Page 1 Points to Remember : 1. Decimal Fractions. A fraction whose denominator is 10 or 100 or 1000 etc. is known as decimal fraction : e.g. 9 10 11 100 33 1000 , , etc. 2. Decimals. The numbers expressed in decimal form are called decimal numbers on simply decimals. e.g. 0·3, 2·8, 21·78, 123·432 etc. A decimal has two parts : (i) whole number (ii) decimal part. In the above examples, 0, 2, 21, 123 .... are whole numbers and ·3, ·8, ·78, ·432 ... are decimal parts. The whole number and decimal parts are separated by a point (·) which is called decimal point. 3. Decimal places. The number of digits contained in the decimal part of a decimal, are the decimal places. 4. Like and unlike decimals : (i) Like decimals. Decimals having the same number of decimal places are called like decimals. (ii) Unlike decimals. Decimals having different places of decimals are called unlike decimals. Note. (i) Putting any numebr of zeros to the extreme right of a decimal part of a decimal number, does not change its value. e.g. 2·34, 2·340, 2·3400, 2·34000 ... are all of the same value. (ii) We put zeros to the extreme right of a decimal part in order to make like decimals. 5. Comparison of decimals : Suppose we have to compare two decimals. Then, we proceed according to the following steps. Step 1. Convert the given decimals into like decimals. Step 2. First compare the whole-number part. The decimal with the greater whole- number part is greater. Step 3. If the whole-number parts are equal, compare the tenths digits. The decimal with the bigger digit in the tenths place is greater. Step 4. If the tenths digits are also equal, compare the hundredths digit, and so on. 6. Converting of a decimal into a fractions : Method : Step 1. Write the given decimal without the decimal point as the numerator of the fraction. Step 2. In the denominator, write 1 followed by as many zeros as there are decimal places in the given decimal. Step 3. Convert the above fraction to the simplest form. 7. Converting of a fraction into a decimal. General Method of Converting a Fraction into a Decimal Step 1. Divide the numerator by the denominator till a nonzero remainder is obtained. Step 2. Put a decimal point in the dividend as well as in the quotient. Step 3. Put a zero on the right of the decimal point in the dividend as well as on the right of the remainder. Step 4. Divide again just as we do in whole numbers. Page 2 Points to Remember : 1. Decimal Fractions. A fraction whose denominator is 10 or 100 or 1000 etc. is known as decimal fraction : e.g. 9 10 11 100 33 1000 , , etc. 2. Decimals. The numbers expressed in decimal form are called decimal numbers on simply decimals. e.g. 0·3, 2·8, 21·78, 123·432 etc. A decimal has two parts : (i) whole number (ii) decimal part. In the above examples, 0, 2, 21, 123 .... are whole numbers and ·3, ·8, ·78, ·432 ... are decimal parts. The whole number and decimal parts are separated by a point (·) which is called decimal point. 3. Decimal places. The number of digits contained in the decimal part of a decimal, are the decimal places. 4. Like and unlike decimals : (i) Like decimals. Decimals having the same number of decimal places are called like decimals. (ii) Unlike decimals. Decimals having different places of decimals are called unlike decimals. Note. (i) Putting any numebr of zeros to the extreme right of a decimal part of a decimal number, does not change its value. e.g. 2·34, 2·340, 2·3400, 2·34000 ... are all of the same value. (ii) We put zeros to the extreme right of a decimal part in order to make like decimals. 