GSEB Class 6 Maths Solutions Chapter 8 Decimals Exercise 8.1

Gujarat Board Textbook Solutions Class 6 Maths Chapter 8 Decimals Ex 8.1

 

Question 1. Write the following as numbers in the given table.
(a) This section shows 3 towers (each of 10 units), one block (equal to 1 unit), and 2 small parts (each equal to 2 tenths).
(b) This section shows 1 hundred, 1 ten, 0 units, and 4 tenths.
Answer:
(a) In the given figure, there are:
3 towers (each of 10 units), one block (equal to 1 unit), and 2 small parts (each equal to 2 tenths). We can therefore write this in the provided table:

(b) In the figure, there are:
1 hundred, 1 ten, 0 units, and 4 tenths. So the given table is written as:
In simple words: Look at the images showing blocks for hundreds, tens, ones, and tenths. Count how many of each block you see, then write these counts into the matching columns in the table.

Exam Tip: Pay close attention to the value each type of block represents (e.g., a tower for tens, a single block for ones, small parts for tenths) to accurately count and record them.

 

Question 2. Write the following decimals in the place value table.
(a) 19.4
(b) 0.3
(c) 10.6
(d) 205.9
Answer:
(a) 19.4
So, \( 19.4 = 1 \times 10 + 9 \times 1 + 4 \times \frac { 1 }{ 10 } \).
Therefore, we have:

(b) 0.3
So, \( 0.3 = 0 + \frac { 3 }{ 10 } \).
Therefore, we have:
(c) 10.6
So, \( 10.6 = 1 \times 10 + 0 \times 1 + 6 \times \frac { 1 }{ 10 } \).
Therefore, we have:
(d) 205.9
So, \( 205.9 = 2 \times 100 + 0 \times 10 + 5 \times 1 + 9 \times \frac { 1 }{ 10 } \).
Therefore, we have:
In simple words: Break down each decimal number into its place values (hundreds, tens, ones, tenths). Then, fill these numbers into the proper columns in the place value table.

Exam Tip: Remember that the first digit after the decimal point is the tenths place, the second is the hundredths, and so on. Also, whole numbers have their places before the decimal point.

 

Question 3. Write each of the following as decimals:
(a) Seven-tenths
(b) Two tens and nine-tenths
(c) Fourteen point six
(d) One hundred and two ones
(e) Six hundred point eight
Answer:
(a) Seven-tenths: We have seven-tenths \( = 7 \times \frac { 1 }{ 10 } = 0.7 \)
(b) Two tens and nine-tenths: We have, two tens and nine-tenths \( = 2 \times 10 + 9 \times \frac { 1 }{ 10 } = 20 + \frac { 9 }{ 10 } = 20 + 0.9 = 20.9 \)
(c) Fourteen point six: We have, fourteen point six \( = 14.6 \)
(d) One hundred and two ones: We have, one hundred and two \( = 1 \text{ hundred} + 0 \text{ tens} + 2 \text{ ones} + 0 \text{ tenths} = 100 + 0 + 2 + \frac { 0 }{ 10 } = 102.0 \)
(e) Six hundred point eight: We have, 6 hundred points 8 \( = 600.8 \)
In simple words: Convert the words describing a number into its decimal form. Think about what each word means for the number's place value.

Exam Tip: Be careful with phrases like "point six" versus "six tenths" – they both represent the same decimal value. Understand the place value of each word.

 

