Beyond the Point: Ultimate Guide to Decimals
Question: Which scale helped you measure the length of the screws accurately? Why?
The scale with the smaller divisions (the one marked in tenths) helped measure the screws accurately. This is because the screw’s length did not end exactly on a whole centimeter mark, so the smaller divisions were needed to measure the fractional part of the length.
Question: What is the meaning of 2 7/10 cm?
It means the length is 2 full centimeters and 7 additional tenths of a centimeter.
Question: Can you explain why the unit was divided into smaller parts to measure the screws?
The unit (centimeter) was divided into smaller parts (tenths or millimeters) to achieve greater precision. The lengths of the screws were not exact whole numbers of centimeters, so a smaller unit was required to measure the leftover part accurately.
Question: Arrange these lengths in increasing order.
First, convert all mixed fractions to improper fractions (tenths) to easily compare them:
- (a) 9/10 = 0.9
- (b) 1 7/10 = 17/10 = 1.7
- (c) 130/10 = 13.0
- (d) 13 1/10 = 131/10 = 13.1
- (e) 10 5/10 = 105/10 = 10.5
- (f) 7 6/10 = 76/10 = 7.6
- (g) 6 7/10 = 67/10 = 6.7
- (h) 4/10 = 0.4
Increasing Order: 0.4, 0.9, 1.7, 6.7, 7.6, 10.5, 13.0, 13.1
Which corresponds to: (h), (a), (b), (g), (f), (e), (c), (d).
Question: Arrange the following lengths in increasing order: 4 1/10, 4/10, 41/10, 41 1/10.
Convert to tenths:
- 4 1/10 = 41/10
- 4/10
- 41/10
- 41 1/10 = 411/10
We are comparing: 41/10, 4/10, 41/10, 411/10.
Increasing Order: 4/10, 4 1/10 (or 41/10), 41 1/10.
Question: What is the total length of Sonu’s arm?
Lower arm = 2 7/10 units. Upper arm = 3 6/10 units.
Total Length = (2 7/10) + (3 6/10) = (2+3) + (7/10 + 6/10) = 5 + 13/10.
Since 13/10 = 1 and 3/10, the total is 5 + 1 + 3/10 = 6 3/10 units.
Question: Find the total length of the honeybee’s body parts.
Total Length = (2 3/10) + (5 4/10) + (7 5/10)
= (2+5+7) + (3/10 + 4/10 + 5/10) = 14 + 12/10
= 14 + 1 + 2/10 = 15 2/10 units.
Question: What is the length of Shylaja’s middle finger?
Finger Length = Hand Length – Palm Length
= (12 4/10) – (6 7/10)
We cannot subtract 7/10 from 4/10. We borrow 1 unit from 12, which is 10/10.
= (11 and 10/10 + 4/10) – (6 7/10) = (11 14/10) – (6 7/10)
= (11 – 6) + (14/10 – 7/10) = 5 + 7/10 = 5 7/10 units.
Question: Difference in length between Celestial Pearl Danio and Philippine Goby.
Difference = (2 4/10) cm – (9/10) cm
Borrow 1 from 2: = (1 and 10/10 + 4/10) – 9/10 = (1 14/10) – 9/10
= 1 + (14/10 – 9/10) = 1 + 5/10 = 1 5/10 cm (or 1.5 cm).
Question: Extend the sequences.
(a) 4, 4 3/10, 4 6/10, … (Adding 3/10 each time) -> 4 9/10, 5 2/10, 5 5/10
(b) 8 2/10, 8 7/10, 9 2/10, … (Adding 5/10 each time) -> 9 7/10, 10 2/10, 10 7/10
(c) 7 6/10, 8 1/10, 8 6/10, … (Adding 5/10 each time) -> 9 1/10, 9 6/10, 10 1/10
(d) 5 7/10, 5 5/10, 5 3/10, … (Subtracting 2/10 each time) -> 5 1/10, 4 9/10, 4 7/10
(e) 13 5/10, 13, 12 5/10, … (Subtracting 5/10 each time) -> 12, 11 5/10, 11
(f) 11 5/10, 10 4/10, 9 3/10, … (Subtracting 1 1/10 each time) -> 8 2/10, 7 1/10, 6
Question: How many one-hundredths make one-tenth?
Since 1 tenth is split into 10 parts to make hundredths, 10 one-hundredths make one-tenth.
Can we also say the length is 4 units and 45 one-hundredths? Yes. 4 tenths are equal to 40 hundredths. So, 4 tenths and 5 hundredths is the same as 40 hundredths + 5 hundredths = 45 hundredths.
Question: Fill the lengths in the empty boxes.
From left to right, the boxes should be filled with:
1/10, 2/10, 30/100 (which is 3/10), 1, 1 30/100 (which is 1.3), 1 99/100, 2 1/100 (or just 2)
Question: Write the measurements for the scales in words.
Top Ruler: The measurement shown is Five and three-tenths.
Bottom Ruler: The measurement shown is Fifteen and two-tenths and five-hundredths (or Fifteen and twenty-five hundredths).
Question: Identify the longest and shortest lengths in each group.
To compare, convert all to hundredths.
(a) 3/10, 3/100, 33/100 -> 30/100, 3/100, 33/100. Shortest: 3/100, Longest: 33/100.
(b) 3 1/10, 30/10, 1/10 -> 31/10, 30/10, 1/10. Shortest: 1/10, Longest: 3 1/10.
(c) 45/100, 54/100, 5/10, 4/10 -> 45/100, 54/100, 50/100, 40/100. Shortest: 4/10, Longest: 54/100.
(d) 3 6/10, 3 6/100, 3 66/100 -> 360/100, 306/100, 366/100. Shortest: 3 6/100, Longest: 3 66/100.
(e) 8/10, 2/100, 9/100, 1 8/100 -> 80/100, 2/100, 9/100, 108/100. Shortest: 2/100, Longest: 1 8/100.
Full Solutions for End-of-Chapter Questions
Page 69: Write decimal quantities
(a) 234 hundredths = 234/100 = 2.34.
(b) 105 tenths = 105/10 = 10.5.
Page 70: Fill in the blanks (mm <-> cm)
70 mm = 7.0 cm
9 mm = 0.9 cm
134 mm = 13.4 cm
2036 mm = 203.6 cm
Page 71: Fill in the blanks (cm <-> m)
36 cm = 0.36 m
50 cm = 0.50 m or 0.5 m
89 cm = 0.89 m
4 cm = 0.04 m
325 cm = 3.25 m
207 cm = 2.07 m
Page 73: Fill in the blanks (g <-> kg)
465 g = 0.465 kg
704 g = 0.704 kg
68 g = 0.068 kg
560 g = 0.56 kg
1560 g = 1.560 kg
2500 g = 2.5 kg
Page 74: Fill in the blanks (rupee <-> paise)
10 p = ₹0.10
5 p = ₹0.05
36 p = ₹0.36
₹0.50 = 50 p
99 p = ₹0.99
250 p = ₹2.50
Page 79: Final Exercise Questions
Q3. Letter on number line: a=6.42, c=6.48, b=6.49.
Q4. Descending order: (a) 11.10, 11.01, 1.101, 1.011, 1.01
Q7. Mahi’s purchases: 0.25+0.3+0.5+0.2+0.05 = 1.30 kg.
Q9. Tinku’s weight: 35.75 – 34.50 = 1.25. He lost 1.25 kg.
Q11. mm in a km: 1 km = 1000 m. 1 m = 1000 mm. So, 1 km = 1000 × 1000 = 1,000,000 mm.
Q12. Railway insurance fee: 1 lakh people = 100,000. Fee per person = 45 paise. Total = 100,000 × 45 = 4,500,000 paise. To convert to rupees, divide by 100. Total = ₹45,000.
Q13. Which is greater? (a) 10/1000 = 0.01. 1/10 = 0.1. So 1/10 is greater. (b) 1/100 = 0.01. 90/1000 = 0.09. So 90 thousandths is greater.
Q16. Fractions to decimals: (a) 1/2=0.5 (b) 3/2=1.5 (c) 1/4=0.25 (d) 3/4=0.75 (e) 1/5=0.2 (f) 4/5=0.8