GSEB Class 6 Maths Solutions Chapter 8 દશાંશ સંખ્યાઓ Exercise 8.1

 

Question 1. નીચે આપેલ કોષ્ટકમાં સંખ્યા લખો:
(a)

(b)

Answer:
(a) In this picture, there are 3 columns in the tens place (each with 10 units), 1 block in the units place, and 2 parts in the tenths place (each one tenth). So, the given table can be shown as follows:Thus, the number is 31.2.
(b) Here, in the given image, there is 1 column in the hundreds place (each 100 units), 1 column in the tens place (each 10 units), no blocks in the units place, and 4 parts in the tenths place (each one tenth). So, the given table can be shown as follows:Thus, the number is 110.4.
In simple words: Look at the pictures and the values in each column (hundreds, tens, units, tenths) to figure out the number. Then, fill in the table with these values to find the final decimal number.

Exam Tip: Carefully observe the number of blocks or columns in each place value (hundreds, tens, units, tenths) to write the correct digits in the table.

 

Question 2. નીચેની દશાંશ સંખ્યાઓને સ્થાનકિંમતના કોષ્ટકમાં લખો:
(a) 19.4
(b) 0.3
(c) 10.6
(d) 205.9
Answer:
(a) 19.4
\( 19.4 = 1 \times 10 + 9 \times 1 + 4 \times \frac{1}{10} \)
Now, let's represent it in the place value table.

(b) 0.3
\( 0.3 = 0 \times 10 + 0 \times 1 + 3 \times \frac{1}{10} \)
Now, let's represent it in the place value table.(c) 10.6
\( 10.6 = 1 \times 10 + 0 \times 1 + 6 \times \frac{1}{10} \)
Now, let's represent it in the place value table.(d) 205.9
\( 205.9 = 2 \times 100 + 0 \times 10 + 5 \times 1 + 9 \times \frac{1}{10} \)
Now, let's represent it in the place value table.In simple words: Break down each decimal number into its place values: hundreds, tens, ones, and tenths. Then, put these values into the correct columns of the place value table.

Exam Tip: Remember that the digit before the decimal point represents whole numbers (units, tens, hundreds), and the digit after the decimal point represents tenths, hundredths, and so on.

 

Question 3. નીચેના દરેકને દશાંશ સ્વરૂપે લખો:
(a) સાત દશાંશ
(b) બે દશક અને નવ દશાંશ
(c) ચૌદ પૉઈન્ટ છ
(d) એક સો અને બે એકમ
(e) છસો પૉઇન્ટ આઠ
Answer:
(a) સાત દશાંશ
Seven tenths \( = 7 \times \frac{1}{10} = 0.7 \)
(b) બે દશક અને નવ દશાંશ
Two tens and nine tenths \( = 2 \times 10 + 9 \times \frac{1}{10} = 20 + \frac{9}{10} = 20 + 0.9 = 20.9 \)
(c) ચૌદ પૉઈન્ટ છ
Fourteen point six \( = 14.6 \)
(d) એક સો અને બે એકમ
One hundred and two units \( = 1 \) hundred \( + 0 \) tens \( + 2 \) units \( + 0 \) tenths
\( = 100 + 0 + 2 + \frac{0}{10} = 102.0 \)
(e) છસો પૉઇન્ટ આઠ
Six hundred point eight \( = 600.8 \)
In simple words: Convert the word descriptions into numbers. "Dashaansh" means tenths, so it goes after the decimal. "Dashak" means tens, and "ekam" means units. "Point" means a decimal point.

Exam Tip: Pay close attention to keywords like "tens," "units," and "tenths" as they tell you the place value of each digit. "Point" always indicates the decimal separator.

