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A Peek Beyond the Point

NCERT Class 7 · Mathematics Based on NCERT Class 7 Mathematics textbook · Free CBSE study kit

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Chapter Notes

Overview
This chapter introduces the concept of decimal numbers as a natural extension of the Indian place value system to measure quantities smaller than one. It explains why we need smaller units (tenths, hundredths, thousandths) for accurate measurement, how to read, write, compare, add, and subtract decimals, and how decimals are used in everyday life (length, weight, money). The one big idea is that just as 10 ones make a ten, 10 tenths make a one, and this pattern continues to the right of the decimal point, allowing us to measure and calculate with precision.
Key Concepts (15)
Tenth (one-tenth)
One of ten equal parts of a unit. 1/10 = 0.1. Ten tenths make one unit.
e.g. If a chocolate bar is divided into 10 equal pieces, each piece is one-tenth (0.1) of the whole bar.
Hundredth (one-hundredth)
One of one hundred equal parts of a unit. 1/100 = 0.01. One hundred hundredths make one unit.
e.g. If a ₹1 coin is divided into 100 equal paise, each paisa is one-hundredth (0.01) of a rupee.
Thousandth (one-thousandth)
One of one thousand equal parts of a unit. 1/1000 = 0.001. One thousand thousandths make one unit.
e.g. 1 milligram is one-thousandth (0.001) of a gram.
Decimal point
A dot (.) used to separate the whole number part from the fractional part in a decimal number.
e.g. In 7.05, the decimal point separates 7 (units) from 05 (5 hundredths).
Decimal notation
A way of writing numbers that uses a decimal point to show tenths, hundredths, thousandths, etc., as an extension of the Indian place value system.
e.g. Writing 2 and 7 tenths as 2.7 instead of 2 7/10.
Place value (decimal extension)
The value of a digit based on its position. To the left of the decimal point: … thousands, hundreds, tens, units. To the right: tenths, hundredths, thousandths. Each place is 10 times the place to its right.
e.g. In 70.5, the digit 7 is in the tens place (7 × 10), 0 is in units (0 × 1), and 5 is in tenths (5 × 1/10).
Conversion (mm to cm)
1 cm = 10 mm, so 1 mm = 1/10 cm = 0.1 cm.
e.g. 5 mm = 5/10 cm = 0.5 cm. 12 mm = 1.2 cm.
Conversion (cm to m)
1 m = 100 cm, so 1 cm = 1/100 m = 0.01 m.
e.g. 15 cm = 15/100 m = 0.15 m.
Conversion (g to kg)
1 kg = 1000 g, so 1 g = 1/1000 kg = 0.001 kg.
e.g. 254 g = 254/1000 kg = 0.254 kg.
Conversion (paise to rupees)
₹1 = 100 paise, so 1 paisa = 1/100 rupee = 0.01 rupee.
e.g. 75 paise = 75/100 rupee = 0.75 rupee.
Equivalent decimals
Decimals that represent the same value even if they have different numbers of digits after the decimal point (e.g., 0.2 = 0.20 = 0.200).
e.g. 0.2, 0.20, and 0.200 all mean 2 tenths. Adding zeros to the right does not change the value.
Comparing decimals
To compare two decimals, start from the leftmost (most significant) digit and compare place by place. The number with the larger digit at the first differing place is greater.
e.g. 6.465 > 6.456 because at the hundredths place, 6 > 5.
Addition of decimals
Align the decimal points vertically and add digits column by column, carrying over when necessary, just like whole numbers.
e.g. 2.7 + 3.5 = 6.2 (7 tenths + 5 tenths = 12 tenths = 1 unit and 2 tenths).
Subtraction of decimals
Align the decimal points vertically. If needed, borrow from the next higher place value (e.g., 1 unit = 10 tenths, 1 tenth = 10 hundredths).