Question 4. Write each of the following as decimals:
(a) \( \frac {5}{ 10 } \)
(b) \( 3 + \frac { 7 }{ 10 } \)
(c) \( 200 + 60 + 5 + \frac { 1 }{ 10 } \)
(d) \( 70 + \frac { 8 }{ 10 } \)
(e) \( \frac { 88 }{ 10 } \)
(f) \( 4\frac { 2 }{ 10 } \)
(g) \( \frac {3}{2} \)
(h) \( \frac { 2 }{ 5 } \)
(i) \( \frac { 12 }{ 5 } \)
(j) \( 3\frac { 3 }{ 5 } \)
(k) \( 4\frac {1}{ 2 } \)
Answer:
(a) \( \frac {5}{ 10 } = 0.5 \)
(b) \( 3 + \frac {7}{10} = 3 + 0.7 = 3.7 \)
(c) \( 200 + 60 + 5 + \frac {1}{10} = 265 + 0.1 = 265.1 \)
(d) \( 70 + \frac {8}{10} = 70 + 0.8 = 70.8 \)
(e) \( \frac { 88 }{ 10 } = \frac{80+8}{10} = \frac { 80 }{ 10 } + \frac { 8 }{ 10 } = 8 + \frac { 8 }{ 10 } = 8.8 \)
(f) \( 4\frac { 2 }{10} = 4 + \frac { 2 }{ 10 } = 4.2 \)
(g) \( \frac { 3 }{ 2 } \)
Since, \( \frac { 3 }{ 2 } = \frac { 3 \times 5 }{ 2 \times 5 } = \frac { 15 }{ 10 } = \frac { 10 + 5 }{ 10} = \frac { 10 }{10} + \frac {5}{10} = 1 + \frac {5}{10} = 1.5 \)
Thus, \( \frac { 3 }{ 2 } = 1.5 \)
(h) \( \frac { 2 }{ 5 } \)
Since, \( \frac { 2 }{ 5 } = \frac { 2 \times 2 }{ 5 \times 2 } = \frac { 4 }{ 10 } = 0.4 \)
So, \( \frac { 2 }{ 5 } = 0.4 \)
(i) \( \frac { 12 }{ 5 } \)
Since, \( \frac {12}{5} = \frac { 12 \times 2 }{ 5 \times 2 } = \frac { 24 }{ 10 } = \frac { 20 + 4 }{ 10 } = \frac { 20 }{10} + \frac { 4 }{ 10 } = 2 + \frac { 4 }{ 10 } = 2.4 \)
(j) \( 3\frac { 3 }{ 5 } \)
Since, \( 3\frac { 3 }{ 5 } = 3 + \frac { 3 }{ 5 } = 3 + \left(\frac{3}{5} \times \frac{2}{2}\right) = 3 + \frac { 6 }{ 10 } = 3.6 \)
(k) \( 4\frac {1}{ 2 } \)
Since, \( 4\frac {1}{2 } = 4 + \frac { 1 }{ 2 } = 4 + \left(\frac{1 \times 5}{2 \times 5}\right) = 4 + \frac {5}{ 10 } = 4.5 \)
In simple words: Convert each fraction or sum into its decimal form. If the denominator is not 10, try to make it 10 by multiplying the top and bottom by the same number.

Exam Tip: To convert any fraction to a decimal, divide the numerator by the denominator. If the denominator can easily be made into a power of 10 (like 10, 100), it often simplifies the process.

 

Note: To convert a decimal to a fraction, we follow these steps:

• Count the number of digits in the decimal part.
• Ignoring the decimal point, write all the given digits as the numerator of the fraction.
• In the denominator, write as many zeros after 1 as there are decimal places in the decimal fraction.
• Reduce the fraction, so obtained, to its simplest form.

 

Question 5. Write the following decimals as fractions. Reduce the fractions to the lowest form.
(a) 0.6
(b) 2.5
(c) 1.0
(d) 3.8
(e) 13.7
(f) 21.2
(g) 6.4
Answer:
(a) 0.6
Here, the number of decimal places \( = 1 \).
The denominator \( = 10 \) and numerator \( = 6 \).
\( \implies \) fraction \( = \frac { 6 }{ 10 } = \frac { 3 }{ 5 } \).
Thus, \( 0.6 = \frac { 3 }{ 5 } \).
(b) 2.5
Here, the number of decimal places \( = 1 \).
The denominator \( = 10 \) and numerator \( = 25 \).
\( \implies \) fraction \( = \frac { 25 }{ 10 } = \frac { 5 }{ 2 } \).
Thus, \( 2.5 = \frac { 5 }{ 2 } \).
(c) 1.0
Here, the number of digits in the decimal part \( = 1 \).
The denominator \( = 10 \) and numerator \( = 10 \).
\( \implies \) fraction \( = \frac { 10 }{ 10 } = 1 \).
(d) 3.8
Here, the number of digits in the decimal part \( = 1 \).
The denominator \( = 10 \) and numerator \( = 38 \).
\( \implies \) fraction \( = \frac { 38 }{ 10 } = \frac { 19 }{ 5 } \).
(e) 13.7
Here, the number of digits in the decimal part \( = 1 \).
The denominator \( = 10 \) and numerator \( = 137 \).
\( \implies \) fraction \( = \frac { 137 }{ 10 } \).
(f) 21.2
Here, the number of digits in the decimal part \( = 1 \).
The denominator \( = 10 \) and numerator \( = 212 \).
\( \implies \) fraction \( = \frac { 212 }{ 10 } = \frac { 106 }{ 5 } \).
(g) 6.4
Here, the number of digits in the decimal part \( = 1 \).
The denominator \( = 10 \) and numerator \( = 64 \).
\( \implies \) fraction \( = \frac { 64 }{ 10 } = \frac { 32 }{ 5 } \).
In simple words: To change a decimal to a fraction, count the digits after the decimal point. Use that count to decide if the denominator should be 10, 100, etc. (one zero for each decimal digit). Then, simplify the fraction.

