Students often refer to Ganita Prakash Class 7 Solutions and Class 7 Maths Chapter 3 A Peek Beyond the Point NCERT Solutions Question Answer to verify their answers.
NCERT Class 7 Maths Chapter 3 A Peek Beyond the Point Solutions Question Answer
Ganita Prakash Class 7 Chapter 3 Solutions A Peek Beyond the Point
NCERT Class 7 Maths Ganita Prakash Chapter 3 A Peek Beyond the Point Solutions Question Answer
3.1 The Need for Smaller Units
Intext Questions (Page 47)
Solution:
Which scale helped you measure the length of the screws accurately? Why?
Solution:
The third scale, divided into 10 equal parts between each centimeter mark, helped us measure the screws accurately. This is because it clearly shows each tenth of a centimeter, helping us measure the screw length precisely.
What is the meaning of 2\(\frac{7}{10}\) cm (the length of the first screw)?
Solution:
Here, the unit length between two consecutive numbers is divided into 10 equal parts.
To get the length 2\(\frac{7}{10}\) cm, we go from 0 to 2 and then take seven parts of \(\frac{1}{10}\).
The length of the screw is 2 cm and \(\frac{7}{10}\).
We read 2\(\frac{7}{10}\) cm as two and seven-tenth centimeters.
Can you explain why the unit was divided into smaller parts to measure the screws?
Solution:
The units are divided into smaller parts to get accurate measurement of the screw.
Measure the following objects using a scale and write their measurements in centimeters (as shown earlier for the lengths of the screws): pen, sharpener, and any other object of your choice.
Solution:
Do it yourself.
Write the measurements of the objects shown in the picture:
Solution:
Eraser = 2\(\frac{4}{10}\) cm
Pencil = 4\(\frac{5}{10}\) cm
Cap of sketch pen = 1\(\frac{4}{10}\) cm
3.2 A Tenth Part
Arrange these lengths in increasing order: (Page 49)
(a) \(\frac{9}{10}\)
(b) 1\(\frac{7}{10}\)
(c) \(\frac{130}{10}\)
(d) 13\(\frac{1}{10}\)
(e) 10\(\frac{5}{10}\)
(f) 7\(\frac{6}{10}\)
(g) 6\(\frac{7}{10}\)
(h) \(\frac{4}{10}\)
Solution:
We observe \(\frac{131}{10}\) is the largest number and \(\frac{4}{10}\) the smallest.
So increasing the order of the given lengths is
Arrange the following lengths in increasing order: (Page 50)
\(4 \frac{1}{10}, \frac{4}{10}, \frac{41}{10}, 41 \frac{1}{10}\)
Solution:
The lengths of the body parts of a honeybee are given. Find its total length. (Page 51)
Head: 2\(\frac{3}{10}\) units
Thorax: 5\(\frac{4}{10}\) units
Abdomen: 7\(\frac{5}{10}\) units
Solution:
Given, Length of Head = 2\(\frac{3}{10}\) units
Length of Thorax = 5\(\frac{4}{10}\) units
Length of Abdomen = 7\(\frac{5}{10}\) units
Total length of honeybee = \(2 \frac{3}{10}+5 \frac{4}{10}+7 \frac{5}{10}\)
A Celestial Pearl Danio’s length is 2\(\frac{4}{10}\) cm, and the length of a Philippine Goby is \(\frac{9}{10}\) cm. What is the difference in their lengths? (Page 52)
Solution:
How big are these fish compared to your finger? (Page 52)
Solution:
Do it yourself.
Observe the given sequences of numbers. Identify the change after each term and extend the pattern:
Solution:
3.3 A Hundredth Part
Observe the figure below. Notice the markings and the corresponding lengths written in the boxes when measured from 0. Fill in the lengths in the empty boxes.