e.g. 3.5 - 2.7 = 0.8 (borrow 1 unit from 3 to make 15 tenths, subtract 7 tenths to get 8 tenths).
Number line (decimals)
A line on which decimal numbers are located by dividing the interval between two whole numbers into 10 equal parts (for tenths), and further subdividing for hundredths, etc.
e.g. 1.4 lies between 1 and 2, four-tenths of the way from 1 to 2.
Formulas & Equations (10)
1 cm = 10 mm
Reads as: One centimetre equals ten millimetres.
Variables: cm = centimetre, mm = millimetre
Use when: Converting length from centimetres to millimetres or vice versa.
⚠ Writing 1 mm = 10 cm instead of 1 mm = 0.1 cm.
1 m = 100 cm
Reads as: One metre equals one hundred centimetres.
Variables: m = metre, cm = centimetre
Use when: Converting length from metres to centimetres or vice versa.
⚠ Forgetting that 1 cm = 0.01 m, not 0.1 m.
1 kg = 1000 g
Reads as: One kilogram equals one thousand grams.
Variables: kg = kilogram, g = gram
Use when: Converting weight from kilograms to grams or vice versa.
⚠ Writing 1 g = 0.01 kg instead of 0.001 kg.
₹1 = 100 paise
Reads as: One rupee equals one hundred paise.
Variables: ₹ = rupee, p = paisa
Use when: Converting money from rupees to paise or vice versa.
⚠ Writing 1 paisa = 0.1 rupee instead of 0.01 rupee.
1 g = 1000 mg
Reads as: One gram equals one thousand milligrams.
Variables: g = gram, mg = milligram
Use when: Converting weight from grams to milligrams or vice versa.
⚠ Writing 1 mg = 0.1 g instead of 0.001 g.
10 tenths = 1 unit
Reads as: Ten tenths make one unit.
Variables: tenths = 1/10, unit = 1
Use when: Understanding place value and carrying over in decimal addition/subtraction.
⚠ Thinking 10 tenths = 10 units.
10 hundredths = 1 tenth
Reads as: Ten hundredths make one tenth.
Variables: hundredths = 1/100, tenth = 1/10
Use when: Converting between tenths and hundredths in decimal operations.
⚠ Thinking 100 hundredths = 1 tenth instead of 1 unit.
100 hundredths = 1 unit
Reads as: One hundred hundredths make one unit.
Variables: hundredths = 1/100, unit = 1
Use when: Understanding place value and regrouping in decimal arithmetic.
⚠ Thinking 100 hundredths = 10 units.
10 thousandths = 1 hundredth
Reads as: Ten thousandths make one hundredth.
Variables: thousandths = 1/1000, hundredth = 1/100
Use when: Converting between thousandths and hundredths.
⚠ Thinking 1000 thousandths = 1 hundredth.
1000 thousandths = 1 unit
Reads as: One thousand thousandths make one unit.
Variables: thousandths = 1/1000, unit = 1
Use when: Understanding the relationship between thousandths and whole units.
⚠ Thinking 1000 thousandths = 10 units.
How It Works
Measuring length using a scale with tenths
  1. Place the object along the scale from the 0 mark.
  2. Count the number of whole centimetres (units) from 0 to the end of the object.
  3. Observe the smaller divisions between two consecutive centimetre marks. Each division is 1/10 cm (one-tenth of a cm).
  4. Count how many tenths beyond the last whole centimetre the object extends.
  5. Write the length as: whole centimetres + number of tenths, e.g., 2 cm and 7 tenths = 2 7/10 cm = 2.7 cm.
💡 Students often forget to count the tenths correctly or misread the scale. Always start from 0, not from the edge of the ruler.
Adding two decimal numbers (e.g., 2.7 + 3.5)
  1. Write the numbers one below the other, aligning the decimal points vertically.
  2. Add the tenths column: 7 tenths + 5 tenths = 12 tenths.
  3. Since 10 tenths = 1 unit, carry over 1 unit to the units column, leaving 2 tenths.
