HBSE Solutions

A Peek Beyond the Point Class 7 Solutions Maths Ganita Prakash Chapter 3✦ Step-by-step

HBSE 7th Class Maths Solutions Chapter 3 A Peek Beyond the Point (NCERT Ganita Prakash) — every question worked out step by step. Tap Solve step by step to reveal the working one line at a time.

Class 7 Maths Chapter 3 A Peek Beyond the Point Solutions

Key ideas used in this chapter

Place value1 unit = 10 tenths = 100 hundredths = 1000 thousandths
Fraction → decimaldenominator 10, 100, 1000 → 1, 2, 3 decimal places
Add / subtractline up the decimal points, then work place by place
Equal placesfill missing places with zeros, e.g. \(0.6=0.600\)

3.1 The Need for Smaller Units (Pages 47–48)

Q.Read the picture
On the scale, each 1 cm is split into 10 equal parts (tenths). Write the measurements of the eraser, the pencil and the cap of the sketch pen.
Step 1
Count the whole centimetres, then the extra tenths for each object.
Eraser
2 whole cm and 4 tenths ⇒ \(2\frac{4}{10}\) cm = 2.4 cm
Pencil
4 whole cm and 5 tenths ⇒ \(4\frac{5}{10}\) cm = 4.5 cm
Cap
1 whole cm and 4 tenths ⇒ \(1\frac{4}{10}\) cm = 1.4 cm

3.2 A Tenth Part (Pages 49–52)

Q.Increasing order
Arrange in increasing order: \(\frac{9}{10},\ 1\frac{7}{10},\ \frac{130}{10},\ 13\frac{1}{10},\ 10\frac{5}{10},\ 7\frac{6}{10},\ 6\frac{7}{10},\ \frac{4}{10}\).
Step 1
Write each length as a decimal.
Step 2
\(0.9,\ 1.7,\ 13.0,\ 13.1,\ 10.5,\ 7.6,\ 6.7,\ 0.4\)
Step 3
Compare the whole parts first, then the tenths.
Answer
\(\frac{4}{10},\ \frac{9}{10},\ 1\frac{7}{10},\ 6\frac{7}{10},\ 7\frac{6}{10},\ 10\frac{5}{10},\ \frac{130}{10},\ 13\frac{1}{10}\)
Q.Increasing order
Arrange in increasing order: \(4\frac{1}{10},\ \frac{4}{10},\ \frac{41}{10},\ 41\frac{1}{10}\).
Step 1
As decimals: \(4.1,\ 0.4,\ 4.1,\ 41.1\)
Step 2
Notice \(4\frac{1}{10}=\frac{41}{10}=4.1\) — they are equal.
Answer
\(\frac{4}{10} < 4\frac{1}{10}=\frac{41}{10} < 41\frac{1}{10}\)
Q.Honeybee length
Head \(2\frac{3}{10}\), thorax \(5\frac{4}{10}\), abdomen \(7\frac{5}{10}\) units. Find the total length.
Step 1
Total \(= 2.3 + 5.4 + 7.5\)
Step 2
Add the tenths: \(3+4+5 = 12\) tenths = 1.2
Step 3
Add the ones: \(2+5+7 = 14\); total \(14 + 1.2\)
Answer
\(= 15.2\) units \(\left(=15\frac{1}{5}\right)\)
Q.Fish lengths
A Celestial Pearl Danio is \(2\frac{4}{10}\) cm and a Philippine Goby is \(\frac{9}{10}\) cm. Find the difference.
Step 1
Difference \(= 2.4 - 0.9\)
Step 2
\(2.4 - 0.9 = 2.4 - 1 + 0.1 = 1.5\)
Answer
\(= 1.5\) cm \(\left(=1\frac{5}{10}\right)\)

