.75 + 1.5 done as 75 + 5 gives 2.90, which is wrong.

Shortcut

Pad with zeros so every number has the same decimal places. Memory clue: "Points in a line, zeros fill the gap, drop the point straight down."

Spot the difference

PointAddingSubtracting
Alignpoints in a linepoints in a line
Example2.75 + 1.50 = 4.255.20 – 1.75 = 3.45
Checksubtract one partadd the answer back

Practice questions

Easy
  1. 1Find 0.4 + 0.3.
  2. 2Find 0.9 – 0.4.
  3. 3Find 2.75 + 1.5.
Medium
  1. 4Find 5.2 – 1.75.
  2. 5Find ₹12.50 + ₹7.25.
  3. 6Find 10 – 3.062.
Challenging
  1. 7Find 12.08 + 5.9 + 0.375.
  2. 8From a 5 m ribbon 2.35 m and then 1.4 m are cut. How much is left?
Show answers (try them yourself first)
  1. 10.7
  2. 20.5
  3. 34.25
  4. 43.45
  5. 5₹19.75
  6. 66.938
  7. 718.355
  8. 81.25 m (5 – 2.35 – 1.4)