  4. Add the units column: 2 units + 3 units + 1 carried unit = 6 units.
  5. Write the result as 6.2 (6 units and 2 tenths).
💡 The most common mistake is forgetting to carry over when the sum of tenths (or hundredths) reaches 10 or more. Always check if you need to regroup.
Subtracting two decimal numbers (e.g., 3.5 - 2.7)
  1. Write the numbers one below the other, aligning the decimal points.
  2. Start from the tenths column: we cannot subtract 7 tenths from 5 tenths.
  3. Borrow 1 unit from the units column (3 becomes 2). The borrowed 1 unit equals 10 tenths.
  4. Now the tenths column has 5 + 10 = 15 tenths. Subtract 7 tenths = 8 tenths.
  5. Subtract the units column: 2 units - 2 units = 0 units.
  6. Write the result as 0.8.
💡 Students often forget to borrow correctly. Remember: 1 unit = 10 tenths, 1 tenth = 10 hundredths, etc. Always align decimal points.
Comparing two decimal numbers (e.g., 6.456 and 6.465)
  1. Start from the leftmost digit (units place). Compare: both have 6 units — same.
  2. Move to the tenths place: both have 4 tenths — same.
  3. Move to the hundredths place: first number has 5 hundredths, second has 6 hundredths.
  4. Since 5 < 6, the second number (6.465) is greater. Stop comparing further digits.
  5. Conclusion: 6.465 > 6.456.
💡 Do not compare the total number of digits. A number with fewer digits can be larger (e.g., 0.9 > 0.89). Always compare place by place from left to right.
Converting a fraction to a decimal (e.g., 1/2 = 0.5)
  1. Find an equivalent fraction with denominator 10, 100, or 1000.
  2. For 1/2, multiply numerator and denominator by 5 to get 5/10.
  3. Write 5/10 as 0.5 (5 tenths).
  4. Alternatively, for 3/4, multiply by 25 to get 75/100 = 0.75.
💡 Not all fractions can be easily converted to tenths or hundredths. For example, 1/3 does not have a terminating decimal. Stick to fractions with denominators that are factors of 10, 100, or 1000.
Locating a decimal on a number line (e.g., 1.4)
  1. Identify the two whole numbers between which the decimal lies. 1.4 lies between 1 and 2.
  2. Divide the segment between 1 and 2 into 10 equal parts (each part = 0.1).
  3. Count 4 parts from 1 towards 2. Mark that point as 1.4.
  4. For more precise decimals like 1.45, further divide the segment between 1.4 and 1.5 into 10 equal parts (each = 0.01).
💡 Students often misplace decimals by counting from 0 instead of from the lower whole number. Always start counting tenths from the lower whole number.
Exam Traps — Don't Make These Mistakes
⚠ Adding zeros to the right changes the value
✗ 0.2 is smaller than 0.20 because 0.20 has more digits.
→ 0.2 and 0.20 are equivalent decimals. Adding zeros to the right of the last digit after the decimal point does not change the value. 0.2 = 2 tenths = 20 hundredths = 0.20.
✓ 0.2 = 0.20 = 0.200. They all represent the same quantity.
⚠ Comparing decimals by total digits
✗ 0.99 is greater than 1.0 because 99 is greater than 1.
→ Comparing the whole number part first: 0.99 has 0 units, 1.0 has 1 unit. Since 0 < 1, 0.99 < 1.0 regardless of the digits after the decimal.
✓ Always compare the whole number part first. 1.0 > 0.99 because 1 unit is greater than 0 units.
⚠ Misaligning decimal points in addition/subtraction
✗ 2.7 + 3.5 = 5.12 (adding 2+3=5 and 7+5=12, writing 5.12).
→ The decimal points were not aligned. 7 tenths + 5 tenths = 12 tenths = 1 unit and 2 tenths, not 12 hundredths. The correct sum is 6.2.