3.3 A Hundredth Part (Pages 54–57)

Q.One length, three ways
Show how \(2\frac{4}{10}\) can be written in three equal ways.
Step 1
\(2\frac{4}{10}=2+\frac{4}{10}\)
Step 2
\(\frac{4}{10}=\frac{40}{100}\), so it is also \(2+\frac{40}{100}\)
Step 3
\(2=\frac{200}{100}\), so \(2+\frac{40}{100}=\frac{240}{100}\)
Answer
\(2+\frac{4}{10}=2+\frac{40}{100}=\frac{240}{100}=2.4\)
Q.Write & read in words
Write each measurement as a decimal and read it: (a) 5 and 3 tenths 7 hundredths (b) 15 and 3 hundredths (c) 7 and 5 tenths 2 hundredths (d) 9 and 8 tenths.
(a)
\(5+\frac{3}{10}+\frac{7}{100}=5.37=\frac{537}{100}\)
five and thirty-seven hundredths.
(b)
\(15+\frac{3}{100}=15.03=\frac{1503}{100}\)
fifteen and three hundredths.
(c)
\(7+\frac{5}{10}+\frac{2}{100}=7.52=\frac{752}{100}\)
seven and fifty-two hundredths.
(d)
\(9+\frac{8}{10}=9.80=\frac{980}{100}\)
nine and eighty hundredths.

Figure It Out (Page 58)

Q1Add / subtract tenths & hundredths
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) \(12\frac{6}{10}\frac{2}{100}-\frac{9}{10}\frac{9}{100}\)
(a)
as decimals: \(0.30 + 3.04\); add hundredths 0+4=4, tenths 3+0=3, ones 0+3=3 ⇒ 3.34
(b)
\(9.57 + 2.13\); 7+3=10 (write 0 carry 1), 5+1+1=7, 9+2=11 ⇒ 11.70
(c)
\(15.64 + 14.36\); 4+6=10, 6+3+1=10, 5+4+1=10, 1+1=2 ⇒ 30.00
(d)
\(7.07 - 4.04\); 7−4=3 hundredths, 0−0, 7−4=3 ⇒ 3.03
(e)
\(8.06 - 5.03\); 6−3=3, 8−5=3 ⇒ 3.03
(f)
\(12.62 - 0.99\); make it \(12.62 - 0.99\): 62−99 needs borrowing ⇒ \(=11.63\)

3.4 Decimal Place Value (Pages 61–64)

Q.How many make…
(a) thousandths in one unit (b) in one tenth (c) in one hundredth (d) tenths in one ten (e) hundredths in one ten.
(a),(b)
1 unit = 1000 thousandths; 1 tenth = \(\frac{1000}{10}\) = 100 thousandths
(c)
1 hundredth = \(\frac{1000}{100}\) = 10 thousandths
(d),(e)
1 ten = 100 tenths; 1 ten = 1000 hundredths
Q.Write in decimal form
(a) 234 hundredths (b) 105 tenths.
(a) Step 1
\(234\text{ hundredths}=\frac{234}{100}\)
(a) Step 2
\(=\frac{200}{100}+\frac{30}{100}+\frac{4}{100}=2+\frac{3}{10}+\frac{4}{100}\) = 2.34
(b) Step 1
\(105\text{ tenths}=\frac{105}{10}\)
(b) Step 2
\(=\frac{100}{10}+\frac{5}{10}=10+\frac{5}{10}\) = 10.5

3.5 Units of Measurement (Pages 65–69)

Q.Conversions
How many metres is (a) 10 cm (b) 15 cm? How many mm in 1 m? Fill the g↔kg and rupee↔paise relations.
Step 1
100 cm = 1 m, so 1 cm = \(\frac{1}{100}\) m = 0.01 m
(a),(b)
10 cm = \(\frac{10}{100}\) = 0.10 m;  15 cm = \(\frac{15}{100}\) = 0.15 m
mm in 1 m
10 mm = 1 cm and 100 cm = 1 m ⇒ \(10\times100 = 1000\) mm
g, paise
1000 g = 1 kg (1 g = 0.001 kg);  100 paise = ₹1 (1 paisa = ₹0.01)

Figure It Out (Page 75)