5. Comparison of decimals : Suppose we have to compare two decimals. Then, we proceed according to the following steps. Step 1. Convert the given decimals into like decimals. Step 2. First compare the whole-number part. The decimal with the greater whole- number part is greater. Step 3. If the whole-number parts are equal, compare the tenths digits. The decimal with the bigger digit in the tenths place is greater. Step 4. If the tenths digits are also equal, compare the hundredths digit, and so on. 6. Converting of a decimal into a fractions : Method : Step 1. Write the given decimal without the decimal point as the numerator of the fraction. Step 2. In the denominator, write 1 followed by as many zeros as there are decimal places in the given decimal. Step 3. Convert the above fraction to the simplest form. 7. Converting of a fraction into a decimal. General Method of Converting a Fraction into a Decimal Step 1. Divide the numerator by the denominator till a nonzero remainder is obtained. Step 2. Put a decimal point in the dividend as well as in the quotient. Step 3. Put a zero on the right of the decimal point in the dividend as well as on the right of the remainder. Step 4. Divide again just as we do in whole numbers. Step 5. Repeat step 4 till the remainder is zero. 8. Addition of Decimals Method : Step 1. Convert the given decimals into like decimals. Step 2. Write the addends one under the other in column form, keeping the decimal points of all the addends in the same column and the digits of the same place in the same column. Step 3. Add as in the case of whole numbers. Step 4. In the sum, put the decimal point directly under decimal points in the addonds. 9. Subtraction of Decimals Method : Step 1. Convert the given decimals into like decimals. Step 2. Write the smaller number under the larger one in column form in such a way that the decimal points of both the numbers are in the same column and the digits of the same place lie in the same column. Step 3. Subtract as we do in case of whole numbers. Step 4. In the difference, put the decimal point directly under the decimal points of the given numbers. 10. 1000 g = 1 kg, 100 cm = 1 metre 100 paise = 1 rupee Q. 1. Write each of the following in figures : (i) Fifty-eight point six three (ii) One hundred twenty-four point four two five (iii) Seven point seven six (iv) Nineteen point eight (v) Four hundred four point zero four four (vi) Point one seven three (vii) Point zero one five Sol. (i) Fifty-eight point six three = 58·63 (ii) One hundred twenty-four point four two five = 124·425 (iii) Seven point seven six = 7·76 (iv) Nineteen point eight = 19·8 (v) Four hundred four point zero four four = 404·044 (vi) Point one seven three = ·173 (vii) Point zero one five = ·015 Ans. Q. 2. Write the place value of each digit in each of the following decimals : (i) 14·83 (ii) 275·269 (iii) 46·075 (iv) 302·459 (v) 5370·34 (vi) 186·209 Sol. (i) 14·83 Place value of 1 = 10, Place value of 4 = 4, Place value of 8 = 8 10 , Place value of 3 = 3 100 (ii) 275·269 Place value of 2 = 200, Place value of 7 = 70, Place value of 5 = 5, Place value of 2 = 2 10 , Place value of 6 = 6 100 , Place value of 9 9 1000 (iii) 46·075 Place value of 4 = 40, Place value of 6 = 6, Place value of 0 = 0, Place value of 7 = 7 100 , Place value of 5 = 5 1000 (iv) 302·459 Place value of 3 = 300, Place value of 0 = 0, Place value of 2 = 2, Place value of 4 4 10 Place value of 5 5 100 Place value of 9 9 1000 Page 3 Points to Remember : 1. Decimal Fractions. A fraction whose denominator is 10 or 100 or 1000 etc. is known as decimal fraction : e.g. 9 10 11 100 33 1000 , , etc. 2. Decimals. The numbers expressed in decimal form are called decimal numbers on simply decimals. e.g. 0·3, 2·8, 21·78, 123·432 etc. A decimal has two parts : (i) whole number (ii) decimal part. In the above examples, 0, 2, 21, 123 .... are whole numbers and ·3, ·8, ·78, ·432 ... are decimal parts. The whole number and decimal parts are separated by a point (·) which is called decimal point. 3. Decimal places. The number of digits contained in the decimal part of a decimal, are the decimal places. 