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Another 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 extends students' understanding of decimals to multiplication and division. It builds on the Indian place value system and fraction operations to show that multiplying and dividing decimals is a natural extension of operations with counting numbers. The one big idea a student must walk away with is that decimals are just fractions with denominators like 10, 100, 1000, so all operations follow the same rules as fractions, and the decimal point is placed based on the total number of decimal places or zeroes in the denominator.
Key Concepts (14)
Decimal
A decimal is a representation of a fraction with a denominator of 10, 100, 1000, etc., using a decimal point to separate the whole number part from the fractional part.
e.g. 27.53 means 2 Tens, 7 Ones, 5 Tenths, 3 Hundredths.
Tenths
One-tenth (0.1) is one part out of ten equal parts of a whole.
e.g. 0.3 is 3 tenths, like 3 out of 10 slices of a pizza.
Hundredths
One-hundredth (0.01) is one part out of one hundred equal parts of a whole.
e.g. 0.05 is 5 hundredths, like 5 paise out of 1 rupee (100 paise).
Thousandths
One-thousandth (0.001) is one part out of one thousand equal parts of a whole.
e.g. 0.004 is 4 thousandths, like 4 grams out of 1 kilogram (1000 g).
Multiplicand
The number that is being multiplied in a multiplication operation.
e.g. In 9.5 × 5, 9.5 is the multiplicand.
Multiplier
The number by which the multiplicand is multiplied.
e.g. In 9.5 × 5, 5 is the multiplier.
Product
The result obtained after multiplying two or more numbers.
e.g. In 9.5 × 5 = 47.5, 47.5 is the product.
Dividend
The number that is being divided in a division operation.
e.g. In 126 ÷ 2.5, 126 is the dividend.
Divisor
The number by which the dividend is divided.
e.g. In 126 ÷ 2.5, 2.5 is the divisor.
Quotient
The result obtained after dividing one number by another.
e.g. In 126 ÷ 2.5 = 50.4, 50.4 is the quotient.
Long Division (Division using Place Value)
A method of dividing numbers by breaking them into place values (Thousands, Hundreds, Tens, Ones, Tenths, Hundredths, etc.) and regrouping remainders to continue division.
e.g. 1324 ÷ 4: 1 Thousand is regrouped as 10 Hundreds, then 13 Hundreds ÷ 4 = 3 Hundreds each, remainder 1 Hundred, and so on.
Reciprocal
The reciprocal of a number is 1 divided by that number. For a fraction, it is obtained by swapping numerator and denominator.
e.g. The reciprocal of 10 is 1/10. Dividing by 10 is the same as multiplying by 1/10.
Leap Year
A year that has 366 days instead of 365, occurring every 4 years, except for years divisible by 100 but not by 400.
e.g. Year 2024 is a leap year (divisible by 4, not by 100). Year 1900 was not a leap year (divisible by 100 but not by 400).
Cyclic Number
A number whose digits cycle when multiplied by certain numbers. For example, 142857 is a cyclic number from 1 ÷ 7.
e.g. 142857 × 2 = 285714 (digits cycled).
Formulas & Equations (5)
Product of two decimals = (Product of numbers without decimal points) ÷ (10^(total decimal places))
Reads as: Product of two decimals equals the product of the numbers without decimal points divided by 10 raised to the total number of decimal places.
Variables: Total decimal places = number of digits after decimal in multiplier + number of digits after decimal in multiplicand.
Use when: When multiplying two decimal numbers directly without converting to fractions.
⚠ Students forget to count decimal places correctly, e.g., for 5.8 × 1.24, they may place decimal after 2 digits instead of 3 (since 5.8 has 1 digit, 1.24 has 2 digits, total = 3).