Question 1Adding decimals
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) \(6.236+0.487\).
Method
Line up the points, give both the same number of decimal places, add from the right, carrying where needed.
(a)
\(5.3+2.6\): tenths 3+6=9, ones 5+2=7 ⇒ 7.9
(b)
\(18.0+8.8\): 0+8=8, 8+8=16 (write 6 carry 1), 1+1=2 ⇒ 26.8
(c)
\(2.15+5.26\): 5+6=11, 1+2+1=4, 2+5=7 ⇒ 7.41
(d)
\(9.01+9.10\): 1+0=1, 0+1=1, 9+9=18 ⇒ 18.11
(e)
\(29.19+9.91\): 9+1=10, 1+9+1=11, 9+9+1=19, 2+1=3 ⇒ 39.10
(f),(g)
\(0.934+0.600=1.534\);  \(0.75+0.03=0.78\)
(h)
\(6.236+0.487\): 6+7=13, 3+8+1=12, 2+4+1=7, 6+0=6 ⇒ 6.723
Question 2Subtracting decimals
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) \(6.236-0.487\).
Method
Add trailing zeros so both numbers have equal decimal places, then subtract, borrowing where needed.
(a),(b)
\(5.6-2.3=3.3\);  \(18.0-8.8=9.2\)
(c)
\(10.4-4.5\): borrow ⇒ 5.9
(d)
\(17.000-16.198\): 000−198 borrow ⇒ 0.802
(e),(f)
\(17.00-0.05=16.95\);  \(34.505-18.100=16.405\)
(g),(h)
\(9.90-9.09=0.81\);  \(6.236-0.487=5.749\)

Figure It Out (Pages 78–80)