4. Like and unlike decimals : (i) Like decimals. Decimals having the same number of decimal places are called like decimals. (ii) Unlike decimals. Decimals having different places of decimals are called unlike decimals. Note. (i) Putting any numebr of zeros to the extreme right of a decimal part of a decimal number, does not change its value. e.g. 2·34, 2·340, 2·3400, 2·34000 ... are all of the same value. (ii) We put zeros to the extreme right of a decimal part in order to make like decimals. 5. Comparison of decimals : Suppose we have to compare two decimals. Then, we proceed according to the following steps. Step 1. Convert the given decimals into like decimals. Step 2. First compare the whole-number part. The decimal with the greater whole- number part is greater. Step 3. If the whole-number parts are equal, compare the tenths digits. The decimal with the bigger digit in the tenths place is greater. Step 4. If the tenths digits are also equal, compare the hundredths digit, and so on. 6. Converting of a decimal into a fractions : Method : Step 1. Write the given decimal without the decimal point as the numerator of the fraction. Step 2. In the denominator, write 1 followed by as many zeros as there are decimal places in the given decimal. Step 3. Convert the above fraction to the simplest form. 7. Converting of a fraction into a decimal. General Method of Converting a Fraction into a Decimal Step 1. Divide the numerator by the denominator till a nonzero remainder is obtained. Step 2. Put a decimal point in the dividend as well as in the quotient. Step 3. Put a zero on the right of the decimal point in the dividend as well as on the right of the remainder. Step 4. Divide again just as we do in whole numbers. Step 5. Repeat step 4 till the remainder is zero. 8. Addition of Decimals Method : Step 1. Convert the given decimals into like decimals. Step 2. Write the addends one under the other in column form, keeping the decimal points of all the addends in the same column and the digits of the same place in the same column. Step 3. Add as in the case of whole numbers. Step 4. In the sum, put the decimal point directly under decimal points in the addonds. 9. Subtraction of Decimals Method : Step 1. Convert the given decimals into like decimals. Step 2. Write the smaller number under the larger one in column form in such a way that the decimal points of both the numbers are in the same column and the digits of the same place lie in the same column. Step 3. Subtract as we do in case of whole numbers. Step 4. In the difference, put the decimal point directly under the decimal points of the given numbers. 10. 1000 g = 1 kg, 100 cm = 1 metre 100 paise = 1 rupee Q. 1. Write each of the following in figures : (i) Fifty-eight point six three (ii) One hundred twenty-four point four two five (iii) Seven point seven six (iv) Nineteen point eight (v) Four hundred four point zero four four (vi) Point one seven three (vii) Point zero one five Sol. (i) Fifty-eight point six three = 58·63 (ii) One hundred twenty-four point four two five = 124·425 (iii) Seven point seven six = 7·76 (iv) Nineteen point eight = 19·8 (v) Four hundred four point zero four four = 404·044 (vi) Point one seven three = ·173 (vii) Point zero one five = ·015 Ans. Q. 2. Write the place value of each digit in each of the following decimals : (i) 14·83 (ii) 275·269 (iii) 46·075 (iv) 302·459 (v) 5370·34 (vi) 186·209 Sol. (i) 14·83 Place value of 1 = 10, Place value of 4 = 4, Place value of 8 = 8 10 , Place value of 3 = 3 100 (ii) 275·269 Place value of 2 = 200, Place value of 7 = 70, Place value of 5 = 5, Place value of 2 = 2 10 , Place value of 6 = 6 100 , Place value of 9 9 1000 (iii) 46·075 Place value of 4 = 40, Place value of 6 = 6, Place value of 0 = 0, Place value of 7 = 7 100 , Place value of 5 = 5 1000 (iv) 302·459 Place value of 3 = 300, Place value of 0 = 0, Place value of 2 = 2, Place value of 4 4 10 Place value of 5 5 100 Place value of 9 9 1000 (v) 5370·34 Place value of 5 = 5000, Place value of 3 = 300, Place value