Dividing by 10, 100, 1000: Move decimal point left by as many places as there are zeroes in the divisor.
Reads as: To divide a decimal by 10, move the decimal point one place to the left; by 100, two places; by 1000, three places.
Variables: Divisor = 10 (1 zero), 100 (2 zeroes), 1000 (3 zeroes).
Use when: When dividing any decimal by 10, 100, 1000, etc.
⚠ Students move the decimal point to the right instead of left, e.g., 3.9 ÷ 10 = 39 instead of 0.39.
Dividing by a decimal: Multiply both dividend and divisor by 10, 100, or 1000 to make the divisor a whole number, then divide.
Reads as: To divide by a decimal, multiply both the dividend and divisor by the same power of 10 to convert the divisor into a whole number, then perform the division.
Variables: Example: 4.68 ÷ 0.13 = (4.68 × 100) ÷ (0.13 × 100) = 468 ÷ 13.
Use when: When the divisor is a decimal (e.g., 126 ÷ 2.5, 4.68 ÷ 1.3).
⚠ Students multiply only the divisor by 10/100/1000 but forget to multiply the dividend by the same number, changing the value.
Converting a fraction to a decimal: Find an equivalent fraction with denominator 10, 100, or 1000, then write the numerator with the decimal point placed according to the number of zeroes.
Reads as: To convert a fraction to a decimal, multiply numerator and denominator by a number that makes the denominator 10, 100, or 1000, then write the numerator with the decimal point placed according to the number of zeroes.
Variables: Example: 1/2 = (1×5)/(2×5) = 5/10 = 0.5.
Use when: When converting fractions like 1/2, 3/4, 7/8 to decimals.
⚠ Students forget to multiply both numerator and denominator by the same number, e.g., 1/2 = 1/10 (wrong).
Number of days in 100 calendar years (with leap every 4 years) = (100 × 365) + (100/4 × 1) = 36,525 days.
Reads as: Number of days in 100 calendar years equals 100 times 365 plus the number of leap years (100 divided by 4) times 1 extra day.
Variables: 100 = number of years, 365 = days in a common year, 100/4 = number of years divisible by 4 (25).
Use when: When calculating calendar days over a century with simple leap year rule.
⚠ Students forget that the 100th year is also divisible by 4 and count it as a leap year, but later adjustments exclude it.
How It Works
Multiplying a Decimal by a Whole Number
  1. Step 1: Write the decimal as a fraction with denominator 10, 100, or 1000 (e.g., 9.5 = 95/10).
  2. Step 2: Multiply the numerators and denominators separately (e.g., 5/1 × 95/10 = 475/10).
  3. Step 3: Convert the resulting fraction back to a decimal (e.g., 475/10 = 47.5).
💡 The number of decimal places in the product equals the number of decimal places in the decimal number (since the whole number has 0 decimal places).
Multiplying Two Decimals
  1. Step 1: Remove the decimal points and multiply the numbers as whole numbers (e.g., 5.8 × 1.24 → 58 × 124 = 7192).
  2. Step 2: Count the total number of digits after the decimal point in both numbers (5.8 has 1 digit, 1.24 has 2 digits, total = 3).
  3. Step 3: Place the decimal point in the product so that it has that many digits after it (7192 → 7.192).
💡 Always count the decimal places from the rightmost digit of the product. If the product has fewer digits, add zeroes to the left (e.g., 0.2 × 0.3 = 0.06, not 0.6).
Dividing a Decimal by 10, 100, 1000
  1. Step 1: Write the dividend with a decimal point at the end if it is a whole number (e.g., 123 → 123.).
  2. Step 2: Count the number of zeroes in the divisor (10 has 1 zero, 100 has 2, 1000 has 3).
  3. Step 3: Move the decimal point to the left by that many places, adding zeroes in front if needed (e.g., 123 ÷ 1000 = 0.123).