✓ Align decimal points vertically: 2.7 + 3.5 = 6.2. Carry over 1 from tenths to units.
⚠ Borrowing incorrectly in subtraction
✗ 3.5 - 2.7 = 1.2 (subtracting 2 from 3 and 7 from 5, ignoring the borrow).
→ You cannot subtract 7 tenths from 5 tenths without borrowing. You must borrow 1 unit (10 tenths) from the units column, making it 15 tenths - 7 tenths = 8 tenths.
✓ 3.5 - 2.7 = 0.8. Borrow 1 unit from 3, making it 2 units and 15 tenths. Then subtract.
⚠ Confusing 0.5 hours with 50 minutes
✗ 4.5 hours after noon means 4:50 PM.
→ 0.5 hours means half an hour (30 minutes), not 50 minutes. 0.5 = 5/10 = 1/2 hour = 30 minutes.
✓ 4.5 hours after noon = 4:30 PM. 0.5 hour = 30 minutes.
⚠ Incorrect unit conversion (mm to cm)
✗ 5 mm = 5.0 cm.
→ 1 mm = 0.1 cm, so 5 mm = 0.5 cm, not 5.0 cm. The decimal point shifts one place to the left.
✓ 5 mm = 5/10 cm = 0.5 cm.
Compare & Contrast
Value and place value
0.2
0.02
0.2 is 2 tenths (2/10), while 0.02 is 2 hundredths (2/100). 0.2 is ten times larger than 0.02.
Meaning of the fractional part
0.5 hour
0.5 as a decimal in cricket (5.5 overs)
0.5 hour means half an hour (30 minutes) because 1 hour = 60 minutes, and 0.5 × 60 = 30. In cricket, 5.5 overs means 5 overs and 5 balls (not 5.5 × 6 = 33 balls), because the decimal notation is non-standard and means 5 overs + 5 balls.
Place value of digits
70.5
7.05
In 70.5, the 7 is in the tens place (7 × 10 = 70), and the 5 is in the tenths place (5 × 1/10 = 0.5). In 7.05, the 7 is in the units place (7 × 1 = 7), and the 5 is in the hundredths place (5 × 1/100 = 0.05). 70.5 is ten times larger than 7.05.
Exam Focus
Marks pattern: 2 MCQs (1 mark each), 1 Very Short Answer (1 mark), 1 Short Answer (2 marks) on conversion or comparison, 1 Long Answer (3 marks) on addition/subtraction word problem.
Most asked: Conversion between fractions and decimals (e.g., 1/2 = 0.5, 3/4 = 0.75) · Addition and subtraction of decimals with carrying/borrowing · Comparing and ordering decimal numbers · Place value of digits in decimal numbers · Real-life applications: length (cm/mm), weight (kg/g), money (rupees/paise)
Don't skip: Word problems involving addition and subtraction of decimals (e.g., total cloth needed, difference in lengths); Arranging decimal numbers in ascending or descending order; Converting between fractions and decimals (e.g., 1/4, 3/5)
One-Line Revision (read aloud before the exam)
  1. A decimal point separates the whole number part from the fractional part.
  2. One-tenth (0.1) is one of ten equal parts of a unit.
  3. One-hundredth (0.01) is one of one hundred equal parts of a unit.
  4. One-thousandth (0.001) is one of one thousand equal parts of a unit.
  5. 10 tenths make 1 unit; 10 hundredths make 1 tenth; 10 thousandths make 1 hundredth.
  6. To add or subtract decimals, always align the decimal points vertically.
  7. When subtracting decimals, borrow 1 unit as 10 tenths if needed.
  8. To compare decimals, compare digits from left to right, place by place.
  9. 0.2 is equal to 0.20 and 0.200 — adding zeros to the right does not change the value.
  10. 1 cm = 10 mm, so 1 mm = 0.1 cm.
  11. 1 m = 100 cm, so 1 cm = 0.01 m.
  12. 1 kg = 1000 g, so 1 g = 0.001 kg.
  13. ₹1 = 100 paise, so 1 paisa = 0.01 rupee.
  14. 0.5 hour = 30 minutes, not 50 minutes.
  15. A decimal number with more digits is not necessarily larger — compare place by place.
  16. To locate a decimal on a number line, divide the segment between whole numbers into 10 equal parts for tenths.