The length of the wire in the first picture is given in three different ways. Can you see how they denote the same length? (Page 54)
Solution:
Fifth box = 2 + \(\frac{4}{10}\) = \(\frac{240}{100}\)
For the lengths shown below, write the measurements and read out the measures in words. (Pages 54 – 55)
Solution:
(a) Length = \(5 \frac{3}{10} \frac{7}{100}\) → Five and three tenths and seven-hundredths
= 5 + \(\frac{37}{100}\) → Five and thirty seven-hundredths
= \(\frac{537}{100}\) → Five hundred thirty seven hundredths
(b) Length = 15 + \(\frac{3}{100}\) → Fifteen and three-hundredths
= \(\frac{1500}{100}+\frac{3}{100}=\frac{1503}{100}\) → Fifteen hundred three-hundredths
(c) Length = \(7 \frac{5}{10} \frac{2}{100}\) → Seven and fifty-two hundredths
= \(7+\frac{5}{10}+\frac{2}{100}=7 \frac{52}{100}\) → Seven and fifty-two-hundredths
= \(\frac{700}{100}+\frac{52}{100}=\frac{752}{100}\) → Seven hundred fifty-two-hundredths
(d) Length = 9\(\frac{8}{10}\) → Nine and eight-tenths
= 9\(\frac{80}{100}\) → Nine and eighty hundredths
= \(\frac{980}{100}\) → Nine hundred eighty hundredths
In each group, identify the longest and the shortest lengths. Mark each length on the scale. (Pages 55 – 56)
Solution:
Identify the longest and shortest lengths:
Are both of these methods different? (Page 57)
Solution:
Yes, both methods are different.
Method 1 is row/line addition, and Method 2 is the column method.
We can also add by changing tenths to hundredths first and then adding the numbers, before converting back to tenths and hundredths.
Figure It Out (Page 58)
Question 1.
Find the sums and differences:
(a) \(\frac{3}{10}+3 \frac{4}{100}\)
(b) \(9 \frac{5}{10} \frac{7}{100}+2 \frac{1}{10} \frac{3}{100}\)
(c) \(15 \frac{6}{10} \frac{4}{100}+14 \frac{3}{10} \frac{6}{100}\)
(d) \(7 \frac{7}{100}-4 \frac{4}{100}\)
(e) \(8 \frac{6}{100}-5 \frac{3}{100}\)
(f) \(12 \frac{6}{100} \frac{2}{100}-\frac{9}{10} \frac{9}{100}\)
Solution:
3.4 Decimal Place Value
We can ask similar questions about fractional parts: (Page 61)
(a) How many thousandths make one unit?
(b) How many thousandths make one tenth?
(c) How many thousandths make one hundredth?
(d) How many tenths make one ten?
(e) How many hundredths make one ten?
Solution:
(a) 1000 thousandths make one unit.
(b) 100 thousandths make one tenth
(c) 10 thousandths make one-hundredth
(d) 100 tenths make one tenth
(e) 1000 hundredths make one tenth
Make a place value table similar to the one above. Write each quantity in decimal form and terms of place value, and read the number: (Page 63)
(a) 2 ones, 3 tenths, and 5 hundredths
(b) 1 ten and 5 tenths
(c) 4 ones and 6 hundredths
(d) 1 hundred, 1 on,e and 1 hundredth
(e) \(\frac{8}{100}\) and \(\frac{9}{10}\)
(f) \(\frac{5}{100}\)
(g) \(\frac{1}{10}\)
(h) 2\(\frac{1}{100}\), 4\(\frac{1}{10}\) and 7\(\frac{7}{1000}\)
Solution:
Make a place value table
Write these quantities in decimal form: (Page 64)
(a) 234 hundredths
(b) 105 tenths
Solution:
(a) 234 hundredths = \(\frac{234}{100}\)
= \(\frac{200+30+4}{100}\)
= \(\frac{200}{100}+\frac{30}{100}+\frac{4}{100}\)
= 2 + \(\frac{3}{10}\) + \(\frac{4}{100}\)
= 2 ones + 3 tenths + 4 hundredths
= 2.34
(b) 105 tenths = \(\frac{105}{10}\)
= \(\frac{100+5}{10}\)
= \(\frac{100}{10}+\frac{5}{10}\)
= 10 + \(\frac{5}{10}\)
= 1 ten + 5 tenth
= 1.5
3.5 Units of Measurement
Fill in the blanks below (mm ↔ cm) (Page 65)
Solution:
How many m is (a) 10 cm? (b) 15 cm?
Solution:
Fill in the blanks below (cm ↔ m): (Page 66)
Solution:
How many mm does 1 meter have? (Page 66)
Solution:
1000 mm = 1 m
Fill in the blanks (g ↔ kg) (Pages 67 – 68)
Solution:
Fill in the blanks below (rupee ↔ paise) (Page 69)
Solution:
3.6 Locating and Comparing Decimals, 3.7 Addition and Subtraction of Decimals
Figure It Out (Page 75)
Question 1.