💡 Moving the decimal point left makes the number smaller. For 12 ÷ 1000, move 3 places left: 12 → 0.012.
Long Division with Decimal Quotient (Dividing Whole Numbers)
  1. Step 1: Set up the division as usual (e.g., 1325 ÷ 4).
  2. Step 2: Divide the largest place value first. If a digit cannot be divided, regroup it into the next smaller place (e.g., 1 Thousand → 10 Hundreds).
  3. Step 3: Continue until you reach the Ones place. If there is a remainder, place a decimal point in the quotient and regroup the remainder Ones into Tenths (e.g., 1 One → 10 Tenths).
  4. Step 4: Continue dividing Tenths, then Hundredths, then Thousandths, etc., until the remainder is zero or you decide to stop.
💡 When you regroup Ones into Tenths, you MUST place a decimal point in the quotient immediately. Forgetting the decimal point is a common error.
Dividing a Decimal by a Whole Number
  1. Step 1: Place the decimal point in the quotient directly above the decimal point in the dividend.
  2. Step 2: Divide as if it were a whole number, bringing down digits after the decimal point.
  3. Step 3: If there is a remainder, add zeroes to the dividend and continue dividing (e.g., 9.5 ÷ 4 = 2.375).
💡 The decimal point in the quotient must be placed before you start dividing the tenths. Do not wait until the end.
Dividing by a Decimal Divisor
  1. Step 1: Convert the divisor into a whole number by multiplying it by 10, 100, or 1000 (e.g., 2.5 × 10 = 25).
  2. Step 2: Multiply the dividend by the same number (e.g., 126 × 10 = 1260).
  3. Step 3: Now divide the new dividend by the new whole number divisor using long division (e.g., 1260 ÷ 25 = 50.4).
💡 You must multiply BOTH the dividend and divisor by the same power of 10. Multiplying only the divisor changes the value of the division.
Leap Year Calculation (Gregorian Calendar)
  1. Step 1: A year is a leap year if it is divisible by 4.
  2. Step 2: Exception: If the year is divisible by 100, it is NOT a leap year unless it is also divisible by 400.
  3. Step 3: So, years divisible by 400 are leap years; years divisible by 100 but not by 400 are not leap years; all other years divisible by 4 are leap years.
💡 The rule is: divisible by 4 → leap year, except if divisible by 100 → not leap year, except if divisible by 400 → leap year. This is a three-step check.
Exam Traps — Don't Make These Mistakes
⚠ Misplacing the decimal point in multiplication
✗ 5.8 × 1.24 = 71.92 (placing decimal after 2 digits instead of 3)
→ 5.8 has 1 decimal place, 1.24 has 2 decimal places, total = 3 decimal places. The product 7192 must have 3 digits after decimal, so it is 7.192, not 71.92.
✓ 5.8 × 1.24 = 7.192
⚠ Moving decimal point wrong direction when dividing by 10, 100, 1000
✗ 3.9 ÷ 10 = 39
→ Dividing by 10 makes the number smaller, so the decimal point must move LEFT, not right. 3.9 ÷ 10 = 0.39.
✓ 3.9 ÷ 10 = 0.39
⚠ Forgetting to multiply both dividend and divisor when dividing by a decimal
✗ 126 ÷ 2.5 = 126 ÷ 25 = 5.04
→ Only the divisor was multiplied by 10 (2.5 → 25), but the dividend was not multiplied by 10. This changes the value. Correct: 126 ÷ 2.5 = (126 × 10) ÷ (2.5 × 10) = 1260 ÷ 25 = 50.4.
✓ 126 ÷ 2.5 = 50.4
⚠ Not placing decimal point in quotient when regrouping Ones to Tenths in long division
✗ 237 ÷ 8 = 29625 (no decimal point)
→ When you regroup the remainder 5 Ones into 50 Tenths, you must place a decimal point in the quotient. The correct quotient is 29.625, not 29625.
✓ 237 ÷ 8 = 29.625
⚠ Confusing when product is greater or less than the numbers multiplied
✗ 0.25 × 0.8 = 2.0 (thinking product is always greater)
→ When both numbers are between 0 and 1, the product is less than both. 0.25 × 0.8 = 0.2, which is less than 0.25 and 0.8.
