RBSE Solutions Class 6 Maths Chapter 6 Decimal Numbers Exercise 6.1

Question 1. Write the numbers for the following in the table given below:
(i) 1 tens 2 unit 3 one-tenths
(ii) 1 hundred 3 tens 7 one-tenths
(iii) 2 hundred 5 tens 1 unit 2 one-tenths
Answer: The numbers are written in the table as decimals by combining the place values. For example, '1 tens 2 units 3 one-tenths' means \(1 \times 10 + 2 \times 1 + 3 \times \frac{1}{10}\), which is \(10 + 2 + 0.3 = 12.3\).

🎯 Exam Tip: Remember that "one-tenths" refers to the first digit after the decimal point, so be careful with place values.

Question 2. Write the place value of following decimal numbers in a table.
(i) 19.4
(ii) 0.5
(iii) 10.9
(iv) 205.9
Answer: We separate each digit of the decimal number into its correct place value column: Hundred, Tens, Unit, and Tenth. The number 19.4, for example, has 1 ten, 9 units, and 4 tenths.

🎯 Exam Tip: Always pay attention to the decimal point to correctly identify the unit and tenth places. A '0' must be placed in a column if there is no value for that place.

Question 3. Write each of the following in the form of decimal:
(i) 7 one-tenths
(ii) 2 tens 4 one-tenths
(iii) Fourteen decimal nine
(iv) Six hundred point three
Answer: We convert the given descriptions into decimal numbers by understanding their place values. For instance, "7 one-tenths" means \( \frac{7}{10} \), which is \( 0.7 \).
(i) 7 one-tenths \( = \frac{7}{10} = 0.7 \)
(ii) 2 tens 4 one-tenths:
2 tens \( = 2 \times 10 = 20 \)
4 one-tenths \( = \frac{4}{10} = 0.4 \)
Thus, 2 tens 4 one-tenths \( = 20 + 0.4 = 20.4 \)
(iii) Fourteen decimal nine \( = 14.9 \)
(iv) Six hundred point three \( = 600.3 \)
In simple words: We turn words describing numbers into actual decimal numbers. If it says "tenths," it means the number after the decimal point.

🎯 Exam Tip: Pay close attention to keywords like "tens," "units," and "one-tenths" as they tell you where each digit goes in the decimal number.

Question 4. Represent the following in form of decimal fractions:
(i) \( \frac {3}{10} \)
(ii) \( 4 + \frac {8}{10} \)
(iii) \( 300 + 50 + 8 + \frac {1}{10} \)
(iv) \( 90 + \frac {3}{10} \)
(v) \( \frac {3}{2} \)
(vi) \( \frac {2}{5} \)
(vii) \( 4 \frac {1}{2} \)
(viii) \( 3 \frac {3}{5} \)
Answer: We convert the given fractions and mixed numbers into decimal form. For fractions with a denominator of 10, it's straightforward. For others, we convert them to an equivalent fraction with a denominator of 10 or 100 first.
(i) \( \frac {3}{10} = 0.3 \)
(ii) \( 4 + \frac {8}{10} = 4 + 0.8 = 4.8 \)
(iii) \( 300 + 50 + 8 + \frac {1}{10} = 358 + 0.1 = 358.1 \)
(iv) \( 90 + \frac {3}{10} = 90 + 0.3 = 90.3 \)
(v) For \( \frac {3}{2} \):
To make the denominator 10, multiply numerator and denominator by 5:
\( \frac {3}{2} \times \frac {5}{5} = \frac {15}{10} = 1.5 \)
(vi) For \( \frac {2}{5} \):
To make the denominator 10, multiply numerator and denominator by 2:
\( \frac {2}{5} \times \frac {2}{2} = \frac {4}{10} = 0.4 \)
(vii) For \( 4 \frac {1}{2} \):
First, convert the mixed fraction to an improper fraction: \( 4 \frac {1}{2} = \frac {(4 \times 2) + 1}{2} = \frac {9}{2} \)
Now, to make the denominator 10, multiply numerator and denominator by 5:
\( \frac {9}{2} \times \frac {5}{5} = \frac {45}{10} = 4.5 \)
(viii) For \( 3 \frac {3}{5} \):
First, convert the mixed fraction to an improper fraction: \( 3 \frac {3}{5} = \frac {(3 \times 5) + 3}{5} = \frac {18}{5} \)
Now, to make the denominator 10, multiply numerator and denominator by 2:
\( \frac {18}{5} \times \frac {2}{2} = \frac {36}{10} = 3.6 \)
In simple words: We change fractions into decimal numbers. If the bottom number of the fraction is 10, it's easy. If not, we make it 10 by multiplying both top and bottom by the same number, then convert.

🎯 Exam Tip: To convert a fraction to a decimal, ensure the denominator is a power of 10 (like 10, 100). If not, find an equivalent fraction that has such a denominator.

Question 5. Write fractions for the following decimal numbers and convert in the simplest form.
(i) 0.6
(ii) 2.5
(iii) 2.8
(iv) 13.7
(v) 21.2
(vi) 1.0
(vii) 6.4
Answer: We convert each decimal number into a fraction and then simplify it to its lowest terms. To do this, we write the decimal as a fraction with a denominator of 10 (or 100, etc.) and then divide the numerator and denominator by their greatest common divisor.
(i) \( 0.6 = \frac{6}{10} \)
To simplify, divide numerator and denominator by 2: \( \frac{6 \div 2}{10 \div 2} = \frac{3}{5} \)
(ii) \( 2.5 = \frac{25}{10} \)
To simplify, divide numerator and denominator by 5: \( \frac{25 \div 5}{10 \div 5} = \frac{5}{2} \)
(iii) \( 2.8 = \frac{28}{10} \)
To simplify, divide numerator and denominator by 2: \( \frac{28 \div 2}{10 \div 2} = \frac{14}{5} \)
(iv) \( 13.7 = \frac{137}{10} \)
This fraction cannot be simplified further as 137 is a prime number.
(v) \( 21.2 = \frac{212}{10} \)
To simplify, divide numerator and denominator by 2: \( \frac{212 \div 2}{10 \div 2} = \frac{106}{5} \)
(vi) \( 1.0 = \frac{10}{10} = 1 \)
(vii) \( 6.4 = \frac{64}{10} \)
To simplify, divide numerator and denominator by 2: \( \frac{64 \div 2}{10 \div 2} = \frac{32}{5} \)
In simple words: We change decimal numbers back into fractions. Then, we make these fractions as simple as possible by dividing the top and bottom numbers by their biggest common factor.

🎯 Exam Tip: Always remember to simplify the fraction to its lowest terms by dividing the numerator and denominator by their greatest common divisor.

Question 6. Convert the following measurements into centimeters:
(iv) 5 cm 2 mm
(v) 95 mm
(vi) 19 cm 1 mm
Answer: We convert the given measurements into centimeters using the conversion factor that 10 millimeters (mm) equals 1 centimeter (cm). So, to convert mm to cm, we divide by 10.
(i) For 2 mm:
\( 10 \text{ mm} = 1 \text{ cm} \)
\( 2 \text{ mm} = \frac{1}{10} \times 2 \text{ cm} = 0.2 \text{ cm} \)
(ii) For 30 mm:
\( 10 \text{ mm} = 1 \text{ cm} \)
\( 30 \text{ mm} = 30 \times \frac{1}{10} \text{ cm} = 3 \text{ cm} \)
(iii) For 116 mm:
\( 10 \text{ mm} = 1 \text{ cm} \)
\( 116 \text{ mm} = \frac{116}{10} \text{ cm} = 11.6 \text{ cm} \)
(iv) For 5 cm 2 mm:
First, convert 2 mm to cm: \( 2 \text{ mm} = 0.2 \text{ cm} \)
Then, add to the existing cm: \( 5 \text{ cm} + 0.2 \text{ cm} = 5.2 \text{ cm} \)
(v) For 95 mm:
\( 10 \text{ mm} = 1 \text{ cm} \)
\( 95 \text{ mm} = 95 \times \frac{1}{10} \text{ cm} = 9.5 \text{ cm} \)
(vi) For 19 cm 1 mm:
First, convert 1 mm to cm: \( 1 \text{ mm} = \frac{1}{10} \times 1 \text{ cm} = 0.1 \text{ cm} \)
Then, add to the existing cm: \( 19 \text{ cm} + 0.1 \text{ cm} = 19.1 \text{ cm} \)
In simple words: To change millimeters to centimeters, you divide the number of millimeters by 10. If a measurement has both cm and mm, convert the mm part to cm first, then add it to the existing cm value.

🎯 Exam Tip: Always remember that 1 cm is equal to 10 mm. This simple conversion factor is crucial for all such problems.

Question 7. On the number line, between which two whole numbers are the following numbers marked? Which whole number of these is closer to the decimal number?
(i) 0.5
(ii) 5.3
(iii) 9.0
(iv) 4.9
(v) 3.8
Answer: We find the two whole numbers that each decimal number falls between and then determine which whole number is closest. If the decimal is .5 or greater, it rounds up.
(i) 0.5 lies between 0 and 1.
Since 0.5 is exactly in the middle, it is considered closer to 1 (following standard rounding rules where .5 rounds up).
(ii) 5.3 lies between 5 and 6.
5.3 is closer to 5 because 0.3 is less than 0.5.
(iii) 9.0 or 9 lies between 9 and 10.
9.0 is exactly 9, so it is closer to 9.
(iv) 4.9 lies between 4 and 5.
4.9 is closer to 5 because 0.9 is greater than 0.5.
(v) 3.8 lies between 3 and 4.
3.8 is closer to 4 because 0.8 is greater than 0.5.
In simple words: For each decimal, find the two whole numbers it sits between. Then, see which whole number it is closest to. If the decimal part is 0.5 or more, round up to the bigger whole number. Otherwise, round down.

🎯 Exam Tip: When rounding a decimal to the nearest whole number, look at the first digit after the decimal point. If it's 5 or more, round up; if it's less than 5, round down.

Question 8. Show the following on a number line.
(i) 0.3
(ii) 1.7
(iii) 3.4
(iv) 2.5
Answer: We mark the position of each decimal number accurately on its respective number line. Each number line is divided into tenths to show the exact location of the decimal.
(i) 0.3
(ii) 1.7
(iii) 3.4
(iv) 2.5
In simple words: We draw a straight line and mark whole numbers. Then, we find the exact spot for each decimal by dividing the space between whole numbers into ten small parts and pointing to the right spot.

🎯 Exam Tip: Ensure your number line is clearly labeled with whole numbers and accurately divided into tenths for precise placement of decimal numbers.

Question 9. The length of Tulsi's hand grip is 95 mm. Write it in cm form.
Answer: To convert millimeters (mm) to centimeters (cm), we use the conversion factor that 10 mm equals 1 cm. Therefore, to convert 95 mm to cm, we divide by 10.
Length of Tulsi's hand grip \( = 95 \text{ mm} \)
We know that \( 10 \text{ mm} = 1 \text{ cm} \)
So, \( 95 \text{ mm} = \frac{95}{10} \text{ cm} = 9.5 \text{ cm} \)
Thus, the length of Tulsi's hand grip is \( 9.5 \text{ cm} \).
In simple words: We convert millimeters to centimeters by dividing the number of millimeters by 10. So, 95 mm becomes 9.5 cm.

🎯 Exam Tip: Always remember the basic unit conversion: 10 mm = 1 cm. This allows you to easily convert between these two length units by either multiplying or dividing by 10.

Question 10. Deepu has a 6 cm. scale. It has been broken at 4.4 cm. What is the length of the left piece of scale?
Answer: To find the length of the remaining piece of the scale, we subtract the length at which it was broken from its total original length. This will show us how much of the scale is left.
Total length of scale \( = 6 \text{ cm} \)
Length at which scale was broken \( = 4.4 \text{ cm} \)
Length of remaining scale \( = (\text{Total length} - \text{Broken length}) \)
\( = (6 - 4.4) \text{ cm} \)
To subtract, align the decimal points: \( (6.0 - 4.4) \text{ cm} \)
\( = 1.6 \text{ cm} \)
Thus, the length of the left piece of scale is \( 1.6 \text{ cm} \).
In simple words: To find how much of the scale is left, we take the original total length and subtract the part that broke off. The leftover piece is 1.6 cm long.

🎯 Exam Tip: When subtracting decimal numbers, always align the decimal points to ensure correct placement of digits and accurate calculation.