Find the sums.
(a) 5.3 + 2.6
(b) 18 + 8.8
(c) 2.15 + 5.26
(d) 9.01 + 9.10
(e) 29.19 + 9.91
(f) 0.934 + 0.6
(g) 0.75 + 0.03
(h) 6.236 + 0.487
Solution:
Question 2.
Find the differences.
(a) 5.6 – 2.3
(b) 18 – 8.8
(c) 10.4 – 4.5
(d) 17 – 16.198
(e) 17 – 0.05
(f) 34.505 – 18.1
(g) 9.9 – 9.09
(h) 6.236 – 0.487
Solution:
3.8 More on the Decimal System
Figure It Out (Pages 78 – 80)
Question 1.
Convert the following fractions into decimals:
(a) \(\frac{5}{100}\)
(b) \(\frac{16}{1000}\)
(c) \(\frac{12}{10}\)
(d) \(\frac{254}{1000}\)
Solution:
Question 2.
Convert the following decimals into a sum of tenths, hundredths, and thousandths:
(a) 0.34
(b) 1.02
(c) 0.8
(d) 0.362
Solution:
Question 3.
What decimal number does each letter represent in the number line below?
Solution:
There are 4 divisions between 6.4 and 6.5.
∴ Each division is \(\frac{1}{4}\) part of 0.1 or \(\frac{1}{10}\) then \(\frac{1}{40}\) = 0.025
∴ Second division after 6.4 = a
= 6.4 + 2 × 0.025
= 6.4 + 0.05
= 6.45
Now the first division after 6.5 = c
= 6.5 + 0.025
= 6.525
and second division after 6.5 = b
= 6.5 + 2 × 0.025
= 6.5 + 0.05
= 6.55
Question 4.
Arrange the following quantities in descending order:
(a) 11.01, 1.011, 1.101, 11.10, 1.01
(b) 2.567, 2.675, 2.768, 2.499, 2.698
(c) 4.678 g, 4.595 g, 4.600 g, 4.656 g, 4.666 g
(d) 33.13 m, 33.31 m, 33.133 m, 33.331 m, 33.313 m
Solution:
(a) Given 11.01, 1.011, 1.101, 11.10, 1.01
Now, arrange the above quantities in decreasing order
11.10, 11.01, 1.101, 1.011, 1.01.
(b) Given 2.567, 2.675, 2.768, 2.499, 2.698
Now, arrange the above quantities in decreasing order
2.768, 2.698, 2.675, 2.567, 2.499.
(c) Here 4.678 g, 4.595 g, 4.600 g, 4.656 g, 4.666 g.
Now, arranging the above quantities in decreasing order
we get 4.678 g, 4.666 g, 4.656 g, 4.600g, 4.595 g.
(d) Here 33.13 m, 33.31 m, 33.133 m, 33.331 m, 33.313 m.
Now, arranging the above quantities in decreasing order,
we get 33.331 m, 33.313 m, 33.31 m, 33.133 m, 33.13 m.
Question 5.
Using the digits 1, 4, 0, 8, and 6, make:
(a) The decimal number closest to 30.
(b) The smallest possible decimal number between 100 and 1000.
Solution:
(a) Here, we will form a number using these digits that is closest to 30.
Options like 18…… 40…., 16…. are possible.
Closest above 30, 40.168
Difference = 40.168 – 30 = 10.168
and closest below 30, 18.640
Difference = 30 – 18.640 = 11.360
The closest number seems to be 40.168.
(b) Clearly, the number must have 3 digits before the decimal point.
Now to be smallest, it should start with the smallest possible digit (1), followed by the next smallest (0), then the next 4, and arrange 6 and 8 after the decimal.
∴ Smallest number = 104.68.
Question 6.
Will a decimal number with more digits be greater than a decimal number with fewer digits?
Solution:
Not necessarily. Comparison depends on the place value of the digits, not the total number of digits.
For example, 0.5 is greater than 0.134. However, if comparing whole number parts, 13 (two digits) is greater than 9 (one digit).
Question 7.
Mahi purchases 0.25 kg of beans, 0.3 kg of carrots, 0.5 kg of potatoes, 0.2 kg of capsicums, and 0.05 kg of ginger. Calculate the total weight of the items she bought.
Solution:
Total Weight = (0.25 + 0.3 + 0.5 + 0.2 + 0.05) kg
= (0.25 + 0.30 + 0.50 + 0.20 + 0.05) kg
= 1.30 kg.