✓ 0.25 × 0.8 = 0.2
⚠ Incorrectly applying leap year rule for century years
✗ Year 1900 was a leap year because it is divisible by 4.
→ Years divisible by 100 are not leap years unless also divisible by 400. 1900 is divisible by 100 but not by 400, so it was not a leap year.
✓ Year 1900 was not a leap year. Year 2000 was a leap year because it is divisible by 400.
Compare & Contrast
Method of multiplication
Multiplying decimals by converting to fractions
Multiplying decimals by counting decimal places
In the fraction method, you convert decimals to fractions (e.g., 9.5 = 95/10), multiply numerators and denominators, then convert back. In the decimal place method, you remove decimal points, multiply as whole numbers, then place the decimal point so that the product has as many decimal places as the sum of decimal places in the multiplier and multiplicand.
Type of divisor
Dividing by a whole number (e.g., 9.5 ÷ 4)
Dividing by a decimal (e.g., 126 ÷ 2.5)
When dividing by a whole number, you place the decimal point in the quotient directly above the dividend's decimal point and divide normally. When dividing by a decimal, you must first multiply both dividend and divisor by a power of 10 to make the divisor a whole number, then proceed with the division.
Exam Focus
Marks pattern: 2 MCQs (1 mark each), 1 Very Short Answer (1 mark), 1 Short Answer (2-3 marks), 1 Long Answer (4-5 marks) — typically 8-10 marks total
Most asked: Multiplication of decimals (especially word problems involving money, length, area) · Division of decimals by 10, 100, 1000 · Long division with decimal quotients · Converting fractions to decimals · Word problems on profit, distance, and sharing
Don't skip: Word problems involving multiplication and division of decimals in real-life contexts (e.g., cost of items, distance travelled, sharing quantities); Long division with decimal quotient (e.g., 1325 ÷ 4, 237 ÷ 8)
One-Line Revision (read aloud before the exam)
  1. A decimal is a fraction with denominator 10, 100, or 1000, written using a decimal point.
  2. To multiply decimals, multiply as whole numbers, then place the decimal point so that the product has as many decimal places as the total in both factors.
  3. To divide a decimal by 10, move the decimal point one place to the left; by 100, two places; by 1000, three places.
  4. In long division, when you regroup Ones into Tenths, place a decimal point in the quotient immediately.
  5. To divide by a decimal, multiply both dividend and divisor by 10, 100, or 1000 to make the divisor a whole number.
  6. The product of two decimals can be greater than, less than, or between the two numbers, depending on whether each is greater than 1 or between 0 and 1.
  7. A leap year occurs every 4 years, except for years divisible by 100 but not by 400.
  8. The Earth takes 365.2422 days to revolve around the Sun, which is why we have leap years.
  9. To convert a fraction to a decimal, find an equivalent fraction with denominator 10, 100, or 1000.
  10. The reciprocal of a number is 1 divided by that number; dividing by a number is the same as multiplying by its reciprocal.
  11. In division, the quotient can be greater than the dividend when the divisor is less than 1.
  12. Some decimal divisions never end, like 10 ÷ 3 = 3.333..., because the remainder repeats forever.
  13. The number 142857 is a cyclic number that appears when dividing 1 by 7.
  14. Always count the total number of decimal places in both factors before placing the decimal point in the product.
  15. When regrouping in long division, remember that 1 One = 10 Tenths, 1 Tenth = 10 Hundredths, and so on.