Question 1Fraction → decimal
Convert into decimals: (a) \(\frac{5}{100}\) (b) \(\frac{16}{1000}\) (c) \(\frac{12}{10}\) (d) \(\frac{254}{1000}\).
Rule
count the zeros in the denominator = number of decimal places.
(a)
\(\frac{5}{100}\): 2 zeros ⇒ 2 places ⇒ 0.05
(b)
\(\frac{16}{1000}\): 3 places ⇒ 0.016
(c)
\(\frac{12}{10}\): 1 place ⇒ 1.2
(d)
\(\frac{254}{1000}\): 3 places ⇒ 0.254
Question 2Decimal → place values
Write as a sum of tenths, hundredths and thousandths: (a) 0.34 (b) 1.02 (c) 0.8 (d) 0.362.
(a)
digits 3, 4 ⇒ \(\frac{3}{10}+\frac{4}{100}\)
(b)
1 whole, tenths 0, hundredths 2 ⇒ \(1+\frac{2}{100}\)
(c)
tenths 8 ⇒ \(\frac{8}{10}\)
(d)
digits 3, 6, 2 ⇒ \(\frac{3}{10}+\frac{6}{100}+\frac{2}{1000}\)
Question 3Number line
Between 6.4 and 6.5 the line is split into 4 equal parts. Find the decimals at a (2nd mark after 6.4), c (1st mark after 6.5) and b (2nd mark after 6.5).
Step 1
Each small part \(=\frac{0.1}{4}=0.025\)
a
\(6.4 + 2\times0.025 = 6.4 + 0.05 = 6.45\)
c
\(6.5 + 1\times0.025 = 6.525\)
b
\(6.5 + 2\times0.025 = 6.55\)
Question 4Descending order
Arrange 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, 4.595, 4.600, 4.656, 4.666 (d) 33.13, 33.31, 33.133, 33.331, 33.313.
How
Compare the whole part first, then tenths, then hundredths, then thousandths.
(a)
11.10, 11.01, 1.101, 1.011, 1.01
(b)
2.768, 2.698, 2.675, 2.567, 2.499
(c)
4.678, 4.666, 4.656, 4.600, 4.595
(d)
33.331, 33.313, 33.31, 33.133, 33.13
Question 5Make a number
Using the digits 1, 4, 0, 8, 6: (a) the decimal closest to 30 (b) the smallest decimal between 100 and 1000.
(a) Step 1
There is no digit 3, so the whole part is either in the teens (1_) or the forties (4_).
(a) Step 2
closest teen 18.640 ⇒ \(30-18.640=11.360\); closest forty 40.168 ⇒ \(40.168-30=10.168\)
(a) Answer
\(10.168 < 11.360\) ⇒ 40.168
(b)
smallest hundreds digit 1, then 0, then 4 ⇒ 104, put 6, 8 after the point ⇒ 104.68
Question 6Reasoning
Is a decimal with more digits always greater than one with fewer digits?
Step 1
Compare 0.5 and 0.134: 0.5 has fewer digits but 5 tenths > 1 tenth.
Answer
No — place value decides, not the number of digits.
Question 7Word problem
Mahi buys 0.25 kg beans, 0.3 kg carrots, 0.5 kg potatoes, 0.2 kg capsicum, 0.05 kg ginger. Find the total weight.
Step 1
Same places: \(0.25+0.30+0.50+0.20+0.05\)
Step 2
hundredths: 5+0+0+0+5 = 10 ⇒ write 0 carry 1
Step 3
tenths: 2+3+5+2+0+1 = 13 ⇒ write 3 carry 1; ones: 0+…+1 = 1
Answer
\(= 1.30\) kg
Question 8Word problem
Pinto supplies 3.79 L, 4.2 L and 4.25 L in the first three days. In 6 days he supplies 25 L. Find the total for the last three days.
Step 1
First 3 days \(= 3.79 + 4.20 + 4.25 = 12.24\) L
Step 2
Last 3 days = total − first 3 = \(25 - 12.24\)
Answer
\(25.00 - 12.24 = 12.76\) L
Question 9Word problem
Tinku weighed 35.75 kg in January and 34.50 kg in February. Gained or lost, and by how much?
Step 1
\(34.50 < 35.75\), so his weight went down — he lost weight.
Step 2
Change \(= 35.75 - 34.50\)
Answer
\(= 1.25\) kg lost
Question 10Pattern
Extend: 5.5, 6.4, 6.39, 7.29, 7.28, 8.18, 8.17, ___, ___.
Step 1
5.5 → 6.4 is \(+0.9\); 6.4 → 6.39 is \(-0.01\).
Step 2
The steps alternate \(+0.9\), then \(-0.01\).
Answer
\(8.17 + 0.9 = 9.07\), then \(9.07 - 0.01 = 9.06\)
Question 11Conversion
How many millimetres make 1 kilometre?
Step 1
1 km = 1000 m
Step 2
1 m = 1000 mm
Answer
\(1000 \times 1000 = 10{,}00{,}000\) mm
Question 12Word problem
Travel insurance is 45 paise per passenger. What is the total for 1 lakh passengers?
Step 1
Total paise \(= 1{,}00{,}000 \times 45 = 45{,}00{,}000\) paise
Step 2
100 paise = ₹1, so divide by 100
Answer
\(\frac{45{,}00{,}000}{100}=\)₹45{,}000
Question 13Which is greater?
(a) \(\frac{10}{1000}\) or \(\frac{1}{10}\)? (b) one-hundredth or 90 thousandths? (c) one-thousandth or 90 hundredths?
(a)
\(\frac{10}{1000}=0.010\), \(\frac{1}{10}=0.100\) ⇒ \(\frac{1}{10}\) is greater
(b)
0.010 vs 0.090 ⇒ 90 thousandths is greater
(c)
0.001 vs 0.90 ⇒ 90 hundredths is greater
Question 14Decimal forms
Write in decimal form: (a) 87 ones, 5 tenths, 60 hundredths (b) 12 tens and 12 tenths (c) 10 tens, 10 ones, 10 tenths, 10 hundredths (d) 25 tens, 25 ones, 25 tenths, 25 hundredths.
(a)
60 hundredths = 0.60; \(87 + 0.5 + 0.60 = 88.10\)
(b)
\(12\times10 + 12\times\frac{1}{10} = 120 + 1.2 = 121.2\)
(c)
\(100 + 10 + 1.0 + 0.10 = 111.10\)
(d)
\(250 + 25 + 2.5 + 0.25 = 277.75\)
Question 15Digit puzzle
Using each digit 0–9 once, fill two decimals so their sum is as close as possible to 10.5.
Step 1
Aim for whole parts adding to about 10, then adjust the decimal parts with the leftover digits.
One answer
\(2.0796 + 8.4153 = 10.4949\)
uses each digit 0–9 once, within 0.0051 of 10.5 (other close answers exist).
Question 16Fraction → decimal
Write in decimal form: (a) \(\frac{1}{2}\) (b) \(\frac{3}{2}\) (c) \(\frac{1}{4}\) (d) \(\frac{3}{4}\) (e) \(\frac{1}{5}\) (f) \(\frac{4}{5}\).
Idea
make the denominator 10 or 100, then read off the decimal.
(a),(b)
\(\frac{1}{2}=\frac{5}{10}=0.5\);  \(\frac{3}{2}=\frac{15}{10}=1.5\)
(c),(d)
\(\frac{1}{4}=\frac{25}{100}=0.25\);  \(\frac{3}{4}=\frac{75}{100}=0.75\)
(e),(f)
\(\frac{1}{5}=\frac{2}{10}=0.2\);  \(\frac{4}{5}=\frac{8}{10}=0.8\)