Exam Tip: Always reduce the fraction to its lowest form by dividing both the numerator and the denominator by their greatest common divisor.

 

Question 6. Express the following as cm using decimals,
(a) 2 mm
(b) 30 mm
(c) 116 mm
(d) 4 cm 2 mm
(e) 162 mm
(f) 83 mm
Answer:
(a) 2 mm:
We know that 10 mm \( = 1 \) cm or 1 mm \( = \frac { 1 }{ 10 } \) cm.
So, 2 mm \( = \frac { 1 }{ 10 } \times 2 \) cm \( = \frac {2}{10} \) cm \( = 0.2 \) cm.
(b) 30 mm:
We know that 10 mm \( = 1 \) cm or 1 mm \( = \frac { 1 }{ 10 } \) cm.
So, 30 mm \( = \frac { 1 }{ 10 } \times 30 \) cm \( = \frac { 30 }{ 10 } \) cm \( = 3.0 \) cm.
(c) 116 mm:
We know that 10 mm \( = 1 \) cm or 1 mm \( = \frac { 1 }{ 10 } \) cm.
So, 116 mm \( = \frac { 1 }{ 10 } \times 116 \) cm \( = \frac { 116 }{ 10 } \) cm \( = 11.6 \) cm.
(d) 4 cm 2 mm:
We have: 4 cm 2 mm \( = 4 \) cm \( + 2 \) mm
\( = 4 \) cm \( + \frac {2}{10} \) cm
Since \( 1 \text{ mm} = \frac { 1 }{ 10 } \text{ cm} \).
\( = \left(4+\frac{2}{10}\right) \) cm \( = 4.2 \) cm.
(e) 162 mm:
We know that 10 mm \( = 1 \) cm or 1 mm \( = \frac { 1 }{ 10 } \) cm.
So, 162 mm \( = \frac { 1 }{ 10 } \times 162 \) cm \( = \frac { 162 }{ 10 } \) cm \( = 16.2 \) cm.
(f) 83 mm:
We know that 10 mm \( = 1 \) cm or 1 mm \( = \frac { 1 }{ 10 } \) cm.
So, 83 mm \( = \frac { 1 }{ 10 } \times 83 \) cm \( = \frac { 83 }{ 10 } \) cm.
\( = \left(\frac{80}{10} + \frac{3}{10}\right) \) cm \( = \left(8+\frac{3}{10}\right) \) cm \( = 8.3 \) cm.
In simple words: To convert millimeters to centimeters, divide the number of millimeters by 10. If you have both centimeters and millimeters, first change the millimeters to centimeters, then add it to the existing centimeters.

Exam Tip: Remember the basic conversion: 10 mm = 1 cm. This fundamental relationship is key to solving all such problems correctly.

 

Question 7. Between which two whole numbers on the number line are the given numbers lie? Which of these whole numbers is nearer the number?
(a) 0.8
(b) 5.1
(c) 2.6
(d) 6.4
(e) 9.1
(f) 4.9
Answer:

(a) 0.8: The number 0.8 lies between 0 and 1, and 1 is closer to 0.8.
(b) 5.1: The number 5.1 lies between 5 and 6, and 5 is closer to 5.1.
(c) 2.6: The number 2.6 lies between 2 and 3, and 3 is closer to 2.6.
(d) 6.4: The number 6.4 lies between 6 and 7, and 6 is closer to 6.4.
(e) 9.1: The number 9.1 lies between 9 and 10, and 9 is closer to 9.1.
(f) 4.9: The number 4.9 lies between 4 and 5, and 5 is closer to 4.9.
In simple words: For each decimal, find the two whole numbers it sits between on the number line. Then, decide which of those two whole numbers is closer to the decimal.

Exam Tip: To find the nearest whole number, look at the digit in the tenths place. If it's 5 or more, round up; if it's less than 5, round down.

 

Question 8. Show the following numbers on the number line.
(a) 0.2
(b) 1.9
(c) 1.1
(d) 2.5
Answer:
(a) 0.2
The number 0.2 lies between 0 and 1.
So, divide the unit length between 0 and 1 into 10 equal parts and take 2 parts as shown below. This point is labeled A.

(b) 1.9
The number 1.9 lies between 1 and 2.
So, divide the unit length between 1 and 2 into 10 equal parts and take 9 parts as shown below. Point B represents 1.9.

(c) 1.1
The number 1.1 lies between 1 and 2. Divide the unit length between 1 and 2 into 10 equal parts and take 1 part as shown below: Point C represents 1.1.

(d) 2.5
The number 2.5 lies between 2 and 3. Divide the unit length between 2 and 3 into 10 equal parts and take 5 parts as shown below. Thus, point D represents 2.5.

In simple words: Draw a number line. For each decimal, divide the space between the two whole numbers it falls between into ten equal smaller parts. Then, mark the decimal at the correct smaller part.

Exam Tip: When marking decimals, carefully count the tenths divisions between whole numbers. Each small tick mark represents one-tenth.

 

Question 9. Write the decimal number represented by the points A, B, C, D on the given number line.

Answer:
(i) Point A: Point A is located between 0 and 1. The unit distance between 0 and 1 is divided into 10 equal parts. Point A stands at the 8th part.
Thus, 0.8 is represented by A.
(ii) Point B: Point B is located between 1 and 2. The unit distance between 1 and 2 is divided into 10 equal parts, and point B is at the 3rd part.
Thus, 1.3 is represented by B.
(iii) Point C: The point C is located between 2 and 3. The unit distance between 2 and 3 is divided into 10 equal parts, and C is at the 2nd part.
Thus, 2.2 is represented by C.
(iv) Point D: The point D is located between 2 and 3. The distance between 2 and 3 is divided into 10 equal parts, and the 9 parts have been taken.
So, 2.9 is represented by D.
In simple words: For each lettered point on the number line, count how many small marks it is past the previous whole number. Each small mark represents one-tenth.

Exam Tip: Carefully count the subdivisions between whole numbers to accurately determine the decimal value of each marked point. Each segment usually represents 0.1.

 

Question 10.
(a) The length of Ramesh's notebook is 9 cm 5 mm. What will be its length in cm?
(b) The length of a young gram plant is 65 mm. Express its length in cm.
Answer:
(a) Length of Ramesh's notebook \( = 9 \) cm 5 mm.
We know that 1 mm \( = \frac { 1 }{10} \) cm.
So, 5 mm \( = \frac {5}{10} \) cm \( = 0.5 \) cm.
Now, 9 cm 5 mm \( = 9.5 \) cm.
Thus, the length of the notebook \( = 9.5 \) cm.
(b) Given length \( = 65 \) mm.
We know that 1 mm \( = \frac { 1 }{ 10 } \) cm.
So, 65 mm \( = 65 \times \frac {1}{ 10 } \) cm \( = \frac { 65 }{10} \) cm.
\( = \left(\frac{60}{10} + \frac{5}{10}\right) \) cm \( = (6 + 0.5) \) cm \( = 6.5 \) cm.
In simple words: To change millimeters to centimeters, divide by 10. If a measurement has both centimeters and millimeters, convert the millimeters first, then add it to the centimeters.

Exam Tip: Remember the key conversion: 1 cm equals 10 mm. This simple fact helps you easily convert between the two units by either multiplying or dividing by ten.