MCQs — 10 Questions with Answers

Q1. What does the decimal 0.347 represent as a fraction?

  • A. 347/1000 ✓
  • B. 347/100
  • C. 347/10
  • D. 347/10000

Answer: A — 0.347 has 3 decimal places, so it represents 347 parts out of 1000.

Q2. When you divide 45.6 by 100, what is the result?

  • A. 4560
  • B. 4.56
  • C. 0.456 ✓
  • D. 0.0456

Answer: C — Dividing by 100 means moving the decimal point 2 places to the left: 45.6 → 0.456.

Q3. What is 7.3 × 2 in decimal form?

  • A. 14.3
  • B. 14.6 ✓
  • C. 1.46
  • D. 146

Answer: B — 7.3 × 2 = 14.6; the product has 1 decimal place because 7.3 has 1 and 2 has 0.

Q4. If 15 × 12 = 180, what is 1.5 × 1.2?

  • A. 18
  • B. 1.8 ✓
  • C. 0.18
  • D. 180

Answer: B — 1.5 has 1 decimal place and 1.2 has 1 decimal place, so 1 + 1 = 2 places; 180 becomes 1.80 = 1.8.

Q5. Arjun bought 4 notebooks at ₹25.50 each. How much did he spend?

  • A. ₹102 ✓
  • B. ₹102.50
  • C. ₹100.50
  • D. ₹112

Answer: A — 25.50 × 4 = 102.00 = ₹102 (2 decimal places in 25.50, 0 in 4, so 2 total; 2550 ÷ 100 = 25.5, and 25.5 × 4 = 102).

Q6. A ribbon is 12.8 m long and is cut into 8 equal pieces. What is the length of each piece?

  • A. 1.6 m ✓
  • B. 1.5 m
  • C. 2.4 m
  • D. 0.16 m

Answer: A — 12.8 ÷ 8 = 1.6 m because 128 ÷ 80 = 1.6 (or use 12.8 = 128/10, then 128/10 ÷ 8 = 128/80 = 1.6).

Q7. When is the product of two decimals less than both factors?

  • A. When both decimals are greater than 1
  • B. When both decimals are between 0 and 1 ✓
  • C. When one decimal is greater than 1
  • D. When the product is a whole number

Answer: B — Example: 0.5 × 0.4 = 0.2, which is smaller than both 0.5 and 0.4.

Q8. Priya walks 1.25 km daily. How far does she walk in 6 days?

  • A. 7.25 km
  • B. 6.75 km
  • C. 7.5 km ✓
  • D. 8.25 km

Answer: C — 1.25 × 6: multiply 125 × 6 = 750, then place decimal 2 places from right = 7.50 = 7.5 km.

Q9. The thickness of a coin is 2.05 mm. What is the height of 10 coins stacked?

  • A. 20.5 mm ✓
  • B. 2.05 cm
  • C. 205 mm
  • D. 0.205 m

Answer: A — 2.05 × 10 = 20.5 mm (move decimal 1 place right when multiplying by 10); this equals 2.05 cm but the best answer in same units is 20.5 mm.

Q10. Which multiplication will give a product less than 1?

  • A. 2.5 × 0.6
  • B. 0.4 × 0.8 ✓
  • C. 1.2 × 0.9
  • D. 3.5 × 2

Answer: B — 0.4 × 0.8 = 0.32, which is less than 1 because both factors are between 0 and 1; in option A, 2.5 × 0.6 = 1.5 > 1.

Flashcards

What does 27.53 mean in expanded form?

2 tens + 7 ones + 5 tenths + 3 hundredths.

How do you multiply 5.96 × 24.8?

Multiply 596 × 248 = 147,808, then place decimal after 3 places (2+1) to get 147.808.

What is the rule for dividing by 10, 100, or 1000?

Move the decimal point left by the number of zeros in the divisor.

Is the product 0.25 × 0.8 greater than both numbers?

No, the product 0.2 is less than both 0.25 and 0.8 because both factors are less than 1.

Convert the fraction 1/2 to a decimal.

Multiply numerator and denominator by 5 to get 5/10 = 0.5.

How many decimal places are in the product 1.64 × 6?

Two decimal places because 1.64 has 2 decimal places and 6 has 0, so 2 + 0 = 2.

What is 3.9 ÷ 10 in decimal form?

0.39 (move decimal point 1 place to the left).

If a pen costs ₹9.5 and you buy 5 pens, what is the total cost?

₹47.5 because 9.5 × 5 = 47.5.

When multiplying two decimals, where do you place the decimal point in the product?

Count the total number of decimal places in both factors and place the decimal that many places from the right in the product.

Is 2.25 × 8 = 18 greater than 2.25?

Yes, because 8 is greater than 1, so the product is larger than both factors.

Important Board Questions

Convert the decimal 0.8 into a fraction in simplest form. [1 mark]

0.8 = 8/10; then reduce by dividing both by their GCD. GCD of 8 and 10 is 2.

A car uses 5.5 litres of petrol to travel 100 km. How much petrol is needed to travel 250 km? Show your working. [2 marks]

Set up proportion: 5.5 litres → 100 km, so x litres → 250 km. Multiply: (5.5 × 250) ÷ 100 = 1375 ÷ 100 = 13.75 litres.