 

Question 4. નીચેના દરેકને દશાંશ સ્વરૂપે લખો:
(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) \( 4\frac{2}{10} \)
(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) \( \frac{3}{2} = \frac{3 \times 5}{2 \times 5} = \frac{15}{10} = \frac{10+5}{10} = \frac{10}{10} + \frac{5}{10} = 1 + \frac{5}{10} = 1.5 \)
(h) \( \frac{2}{5} = \frac{2 \times 2}{5 \times 2} = \frac{4}{10} = 0.4 \)
(i) \( \frac{12}{5} = \frac{12 \times 2}{5 \times 2} = \frac{24}{10} = \frac{20+4}{10} = \frac{20}{10} + \frac{4}{10} = 2 + \frac{4}{10} = 2.4 \)
(j) \( 3\frac{3}{5} = 3 + \frac{3}{5} = 3 + (\frac{3 \times 2}{5 \times 2}) = 3 + \frac{6}{10} = 3.6 \)
(k) \( 4\frac{1}{2} = 4 + \frac{1}{2} = 4 + (\frac{1 \times 5}{2 \times 5}) = 4 + \frac{5}{10} = 4.5 \)
In simple words: To change fractions into decimals, make the bottom number (denominator) 10. You can do this by multiplying both the top and bottom numbers by the same value. Then, write the top number with a decimal point one place from the right. If it's a mixed number, add the whole number part to the decimal.

Exam Tip: To convert a fraction to a decimal, divide the numerator by the denominator. If the denominator is a power of 10, simply move the decimal point to the left by the number of zeros in the denominator.

 

Question 5. નીચેની દશાંશ સંખ્યાઓને અપૂર્ણાંક સ્વરૂપમાં લખી સરળ સ્વરૂપમાં ફેરવો:
(a) 0.6
(b) 2.5
(c) 1.0
(d) 3.8
(e) 13.7
(f) 21.2
(g) 6.4
Answer:
We will write the given decimal numbers as fractions with a denominator of 10. After that, we will simplify the fraction to its simplest form.
(a) 0.6
\( 0.6 = \frac{6}{10} = \frac{6 \div 2}{10 \div 2} = \frac{3}{5} \) (Here, 2 is the G.C.D. of 6 and 10.)
(b) 2.5
\( 2.5 = \frac{25}{10} = \frac{25 \div 5}{10 \div 5} = \frac{5}{2} \) (Here, 5 is the G.C.D. of 25 and 10.)
(c) 1.0
\( 1.0 = 1 + \frac{0}{10} = 1 + 0 = 1 \)
Now, 1 is already in its simplest form.
(d) 3.8
\( 3.8 = \frac{38}{10} = \frac{38 \div 2}{10 \div 2} = \frac{19}{5} \)
(e) 13.7
\( 13.7 = \frac{137}{10} \)
(f) 21.2
\( 21.2 = \frac{212}{10} = \frac{212 \div 2}{10 \div 2} = \frac{106}{5} \) (Here, 2 is the G.C.D. of 212 and 10.)
(g) 6.4
\( 6.4 = \frac{64}{10} = \frac{64 \div 2}{10 \div 2} = \frac{32}{5} \) (Here, 2 is the G.C.D. of 64 and 10.)
In simple words: To change a decimal to a fraction, write the number without the decimal point as the top number. For the bottom number, use 10 for one decimal place, 100 for two, and so on. Then, make the fraction as simple as possible by dividing the top and bottom by their largest common factor.

Exam Tip: When converting decimals to fractions, remember that the number of digits after the decimal point tells you the power of 10 to use as the denominator. Always simplify the fraction to its lowest terms.

 

Question 6. દશાંશનો ઉપયોગ કરી, નીચેના દરેકને સેમીમાં દર્શાવો:
(a) 2 મિમી
(b) 30 મિમી
(c) 116 મિમી
(d) 4 સેમી 2 મિમી
(e) 162 મિમી
(f) 83 મિમી
Answer:
(a) 2 મિમી
10 mm \( = 1 \) cm.
\( \implies 1 \) mm \( = \frac{1}{10} \) cm.
\( \implies 2 \) mm \( = 2 \times \frac{1}{10} \) cm \( = \frac{2}{10} \) cm \( = 0.2 \) cm.
(b) 30 મિમી
10 mm \( = 1 \) cm.
\( \implies 1 \) mm \( = \frac{1}{10} \) cm.
\( \implies 30 \) mm \( = 30 \times \frac{1}{10} \) cm \( = \frac{30}{10} \) cm \( = 3 \) cm.
(c) 116 મિમી
10 mm \( = 1 \) cm.
\( \implies 1 \) mm \( = \frac{1}{10} \) cm.
\( \implies 116 \) mm \( = 116 \times \frac{1}{10} \) cm \( = \frac{116}{10} \) cm.
Now, \( \frac{116}{10} \) cm \( = (\frac{110+6}{10}) \) cm \( = (\frac{110}{10} + \frac{6}{10}) \) cm \( = (11 + \frac{6}{10}) \) cm \( = 11.6 \) cm.
(d) 4 સેમી 2 મિમી
10 mm \( = 1 \) cm.
\( \implies 1 \) mm \( = \frac{1}{10} \) cm.
\( \implies 4 \) cm \( 2 \) mm \( = 4 \) cm \( + 2 \times \frac{1}{10} \) cm
\( = 4 \) cm \( + \frac{2}{10} \) cm \( = (4 + 0.2) \) cm \( = 4.2 \) cm.
(e) 162 મિમી
10 mm \( = 1 \) cm.
\( \implies 1 \) mm \( = \frac{1}{10} \) cm.
\( \implies 162 \) mm \( = 162 \times \frac{1}{10} \) cm \( = \frac{162}{10} \) cm.
Now, \( \frac{162}{10} \) cm \( = (\frac{160+2}{10}) \) cm \( = (\frac{160}{10} + \frac{2}{10}) \) cm \( = (16 + 0.2) \) cm \( = 16.2 \) cm.
(f) 83 મિમી
10 mm \( = 1 \) cm.
\( \implies 1 \) mm \( = \frac{1}{10} \) cm.
\( \implies 83 \) mm \( = 83 \times \frac{1}{10} \) cm \( = \frac{83}{10} \) cm.
Now, \( \frac{83}{10} \) cm \( = (\frac{80+3}{10}) \) cm
\( = (\frac{80}{10} + \frac{3}{10}) \) cm \( = (8 + 0.3) \) cm \( = 8.3 \) cm.
In simple words: To change millimeters (mm) to centimeters (cm), divide the number of millimeters by 10. If you have both centimeters and millimeters, convert the millimeters to centimeters first, then add it to the existing centimeters.

Exam Tip: Remember the basic conversion: 1 cm = 10 mm. This means 1 mm = 0.1 cm. Use this to correctly convert all measurements to centimeters.

 

Question 7. સંખ્યારેખા પર કઈ બે પૂર્ણ સંખ્યાઓની વચ્ચે નીચેની સંખ્યાઓનો સમાવેશ થશે? કઈ પૂર્ણ સંખ્યા આપેલ દશાંશ સંખ્યાની નજીક છે?
(a) 0.8
(b) 5.1
(c) 2.6
(d) 6.4
(e) 9.1
(f) 4.9
Answer:
(a) 0.8
0.8 lies between 0 and 1.
0.8 is closer to 1.
(b) 5.1
5.1 lies between 5 and 6.
5.1 is closer to 5.
(c) 2.6
2.6 lies between 2 and 3.
2.6 is closer to 3.
(d) 6.4
6.4 lies between 6 and 7.
6.4 is closer to 6.
(e) 9.1
9.1 lies between 9 and 10.
9.1 is closer to 9.
(f) 4.9
4.9 lies between 4 and 5.
4.9 is closer to 5.
In simple words: To find which two whole numbers a decimal is between, look at the whole number part. To see which whole number it's closest to, check the digit right after the decimal point. If it's 5 or more, round up; if it's less than 5, round down.

Exam Tip: When determining closeness, mentally check the distance to the lower and upper whole numbers. For example, 0.8 is 0.8 away from 0 and 0.2 away from 1, so it's closer to 1.

 

Question 8. નીચેની સંખ્યાઓને સંખ્યારેખા પર દર્શાવો:
(a) 0.2
(b) 1.9
(c) 1.1
(d) 2.5
Answer:
(a) 0.2
It is clear that 0.2 lies between 0 and 1. Now, divide the number line between 0 and 1 into 10 equal parts. Take the first two parts on the number line. Point A is the point corresponding to 0.2.
(b) 1.9
It is clear that 1.9 lies between 1 and 2. Now, divide the number line between 1 and 2 into 10 equal parts. Take the first nine parts on the number line. Point B is the point corresponding to 1.9.
(c) 1.1
It is clear that 1.1 lies between 1 and 2. Now, divide the number line between 1 and 2 into 10 equal parts. Take the first part on the number line. Point C is the point corresponding to 1.1.
(d) 2.5
It is clear that 2.5 lies between 2 and 3. Now, divide the number line between 2 and 3 into 10 equal parts. Take the first five parts on the number line. Point D is the point corresponding to 2.5.
In simple words: To show a decimal on a number line, first find the two whole numbers it sits between. Then, divide that part of the number line into 10 equal sections. Count over to the right the number of tenths the decimal has, and mark that spot.

Exam Tip: Remember that between any two whole numbers, there are 10 equal divisions representing tenths. Clearly label the whole numbers and the exact position of the decimal point on the number line.

 

Question 9. આપેલ સંખ્યારેખા ઉપર બિંદુઓ A, B, C, D કઈ દશાંશ સંખ્યાનું નિરૂપણ કરે છે?
Answer:
(i) Point A
Point A is between 0 and 1. The number line between 0 and 1 is divided into 10 equal parts. Point A is on the second part.
\( \implies \) Point A corresponds to 0.2.
(ii) Point B
Point B is between 1 and 2. The number line between 1 and 2 is divided into 10 equal parts. Point B is on the third part.
\( \implies \) Point B corresponds to 1.3.
(iii) Point C
Point C is between 2 and 3. The number line between 2 and 3 is divided into 10 equal parts. Point C is on the second part.
\( \implies \) Point C corresponds to 2.2.
(iv) Point D
Point D is between 2 and 3. The number line between 2 and 3 is divided into 10 equal parts. Point D is on the seventh part.
\( \implies \) Point D corresponds to 2.7.
In simple words: Look at the number line to find where each point is. The spaces between whole numbers are divided into 10 small parts, each standing for 0.1. Count how many small parts a point is from the last whole number to find its decimal value.

Exam Tip: Identify the whole number before the point and then count the number of small divisions (tenths) from that whole number to the marked point to determine its decimal value.

 

Question 10. (a) રમેશની નોટબુકની લંબાઈ 9 સેમી અને 5 મિમી છે. સેમીમાં તેની લંબાઈ કેટલી થશે? (b) ચણાના નાના છોડની લંબાઈ 65 મિમી છે. તેની લંબાઈ સેમીમાં દર્શાવો.
Answer:
(a) The length of Ramesh's notebook is 9 cm and 5 mm.
We know that 10 mm \( = 1 \) cm.
\( \implies 1 \) mm \( = \frac{1}{10} \) cm.
So, 5 mm \( = 5 \times \frac{1}{10} \) cm \( = \frac{5}{10} \) cm \( = 0.5 \) cm.
Now, the total length of Ramesh's notebook \( = 9 \) cm \( + 5 \) mm \( = 9 \) cm \( + 0.5 \) cm \( = (9 + 0.5) \) cm \( = 9.5 \) cm.
The length of Ramesh's notebook is 9.5 cm.
(b) The length of a chickpea plant is 65 mm.
We know that 10 mm \( = 1 \) cm.
\( \implies 1 \) mm \( = \frac{1}{10} \) cm.
So, 65 mm \( = 65 \times \frac{1}{10} \) cm \( = \frac{65}{10} \) cm \( = 6.5 \) cm.
The length of the chickpea plant is 6.5 cm.
In simple words: To change millimeters (mm) to centimeters (cm), divide the millimeters by 10. Then, add this decimal part to any existing centimeters to find the total length in centimeters.

Exam Tip: Always remember the conversion factor: 1 cm = 10 mm. This means you divide by 10 to convert millimeters to centimeters and multiply by 10 to convert centimeters to millimeters.