MCQs — 10 Questions with Answers

Q1. Sonu's mother needed screws of different sizes. This teaches us that measuring small differences requires:

  • A. Dividing the unit into smaller equal parts ✓
  • B. Using a longer ruler
  • C. Measuring multiple times
  • D. Asking someone else to measure

Answer: A — The story shows that standard units are too large for precise measurements, so we divide 1 cm into 10 tenths to measure accurately.

Q2. How many one-tenths are there in 27/10?

  • A. 2
  • B. 7
  • C. 27 ✓
  • D. 10

Answer: C — 27/10 means 27 one-tenths, since each 1/10 is one one-tenth.

Q3. Read the measurement 3 5/10 cm aloud:

  • A. Three-five tenths centimeters
  • B. Three and five one-tenths centimeters ✓
  • C. Three and five hundredths centimeters
  • D. Thirty-five tenths centimeters

Answer: B — A mixed number like 3 5/10 is read as 'three and five one-tenths' or 'three and five-tenths.'

Q4. Which represents the smallest length?

  • A. 3/10
  • B. 3/100 ✓
  • C. 30/100
  • D. 33/100

Answer: B — 3/100 is much smaller than 3/10 because hundredths are 10 times smaller than tenths.

Q5. A paper sheet measures 8 9/10 units. When folded in half, its length falls between which two tenths?

  • A. 4 3/10 and 4 4/10
  • B. 4 4/10 and 4 5/10 ✓
  • C. 4 4/10 and 4 6/10
  • D. 4 5/10 and 4 6/10

Answer: B — Half of 8 9/10 is 4 45/100, which lies between 4 4/10 (=4 40/100) and 4 5/10 (=4 50/100).

Q6. Shylaja's hand length is 12 4/10 units and palm length is 6 7/10 units. What is her middle finger length? (Hint: Subtract palm from hand length.)

  • A. 5 7/10 units ✓
  • B. 5 3/10 units
  • C. 6 7/10 units
  • D. 6 3/10 units

Answer: A — 12 4/10 − 6 7/10: Cannot subtract 7/10 from 4/10, so borrow 1 unit to get 11 14/10 − 6 7/10 = 5 7/10.

Q7. In the place value system, if ones place has value 1, tenths place has value:

  • A. 1/10 (one-tenth) ✓
  • B. 1/100 (one-hundredth)
  • C. 10 times the ones place
  • D. Cannot be determined

Answer: A — Each place to the right is 1/10 of the previous place, so tenths = 1/10 of ones place.

Q8. A Celestial Pearl Danio is 2 4/10 cm long and a Philippine Goby is 9/10 cm long. How much longer is the Danio?

  • A. 1 5/10 cm ✓
  • B. 2 5/10 cm
  • C. 1 6/10 cm
  • D. 1 4/10 cm

Answer: A — 2 4/10 − 9/10 = 1 14/10 − 9/10 = 1 5/10 cm. (Converting 2 4/10 to 1 14/10 by borrowing 1 unit.)

Q9. When adding 15 3/10 4/100 and 2 6/10 8/100, why do we group units, tenths, and hundredths separately?

  • A. To make the problem easier to understand
  • B. Because only same place values can be added together, like in whole numbers ✓
  • C. To avoid mistakes in writing
  • D. Because the book says so

Answer: B — Just as we add ones with ones and tens with tens in whole numbers, we add hundredths with hundredths, tenths with tenths, and units with units in decimals.

Q10. If we split a one-tenth into 10 equal parts, what fraction of the unit is each part?

  • A. 1/10 of the unit
  • B. 1/100 of the unit ✓
  • C. 1/1000 of the unit
  • D. 1/20 of the unit

Answer: B — One-tenth divided into 10 parts means 1/10 ÷ 10 = 1/100 of the unit.

Flashcards

How many one-tenths make 1 unit?

10 one-tenths equal 1 unit.

What does 2 7/10 cm mean?

2 cm and 7 one-tenths of a cm, or 27/10 cm total.

How many one-hundredths are in 1 one-tenth?

10 one-hundredths make 1 one-tenth.

Write 45 one-hundredths in two other ways.

45/100 can be written as 4 5/10 (converting 40/100 to tenths) or 0 45/100.

Why do we split units into 10 parts instead of other numbers?

Because 10 is the base of our Indian place value system, making calculations consistent and easy.

If Sonu's lower arm is 2 7/10 units and upper arm is 3 6/10 units, what is the total?

Total is 6 3/10 units: (2+3) units and (7+6) tenths = 5 13/10 = 6 3/10.

Read 15 3/10 4/100 aloud.

Fifteen and three-tenths and four-hundredths, or fifteen and thirty-four-hundredths.

When subtracting 6 7/10 from 12 4/10, why do we need to borrow?

Because we cannot subtract 7 tenths from 4 tenths, so we borrow 1 unit (= 10 tenths) first.

How many one-hundredths equal 1 unit?

100 one-hundredths equal 1 unit.

Convert 34/10 into mixed number form.

34/10 = 3 4/10 (three and four-tenths).

Important Board Questions

What is the length of the Sonu's arm if his lower arm is 2 7/10 units and upper arm is 3 6/10 units? Write your answer as a mixed number. [1 mark]

Add units separately from tenths. If tenths exceed 10, convert 10 tenths to 1 unit.

A pencil is 7 8/10 units long. A sharpener is 2 3/10 units long. Find the total length of both objects. Show your working. [2 marks]

Add the units together and add the tenths together separately. Remember, 10 tenths = 1 unit.

Shylaja's total hand length is 12 4/10 units. Her palm measures 6 7/10 units. Calculate the length of her middle finger by subtracting palm from hand length. Show all steps, including borrowing if needed. [3 marks]

Cannot subtract 7/10 from 4/10 directly. Borrow 1 unit (convert to 10/10) from the whole number part first. Then subtract tenths from tenths and units from units.

A honeybee has three body parts: Head = 2 3/10 units, Thorax = 5 4/10 units, and Abdomen = 7 5/10 units. (a) Find the total length of the honeybee. (b) If a fly is 15 8/10 units long, how much longer is the fly than the honeybee? Show all working and explain each step. [5 marks]

(a) Add all three parts by grouping units and tenths separately. Watch for tenths that sum to 10 or more and convert to units. (b) Subtract the honeybee's total from the fly's length; borrow if the tenths part of honeybee is larger than fly's tenths.

Assertion-Reason Questions for A Peek Beyond the Point

Q1. Assertion (A):

To measure the length of a screw accurately when the measurement falls between 2 cm and 3 cm, we need to divide the unit into smaller parts like tenths.

Reason (R):

Whole unit measurements cannot capture precise values that lie between whole numbers.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true. The reason correctly explains why the assertion is true: we need smaller units (tenths) specifically because measurements often fall between whole units, and whole units alone cannot represent these intermediate values precisely.

Q2. Assertion (A):

The length 3 4/10 units is the same as 34/10 units.

Reason (R):

Because 3 units equal 30 tenths, and 4/10 tenths can be added to get 34/10 tenths.

Answer: B — Both A and R are true, but R is NOT the correct explanation of A — The assertion is true: 3 4/10 = 30/10 + 4/10 = 34/10. The reason is also technically true (3 = 30/10 and adding 4/10 gives 34/10), but it explains the conversion through basic arithmetic rather than explaining why they represent the same length. Both are correct, but the reason isn't the most direct explanation.

Q3. Assertion (A):

When subtracting 6 7/10 from 12 4/10, we must regroup because 4 tenths is less than 7 tenths.

Reason (R):

Regrouping in subtraction is only necessary when the digit in the units place of the minuend is smaller than the digit in the units place of the subtrahend.

Answer: C — A is true but R is false — The assertion is true—we must regroup because we cannot subtract 7 from 4 in the tenths place. However, the reason is false because it only mentions the units place, not the fractional (tenths) place where the actual problem occurs. The reason focuses on the wrong place value.

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Case-Based Questions (CBQ) for A Peek Beyond the Point

Case Study:

Ravi is building a model house and needs to measure the dimensions of different wooden pieces. He has a ruler marked with tenths of a centimeter. The door frame length is 15 7/10 cm, the window frame length is 8 4/10 cm, and the roof piece length is 12 6/10 cm. Ravi needs to check if he has enough wood by calculating the total length of these three pieces. He also needs to find how much longer the roof piece is compared to the window frame.

Q1. What is the total length of the three wooden pieces?

  • A. 35 17/10 cm
  • B. 36 7/10 cm ✓
  • C. 35 27/10 cm
  • D. 37 cm

Answer: B

Q2. How much longer is the roof piece than the window frame?

  • A. 3 2/10 cm
  • B. 4 2/10 cm ✓
  • C. 3 10/10 cm
  • D. 4 6/10 cm

Answer: B

Q3. If Ravi wants to add another piece that is 5 8/10 cm long, what will be the new total?

  • A. 41 15/10 cm
  • B. 41 5/10 cm
  • C. 42 5/10 cm
  • D. 42 3/10 cm ✓

Answer: D

Q4. Which two pieces have a combined length closest to 24 cm?

  • A. Door frame and window frame
  • B. All combinations are equal
  • C. Window frame and roof piece
  • D. Door frame and roof piece ✓

Answer: D

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