A bag of rice costs ₹245.50 and weighs 15.5 kg. If Meena wants to buy 3 bags, how much will she spend? Write your steps clearly. [3 marks]

Step 1: Multiply 245.50 × 3 by removing decimals first (24550 × 3 ÷ 100). Step 2: 24550 × 3 = 73650. Step 3: Place decimal 2 places from right = 736.50 = ₹736.50.

Ajay has 8.4 metres of cloth. He cuts it into 6 equal pieces for a craft project. (a) What is the length of each piece in metres? (b) If he needs 4 such pieces for one project, how much cloth does he need? (c) Express your answer in both decimals and fractions. Show all your work. [5 marks]

(a) Divide: 8.4 ÷ 6 = 84/10 ÷ 6 = 84/60 = 1.4 m. (b) Multiply: 1.4 × 4 = 5.6 m. (c) 5.6 m = 56/10 = 28/5 metres (in fraction, simplified form). Show fraction to decimal conversion using place value: 8.4 = 8 units + 4 tenths.

Assertion-Reason Questions for Another Peek Beyond the Point

Q1. Assertion (A):

When multiplying 12.5 by 7.5, we convert to fractions as 125/10 × 75/10, multiply to get 9375/100, and the product is 93.75.

Reason (R):

Decimals can be written as fractions with denominators as powers of 10, and multiplying fractions means multiplying numerators and denominators separately.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true. The assertion correctly shows the decimal multiplication process via fractions, and the reason correctly explains why this method works—decimals are fractions in disguise, and fraction multiplication rules apply. The reason directly explains why the method in the assertion works.

Q2. Assertion (A):

The product of 0.25 × 0.8 is 0.2, which is less than both 0.25 and 0.8.

Reason (R):

When two numbers are both between 0 and 1, their product is always greater than both numbers.

Answer: C — A is true but R is false — The assertion is true (0.25 × 0.8 = 0.2 is correct and less than both factors). However, the reason is false—it contradicts the assertion by saying the product should be greater than both numbers. The chapter clearly states that when both numbers are between 0 and 1, the product is less than both numbers, not greater.

Q3. Assertion (A):

To find 678 ÷ 1000, we move the decimal point 3 places to the left, giving 0.678.

Reason (R):

The rule is to move the decimal point left by a number of places equal to the number of zeros in the divisor.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true and directly connected. The assertion correctly applies the rule, and the reason correctly states the rule. The reason explains exactly why the assertion works—1000 has 3 zeros, so we move the decimal 3 places left.

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

Case Study:

Rajesh runs a small vegetable market. He purchases oranges at ₹45.50 per kg and sells them at ₹68.75 per kg. On Monday, he sold 12.5 kg of oranges, and on Tuesday, he sold 8.75 kg. He also bought spices: 0.75 kg of turmeric at ₹180.50 per kg and 1.25 kg of pepper at ₹250.75 per kg. Calculate his various transactions using decimal multiplication.

Q1. How much revenue did Rajesh earn from selling oranges on Monday (at ₹68.75 per kg for 12.5 kg)?

  • A. ₹859.375 ✓
  • B. ₹859.38
  • C. ₹860.00
  • D. ₹857.50

Answer: A

Q2. What was the total revenue from oranges sold on both Monday and Tuesday (12.5 kg + 8.75 kg at ₹68.75 per kg)?

  • A. ₹1441.40
  • B. ₹1440.52
  • C. ₹1443.59
  • D. ₹1442.81 ✓

Answer: D

Q3. How much did Rajesh spend on turmeric (0.75 kg at ₹180.50 per kg)?

  • A. ₹135.375 ✓
  • B. ₹135.38
  • C. ₹130.50
  • D. ₹135.50

Answer: A

Q4. What was the total cost of spices purchased (turmeric + pepper)?

  • A. ₹429.88
  • B. ₹431.93 ✓
  • C. ₹430.50
  • D. ₹433.21

Answer: B

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