Question 8.
Pinto supplies 3.79 L, 4.2 L, and 4.25 L of milk to a milk dairy in the first three days. In 6 days, he supplies 25 litres of milk. Find the total quantity of milk supplied to the dairy in the last three days.
Solution:
Here supply in first three days = 3.79 L + 4.20 L + 4.25 L = 12.24 L
Total supply in 6 days = 25 L
∴ Supply in last three days = 25 – 12.24 = 12.76 L
Question 9.
Tinku weighed 35.75 kg in January and 34.50 kg in February. Has he gained or lost weight? How much is the change?
Solution:
34.50 kg (Feb) < 33.75 kg (Jan).
Tinku has lost weight.
Required change = 33.75 – 34.50 = 1.25 kg
He lost 1.25 g.
Question 10.
Extend the pattern: 5.5, 6.4, 6.39, 7.29, 7.28, 6.18, 6.17, ____, _____
Solution:
Pattern alternates +0.9, -0.01, +0.9, -0.01
Next terms 8.17 + 0.9 = 9.07, 9.07 – 0.01 = 9.06
Question 11.
How many millimeters make 1 kilometer?
Solution:
We know 1 km = 1000 m
1 m = 1000 mm
∴ 1 km = 1000 × 1000 mm = 10,00,000 mm
Question 12.
Indian Railways offers optional travel insurance for passengers who book e-tickets. It costs 45 paise per passenger. If 1 lakh people opt for insurance in a day, what is the total insurance fee paid? Solution:
Here, 1 lakh = 1,00,000 people
Total fee = 100,000 people × 45 paise/person
= 4,500,0000 paise (∵ 100 paise = 1 rupee)
= \(\frac{4,500,000}{100}\) = ₹ 45,000
Question 13.
Which is greater?
(a) \(\frac{10}{1000}\) or \(\frac{1}{10}\)?
(b) One-hundredth or 90 thousandths?
(c) One-thousandth or 90 hundredths?
Solution:
(a) Here \(\frac{10}{1000}\) and \(\frac{1}{10}=\frac{100}{1000}\)
∵ 100 is greater than 10 then \(\frac{100}{1000}\) is greater than \(\frac{10}{1000}\)
∵ \(\frac{1}{10}\) is greater than \(\frac{10}{1000}\)
(b) One hundredth = \(\frac{1}{100}\) = 0.01
90 thousandths = \(\frac{90}{1000}\) = 0.090
∵ 0.090 > 0.010
∴ 90 thousandths is greater than one hundredth.
(c) One thousandth = \(\frac{1}{1000}\) = 0.001
and 90 hundredths = \(\frac{90}{100}\) = 0.90
∵ 0.90 > 0.001
∴ 90 hundredths is greater than one-thousandth.
Question 14.
Write the decimal forms of the quantities mentioned (an example is given):
(a) 87 ones, 5 tenths, and 60 hundredths = 88.10
(b) 12 tens and 12 tenths
(c) 10 tens, 10 ones, 10 tenths, and 10 hundredths
(d) 25 tens, 25 ones, 25 tenths, and 25 hundredths
Solution:
(a) 87 ones, 5 tenths, and 60 hundredths = 88.10
(b) Here 12 tens and 12 tenths = 12 × 10 + 12 × \(\frac{1}{10}\)
= 120 + 1.2
= 121.2
(c) Here 10 tens, 10 ones, 10 tenths and 10 hundredths = 10 × 10 + 10 × 1 + 10 × \(\frac{1}{10}\) + 10 × \(\frac{1}{100}\)
= 100 + 10 + 1 + 0.10
= 111.10
(d) 25 tens, 25 ones, 25 tenths and 25 hundredths = 25 × 10 + 25 × 1 + 25 × \(\frac{1}{10}\) + 25 × \(\frac{1}{100}\)
= 250 + 25 + 2.5 + 0.25
= 277.75
Question 15.
Using each digit 0-9 not more than once, fill the boxes below so that the sum is closest to 10.5:
Solution:
Question 16.
Write the following fractions in decimal form:
(a) \(\frac{1}{2}\)
(b) \(\frac{3}{2}\)
(c) \(\frac{1}{4}\)
(d) \(\frac{3}{4}\)
(e) \(\frac{1}{5}\)
(f) \(\frac{4}{5}\)
Solution:
