Another Peek Beyond the Point
1. What is a decimal number? Explain the decimal 48.376 using place values.
Answer:
A decimal number is an extension of the place value system used to represent fractions such as tenths, hundredths and thousandths.
In 48.376:
4 is in the Tens place = 40
8 is in the Ones place = 8
3 is in the Tenths place = 0.3
7 is in the Hundredths place = 0.07
6 is in the Thousandths place = 0.006
So,
48.376 = 40 + 8 + 0.3 + 0.07 + 0.006
Explanation: Digits on the left of the decimal point represent whole numbers, while digits on the right represent parts of a whole.
2. Express 375 g as kilograms in both fraction and decimal form.
Answer:
Since 1 kg = 1000 g,
375 g = 375/1000 kg
= 3/8 kg
Decimal form:
375/1000 = 0.375 kg
Explanation: To convert grams into kilograms, divide by 1000. Therefore, 375 g = 0.375 kg.
3. Write 683/1000 as a sum of fractions and as a decimal.
Answer:
683/1000
= 600/1000 + 80/1000 + 3/1000
= 6/10 + 8/100 + 3/1000
Decimal form:
0.6 + 0.08 + 0.003
= 0.683
Explanation: The numerator is expanded according to place values and then written as a decimal.
4. State the rule for dividing a number by 1000. Use it to find 8452 ÷ 1000.
Answer:
Rule: Move the decimal point three places to the left.
8452.0 ÷ 1000
= 8.452
Explanation: Since 1000 has three zeroes, the decimal point shifts three places left.
5. A pencil costs ₹8.4. What is the cost of 9 pencils?
Answer:
Cost of 1 pencil = ₹8.4
Cost of 9 pencils
= 8.4 × 9
= 84/10 × 9
= 756/10
= ₹75.6
Explanation: Multiplication gives the total cost of several identical items.
6. A vehicle travels 14.5 km per litre. How far will it travel using 6 litres?
Answer:
Distance
= 14.5 × 6
= 87.0 km
Explanation: Distance = Distance per litre × Number of litres.
7. The length of a rectangle is 8.4 cm and its breadth is 2.5 cm. Find its area.
Answer:
Area = Length × Breadth
= 8.4 × 2.5
= 84/10 × 25/10
= 2100/100
= 21 cm²
Explanation: Area of a rectangle is obtained by multiplying length and breadth.
8. Given that 724 × 36 = 26064, find 7.24 × 3.6.
Answer:
7.24 has 2 decimal places.
3.6 has 1 decimal place.
Total decimal places = 3
Using the given product:
724 × 36 = 26064
Therefore,
7.24 × 3.6 = 26.064
Explanation: After multiplying whole numbers, place the decimal point according to the total decimal places.
9. Without multiplying, decide whether the product of 0.45 × 0.8 will be greater or smaller than both numbers.
Answer:
Both numbers are between 0 and 1.
Therefore, the product will be less than both numbers.
Explanation: When two decimals between 0 and 1 are multiplied, the product becomes smaller than each number.
10. Multiply 23.04 by 100 and 1000.
Answer:
23.04 × 100 = 2304
23.04 × 1000 = 23040
Explanation: Multiplying by 100 moves the decimal point 2 places right.
Multiplying by 1000 moves it 3 places right.
11. A ribbon is 5.7 m long. It is cut into 10 equal pieces. Find the length of each piece.
Answer:
Length of each piece
= 5.7 ÷ 10
= 0.57 m
Explanation: Dividing by 10 shifts the decimal point one place left.
12. A 36 m rope is shared equally among 8 children. How much rope does each child get?
Answer:
36 ÷ 8
= 4.5
Each child gets 4.5 m.
Explanation: 32 ÷ 8 = 4 and the remaining 4 m gives 0.5 m to each child.
13. Why do we place a decimal point in the quotient during long division when ones are regrouped into tenths?
Answer:
When ones are regrouped into tenths, we move from whole-number places to decimal places.
Explanation: The decimal point shows that the answer now contains fractional parts of a whole.
14. Find 245 ÷ 4 using decimal division.
Answer:
4 × 61 = 244
Remainder = 1
1 ÷ 4 = 0.25
Therefore,
245 ÷ 4 = 61.25
Explanation: The remainder is converted into decimal form by continuing the division.
15. A shopkeeper packs 7.2 kg sugar equally into 3 bags. What is the weight of each bag?
Answer:
7.2 ÷ 3
= 2.4
Each bag contains 2.4 kg sugar.
Explanation: The total quantity is divided equally among 3 bags.
16. Find 0.08 ÷ 4.
Answer:
0.08 = 8 hundredths
8 hundredths ÷ 4
= 2 hundredths
= 0.02
Explanation: Each group receives 2 hundredths.
17. Find 6.24 ÷ 0.12.
Answer:
Multiply dividend and divisor by 100:
6.24 ÷ 0.12
= 624 ÷ 12
= 52
Explanation: Multiplying both numbers by the same power of 10 makes the divisor a whole number.
18. Why does the decimal expansion of 10 ÷ 3 never end?
Answer:
10 ÷ 3 = 3.3333…
The remainder keeps repeating as 1.
Explanation: Since the remainder never becomes zero, the division continues forever and produces a non-terminating decimal.
19. The Earth takes 365.2422 days to revolve around the Sun. How many extra days accumulate after 100 years if only 365 days are counted each year?
Answer:
Extra days per year
= 365.2422 − 365
= 0.2422
After 100 years:
0.2422 × 100
= 24.22 days
Explanation: Ignoring the decimal part every year causes a difference of 24.22 days after 100 years.
20. According to the leap-year system described in the chapter, how many days are there in 100 calendar years when every 4th year is a leap year except the 100th year?
Answer:
Leap years:
100/4 − 100/100
= 25 − 1
= 24
Days:
(24 × 366) + (76 × 365)
= 8784 + 27740
= 36524 days
Explanation: There are 24 leap years and 76 ordinary years in 100 years under this rule, giving a total of 36,524 days.
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