of 7 = 70, Place value of 0 = 0, Place value of 3 3 10 , Place value of 4 4 100 (vi) 186·209 Place value of 1 = 100, Place value of 8 = 80, Place value of 6 = 6, Place value of 2 = 2 10 , Place value of 0 = 0, Place value of 9 = 9 1000 Q. 3. Write each of the following decimals in the expanded form : (i) 67·83 (ii) 283·61 (iii) 24·675 (iv) 0·294 (v) 8·006 (vi) 4615·72 Sol. (i) 67·83 = (6 × 10) + (7 × 1) + 8 1 10 3 1 100 F H G I K J F H G I K J (ii) 283·61 = (2 × 100) + (8 × 10) + (3 × 1) + 6 1 10 1 1 100 F H G I K J F H G I K J (iii) 24·675 = (2 × 10) + (4 × 1) + 6 1 10 7 1 100 5 1 1000 F H G I K J F H G I K J F H G I K J (iv) 0·294 = 2 1 10 9 1 100 4 1 1000 F H G I K J F H G I K J F H G I K J (v) 8·006 = (8 × 1) + 6 1 1000 F H G I K J (vi) 4615·72 = (4 × 1000) + (6 × 100) + (1 × 10) + (5 × 1) + 7 1 10 2 1 100 F H G I K J F H G I K J Ans. Q. 4. Write each of the following in the decimal form : (i) 40 6 7 10 9 100 (ii) 500 70 8 3 10 1 100 6 1000 (iii) 700 30 1 8 10 4 100 (iv) 600 5 7 100 9 1000 (v) 800 5 8 10 6 1000 (vi) 30 9 4 100 8 1000 Sol. (i) 40 6 7 10 9 100 = 46·79 (ii) 500 70 8 3 10 1 100 6 1000 = 578·316 (iii) 700 30 1 8 10 4 100 = 731·84 (iv) 600 5 7 100 9 1000 = 605·079 (v) 800 5 8 10 6 1000 = 805·806 (vi) 30 9 4 100 8 1000 = 39·048 Ans. Q. 5. Convert each of the following into like decimals : (i) 7·5, 64·23, 0·074 (ii) 0·6, 5·937, 2·36, 4·2 (iii) 1·6, 0·07, 3·58, 2·9 (iv) 2·5, 0·63, 14·08, 1·637 Sol. (i) 7·5, 64·23, 0·074 = 7·500, 64·230, 0·074 (Here, at the most 0·074 has 3 places) (ii) 0·6, 5·937, 2·36, 4·2 = 0·600, 5·937, 2·360, 4·200 (Here, 5·937 has at most 3 places) (iii) 1·6, 0·07, 3·58, 2·9 = 1·60, 0·07, 3·58, 2·90 Page 4 Points to Remember : 1. Decimal Fractions. A fraction whose denominator is 10 or 100 or 1000 etc. is known as decimal fraction : e.g. 9 10 11 100 33 1000 , , etc. 2. Decimals. The numbers expressed in decimal form are called decimal numbers on simply decimals. e.g. 0·3, 2·8, 21·78, 123·432 etc. A decimal has two parts : (i) whole number (ii) decimal part. In the above examples, 0, 2, 21, 123 .... are whole numbers and ·3, ·8, ·78, ·432 ... are decimal parts. The whole number and decimal parts are separated by a point (·) which is called decimal point. 3. Decimal places. The number of digits contained in the decimal part of a decimal, are the decimal places. 4. Like and unlike decimals : (i) Like decimals. Decimals having the same number of decimal places are called like decimals. (ii) Unlike decimals. Decimals having different places of decimals are called unlike decimals. Note. (i) Putting any numebr of zeros to the extreme right of a decimal part of a decimal number, does not change its value. e.g. 2·34, 2·340, 2·3400, 2·34000 ... are all of the same value. (ii) We put zeros to the extreme right of a decimal part in order to make like decimals. 5. Comparison of decimals : Suppose we have to compare two decimals. Then, we proceed according to the following steps. Step 1. Convert the given decimals into like decimals. Step 2. First compare the whole-number part. The decimal with the greater whole- number part is greater. Step 3. If the whole-number parts are equal, compare the tenths digits. The decimal with the bigger digit in the tenths place is greater. Step 4. If the tenths digits are also equal, compare the hundredths digit, and so on. 6. Converting of a decimal into a fractions : Method : Step 1. Write the given decimal without the decimal point as the numerator of the fraction. Step 2. In the denominator, write 1 followed by as many zeros as there are decimal places in the given decimal. Step 3. Convert the above fraction to the simplest form. 7. Converting of a fraction into a decimal. General Method of Converting a Fraction into a Decimal Step 1. Divide the numerator by the denominator till a nonzero remainder is obtained. Step 2. Put a decimal point in the dividend as well as in the quotient. Step 3. Put a zero on the right of the decimal point in the dividend as well as on the right of the remainder. Step 4. Divide again just as we do in whole numbers. Step 5. Repeat step 4 till the remainder is zero. 8. Addition of Decimals Method : Step 1. Convert the given decimals into like decimals. Step 2. Write the addends one under the other in column form, keeping the decimal points of all the addends in the same column and the digits of the same place in the same column. Step 3. Add as in the case of whole numbers. Step 4. In the sum, put the decimal point directly under decimal points in the addonds. 9. Subtraction of Decimals Method : Step 1. Convert the given decimals into like decimals. Step 2. Write the smaller number under the larger one in column form in such a way that the decimal points of both the numbers are in the same column and the digits of the same place lie in the same column. Step 3. Subtract as we do in case of whole numbers. Step 4. In the difference, put the decimal point directly under the decimal points of the given numbers. 10. 1000 g = 1 kg, 100 cm = 1 metre 100 paise = 1 rupee Q. 1. Write each of the following in figures : (i) Fifty-eight point six three (ii) One hundred twenty-four point four two five (iii) Seven point seven six (iv) Nineteen point eight (v) Four hundred four point zero four four (vi) Point one seven three (vii) Point zero one five Sol. (i) Fifty-eight point six three = 58·63 (ii) One hundred twenty-four point four two five = 124·425 (iii) Seven point seven six = 7·76 (iv) Nineteen point eight = 19·8 (v) Four hundred four point zero four four = 404·044 (vi) Point one seven three = ·173 (vii) Point zero one five = ·015 Ans. Q. 2. Write the place value of each digit in each of the following decimals : (i) 14·83 (ii) 275·269 (iii) 46·075 (iv) 302·459 (v) 5370·34 (vi) 186·209 Sol. (i) 14·83 Place value of 1 = 10, Place value of 4 = 4, Place value of 8 = 8 10 , Place value of 3 = 3 100 (ii) 275·269 Place value of 2 = 200, Place value of 7 = 70, Place value of 5 = 5, Place value of 2 = 2 10 , Place value of 6 = 6 100 , Place value of 9 9 1000 (iii) 46·075 Place value of 4 = 40, Place value of 6 = 6, Place value of 0 = 0, Place value of 7 = 7 100 , Place value of 5 = 5 1000 (iv) 302·459 Place value of 3 = 300, Place value of 0 = 0, Place value of 2 = 2, Place value of 4 4 10 Place value of 5 5 100 Place value of 9 9 1000 (v) 5370·34 Place value of 5 = 5000, Place value of 3 = 300, Place value of 7 = 70, Place value of 0 = 0, Place value of 3 3 10 , Place value of 4 4 100 (vi) 186·209 Place value of 1 = 100, Place value of 8 = 80, Place value of 6 = 6, Place value of 2 = 2 10 , Place value of 0 = 0, Place value of 9 = 9 1000 Q. 3. Write each of the following decimals in the expanded form : (i) 67·83 (ii) 283·61 (iii) 24·675 (iv) 0·294 (v) 8·006 (vi) 4615·72 Sol. (i) 67·83 = (6 × 10) + (7 × 1) + 8 1 10 3 1 100 F H G I K J F H G I K J (ii) 283·61 = (2 × 100) + (8 × 10) + (3 × 1) + 6 1 10 1 1 100 F H G I K J F H G I K J (iii) 24·675 = (2 × 10) + (4 × 1) + 6 1 10 7 1 100 5 1 1000 F H G I K J F H G I K J F H G I K J (iv) 0·294 = 2 1 10 9 1 100 4 1 1000 F H G I K J F H G I K J F H G I K J (v) 8·006 = (8 × 1) + 6 1 1000 F H G I K J (vi) 4615·72 = (4 × 1000) + (6 × 100) + (1 × 10) + (5 × 1) + 7 1 10 2 1 100 F H G I K J F H G I K J Ans. Q. 4. Write each of the following in the decimal form : (i) 40 6 7 10 9 100 (ii) 500 70 8 3 10 1 100 6 1000 (iii) 700 30 1 8 10 4 100 (iv) 600 5 7 100 9 1000 (v) 800 5 8 10 6 1000 (vi) 30 9 4 100 8 1000 Sol. (i) 40 6 7 10 9 100 = 46·79 (ii) 500 70 8 3 10 1 100 6 1000 = 578·316 (iii) 700 30 1 8 10 4 100 = 731·84 (iv) 600 5 7 100 9 1000 = 605·079 (v) 800 5 8 10 6 1000 = 805·806 (vi) 30 9 4 100 8 1000 = 39·048 Ans. Q. 5. Convert each of the following into like decimals : (i) 7·5, 64·23, 0·074 (ii) 0·6, 5·937, 2·36, 4·2 (iii) 1·6, 0·07, 3·58, 2·9 (iv) 2·5, 0·63, 14·08, 1·637 Sol. (i) 7·5, 64·23, 0·074 = 7·500, 64·230, 0·074 (Here, at the most 0·074 has 3 places) (ii) 0·6, 5·937, 2·36, 4·2 = 0·600, 5·937, 2·360, 4·200 (Here, 5·937 has at most 3 places) (iii) 1·6, 0·07, 3·58, 2·9 = 1·60, 0·07, 3·58, 2·90 (Here, at the most are two places) (iv) 2·5, 0·63, 14·08, 1·637 = 2·500, 0·630, 14·080, 1·637 Ans. (Here, at the most are three places) Q. 6. Fill in each of the place holders with the correct symbol > or < : (i) 84·23 76·35 (ii) 7·608 7·68 (iii) 8·34 8·43 (iv) 12·06 12·006 (v) 3·85 3·805 (vi) 0·97 1·07 Sol. Making like decimals where ever it is necessary, (i) 84·23 76·35 84·23 > 76·35 (ii) 7·608 7·68 7·608 7·680 7·608 < 7·680 (iii) 8·34 8·43 8·34 < 8·43 (iv) 12·06 12·006 12·060 12·006 12·06 > 12·006 (v) 3·85 3·805 3·850 3·805 3·850 > 3·805 (vi) 0·97 1·07 0·97 < 1·07 Ans. Q. 7. Arrange the following decimals in an ascending order : (i) 5·8, 7·2, 5·69, 7·14, 5·06 (ii) 0·6, 6·6, 6·06, 66·6, 0·06 (iii) 6·54, 6·45, 6·4, 6·5, 6·05 (iv) 3·3, 3·303, 3·033, 0·33, 3·003 Sol. First of all making them in like decimals, (i) 5·8, 7·2, 5·69, 7·14, 5·06 5·80, 7·20, 5·69, 7·14, 5·06 Arranging in ascending order, 5·06 < 5·69 < 5·80 < 7·14 < 7·20 5·06 < 5·69 < 5·8 < 7·14 < 7·2 Ans. (ii) 0·6, 6·6, 6·06, 66·6, 0·06 0·60, 6·60, 6·06, 66·60, 0·06 Arranging in ascending order, 0·06 < 0·60 < 6·06 < 6·60 < 66·60 0·06 < 0·6 < 6·06 < 6·6 < 66·6 Ans. (iii) 6·54, 6·45, 6·4, 6·5, 6·05 6·54, 6·45, 6·4, 6·5, 6·05 Arranging in ascending order, 6·05 < 6·40 < 6·45 < 6·50 < 6·54 6·05 < 6·4 < 6·45 < 6·5 < 6·54 Ans. (iv) 3·3, 3·303, 3·033, 0·33, 3·003 3·300, 3·303, 3·033, 0·330, 3·003 Arranging in descending order, 0·330 < 3·003 < 3·033 < 3·300 < 3·303 0·33 < 3·003 < 3·033 < 3·3 < 3·303 Ans. Q. 8. Arrange the following decimals in a descending order : (i) 7·3, 8·73, 73·03, 7·33, 8·073 (ii) 3·3, 3·03, 30·3, 30·03, 3·003 (iii) 2·7, 7·2, 2·27, 2·72, 2·02, 2·007 (iv) 8·88, 8·088, 88·8, 88·08, 8·008 Sol. Making them in like decimals and them comparing (i) 7·3, 8·73, 73·03, 7·33, 8·073 7·300, 8·730, 73·030, 7·330, 8·073 Arranging in decending order 73·030 > 8·730 > 8·073 > 7·330 > 7·300 73·03 > 8·73 > 8·073 > 7·33 > 7·3 Ans. (ii) 3·3, 3·03, 30·3, 30·03, 3·003 3·300, 3·030, 30·300, 30·030, 3·003 Arranging in descending order 30·300 > 30·030 > 3·300 > 3·030 > 3·003 30·3 > 30·03 > 3·3 > 3·03 > 3·003 Ans. (iii) 2·7, 7·2, 2·27, 2·72, 2·02, 2·007 2·700, 7·200, 2·270, 2·720, 2·020, 2·007 Arranging in descending order 7·200 > 2·720 > 2·700 > 2·270 > 2·020> 2·007 7·2 > 2·72 > 2·7 > 2·27 > 2·02 > 2·007 Ans. (iv) 8·88, 8·088, 88·8, 88·08, 8·008 8·880, 8·088, 88·800, 88·080, 8·008 Arranging in decending order, 88·800 > 88·080 > 8·880 > 8·088 > 8·008 88·8 > 88·08 > 8·88 > 8·088 > 8·008 Ans. Page 5 Points to Remember : 1. Decimal Fractions. A fraction whose denominator is 10 or 100 or 1000 etc. is known as decimal fraction : e.g. 9 10 11 100 33 1000 , , etc. 2. Decimals. The numbers expressed in decimal form are called decimal numbers on simply decimals. e.g. 0·3, 2·8, 21·78, 123·432 etc. A decimal has two parts : (i) whole number (ii) decimal part. In the above examples, 0, 2, 21, 123 .... are whole numbers and ·3, ·8, ·78, ·432 ... are decimal parts. The whole number and decimal parts are separated by a point (·) which is called decimal point. 3. Decimal places. The number of digits contained in the decimal part of a decimal, are the decimal places. 4. Like and unlike decimals : (i) Like decimals. Decimals having the same number of decimal places are called like decimals. (ii) Unlike decimals. Decimals having different places of decimals are called unlike decimals. Note. (i) Putting any numebr of zeros to the extreme right of a decimal part of a decimal number, does not change its value. e.g. 2·34, 2·340, 2·3400, 2·34000 ... are all of the same value. (ii) We put zeros to the extreme right of a decimal part in order to make like decimals. 5. Comparison of decimals : Suppose we have to compare two decimals. Then, we proceed according to the following steps. Step 1. Convert the given decimals into like decimals. Step 2. First compare the whole-number part. The decimal with the greater whole- number part is greater. Step 3. If the whole-number parts are equal, compare the tenths digits. The decimal with the bigger digit in the tenths place is greater. Step 4. If the tenths digits are also equal, compare the hundredths digit, and so on. 6. Converting of a decimal into a fractions : Method : Step 1. Write the given decimal without the decimal point as the numerator of the fraction. Step 2. In the denominator, write 1 followed by as many zeros as there are decimal places in the given decimal. Step 3. Convert the above fraction to the simplest form. 7. Converting of a fraction into a decimal. General Method of Converting a Fraction into a Decimal Step 1. Divide the numerator by the denominator till a nonzero remainder is obtained. Step 2. Put a decimal point in the dividend as well as in the quotient. Step 3. Put a zero on the right of the decimal point in the dividend as well as on the right of the remainder. Step 4. Divide again just as we do in whole numbers. Step 5. Repeat step 4 till the remainder is zero. 8. Addition of Decimals Method : Step 1. Convert the given decimals into like decimals. Step 2. Write the addends one under the other in column form, keeping the decimal points of all the addends in the same column and the digits of the same place in the same column. Step 3. Add as in the case of whole numbers. Step 4. In the sum, put the decimal point directly under decimal points in the addonds. 9. Subtraction of Decimals Method : Step 1. Convert the given decimals into like decimals. Step 2. Write the smaller number under the larger one in column form in such a way that the decimal points of both the numbers are in the same column and the digits of the same place lie in the same column. Step 3. Subtract as we do in case of whole numbers. Step 4. In the difference, put the decimal point directly under the decimal points of the given numbers. 10. 1000 g = 1 kg, 100 cm = 1 metre 100 paise = 1 rupee Q. 1. Write each of the following in figures : (i) Fifty-eight point six three (ii) One hundred twenty-four point four two five (iii) Seven point seven six (iv) Nineteen point eight (v) Four hundred four point zero four four (vi) Point one seven three (vii) Point zero one five Sol. (i) Fifty-eight point six three = 58·63 (ii) One hundred twenty-four point four two five = 124·425 (iii) Seven point seven six = 7·76 (iv) Nineteen point eight = 19·8 (v) Four hundred four point zero four four = 404·044 (vi) Point one seven three = ·173 (vii) Point zero one five = ·015 Ans. Q. 2. Write the place value of each digit in each of the following decimals : (i) 14·83 (ii) 275·269 (iii) 46·075 (iv) 302·459 (v) 5370·34 (vi) 186·209 Sol. (i) 14·83 Place value of 1 = 10, Place value of 4 = 4, Place value of 8 = 8 10 , Place value of 3 = 3 100 (ii) 275·269 Place value of 2 = 200, Place value of 7 = 70, Place value of 5 = 5, Place value of 2 = 2 10 , Place value of 6 = 6 100 , Place value of 9 9 1000 (iii) 46·075 Place value of 4 = 40, Place value of 6 = 6, Place value of 0 = 0, Place value of 7 = 7 100 , Place value of 5 = 5 1000 (iv) 302·459 Place value of 3 = 300, Place value of 0 = 0, Place value of 2 = 2, Place value of 4 4 10 Place value of 5 5 100 Place value of 9 9 1000 (v) 5370·34 Place value of 5 = 5000, Place value of 3 = 300, Place value of 7 = 70, Place value of 0 = 0, Place value of 3 3 10 , Place value of 4 4 100 (vi) 186·209 Place value of 1 = 100, Place value of 8 = 80, Place value of 6 = 6, Place value of 2 = 2 10 , Place value of 0 = 0, Place value of 9 = 9 1000 Q. 3. Write each of the following decimals in the expanded form : (i) 67·83 (ii) 283·61 (iii) 24·675 (iv) 0·294 (v) 8·006 (vi) 4615·72 Sol. (i) 67·83 = (6 × 10) + (7 × 1) + 8 1 10 3 1 100 F H G I K J F H G I K J (ii) 283·61 = (2 × 100) + (8 × 10) + (3 × 1) + 6 1 10 1 1 100 F H G I K J F H G I K J (iii) 24·675 = (2 × 10) + (4 × 1) + 6 1 10 7 1 100 5 1 1000 F H G I K J F H G I K J F H G I K J (iv) 0·294 = 2 1 10 9 1 100 4 1 1000 F H G I K J F H G I K J F H G I K J (v) 8·006 = (8 × 1) + 6 1 1000 F H G I K J (vi) 4615·72 = (4 × 1000) + (6 × 100) + (1 × 10) + (5 × 1) + 7 1 10 2 1 100 F H G I K J F H G I K J Ans. Q. 4. Write each of the following in the decimal form : (i) 40 6 7 10 9 100 (ii) 500 70 8 3 10 1 100 6 1000 (iii) 700 30 1 8 10 4 100 (iv) 600 5 7 100 9 1000 (v) 800 5 8 10 6 1000 (vi) 30 9 4 100 8 1000 Sol. (i) 40 6 7 10 9 100 = 46·79 (ii) 500 70 8 3 10 1 100 6 1000 = 578·316 (iii) 700 30 1 8 10 4 100 = 731·84 (iv) 600 5 7 100 9 1000 = 605·079 (v) 800 5 8 10 6 1000 = 805·806 (vi) 30 9 4 100 8 1000 = 39·048 Ans. Q. 5. Convert each of the following into like decimals : (i) 7·5, 64·23, 0·074 (ii) 0·6, 5·937, 2·36, 4·2 (iii) 1·6, 0·07, 3·58, 2·9 (iv) 2·5, 0·63, 14·08, 1·637 Sol. (i) 7·5, 64·23, 0·074 = 7·500, 64·230, 0·074 (Here, at the most 0·074 has 3 places) (ii) 0·6, 5·937, 2·36, 4·2 = 0·600, 5·937, 2·360, 4·200 (Here, 5·937 has at most 3 places) (iii) 1·6, 0·07, 3·58, 2·9 = 1·60, 0·07, 3·58, 2·90 (Here, at the most are two places) (iv) 2·5, 0·63, 14·08, 1·637 = 2·500, 0·630, 14·080, 1·637 Ans. (Here, at the most are three places) Q. 6. Fill in each of the place holders with the correct symbol > or < : (i) 84·23 76·35 (ii) 7·608 7·68 (iii) 8·34 8·43 (iv) 12·06 12·006 (v) 3·85 3·805 (vi) 0·97 1·07 Sol. Making like decimals where ever it is necessary, (i) 84·23 76·35 84·23 > 76·35 (ii) 7·608 7·68 7·608 7·680 7·608 < 7·680 (iii) 8·34 8·43 8·34 < 8·43 (iv) 12·06 12·006 12·060 12·006 12·06 > 12·006 (v) 3·85 3·805 3·850 3·805 3·850 > 3·805 (vi) 0·97 1·07 0·97 < 1·07 Ans. Q. 7. Arrange the following decimals in an ascending order : (i) 5·8, 7·2, 5·69, 7·14, 5·06 (ii) 0·6, 6·6, 6·06, 66·6, 0·06 (iii) 6·54, 6·45, 6·4, 6·5, 6·05 (iv) 3·3, 3·303, 3·033, 0·33, 3·003 Sol. First of all making them in like decimals, (i) 5·8, 7·2, 5·69, 7·14, 5·06 5·80, 7·20, 5·69, 7·14, 5·06 Arranging in ascending order, 5·06 < 5·69 < 5·80 < 7·14 < 7·20 5·06 < 5·69 < 5·8 < 7·14 < 7·2 Ans. (ii) 0·6, 6·6, 6·06, 66·6, 0·06 0·60, 6·60, 6·06, 66·60, 0·06 Arranging in ascending order, 0·06 < 0·60 < 6·06 < 6·60 < 66·60 0·06 < 0·6 < 6·06 < 6·6 < 66·6 Ans. (iii) 6·54, 6·45, 6·4, 6·5, 6·05 6·54, 6·45, 6·4, 6·5, 6·05 Arranging in ascending order, 6·05 < 6·40 < 6·45 < 6·50 < 6·54 6·05 < 6·4 < 6·45 < 6·5 < 6·54 Ans. (iv) 3·3, 3·303, 3·033, 0·33, 3·003 3·300, 3·303, 3·033, 0·330, 3·003 Arranging in descending order, 0·330 < 3·003 < 3·033 < 3·300 < 3·303 0·33 < 3·003 < 3·033 < 3·3 < 3·303 Ans. Q. 8. Arrange the following decimals in a descending order : (i) 7·3, 8·73, 73·03, 7·33, 8·073 (ii) 3·3, 3·03, 30·3, 30·03, 3·003 (iii) 2·7, 7·2, 2·27, 2·72, 2·02, 2·007 (iv) 8·88, 8·088, 88·8, 88·08, 8·008 Sol. Making them in like decimals and them comparing (i) 7·3, 8·73, 73·03, 7·33, 8·073 7·300, 8·730, 73·030, 7·330, 8·073 Arranging in decending order 73·030 > 8·730 > 8·073 > 7·330 > 7·300 73·03 > 8·73 > 8·073 > 7·33 > 7·3 Ans. (ii) 3·3, 3·03, 30·3, 30·03, 3·003 3·300, 3·030, 30·300, 30·030, 3·003 Arranging in descending order 30·300 > 30·030 > 3·300 > 3·030 > 3·003 30·3 > 30·03 > 3·3 > 3·03 > 3·003 Ans. (iii) 2·7, 7·2, 2·27, 2·72, 2·02, 2·007 2·700, 7·200, 2·270, 2·720, 2·020, 2·007 Arranging in descending order 7·200 > 2·720 > 2·700 > 2·270 > 2·020> 2·007 7·2 > 2·72 > 2·7 > 2·27 > 2·02 > 2·007 Ans. (iv) 8·88, 8·088, 88·8, 88·08, 8·008 8·880, 8·088, 88·800, 88·080, 8·008 Arranging in decending order, 88·800 > 88·080 > 8·880 > 8·088 > 8·008 88·8 > 88·08 > 8·88 > 8·088 > 8·008 Ans. Convert each of the following into a fraction in its simplest form : Q. 1. ·9 Q. 2. 0·6 Q. 3. ·08 Q. 4. 0·15 Q. 5. 0·48 Q. 6. ·053 Q. 7. 0·125 Q. 8. ·224 Sol. 1. ·9 = 9 10 2. 0·6 = 6 10 6 2 10 2 3 5 (Dividing by 2, the HCF of 6, 10) 3. ·08 = 8 100 8 4 100 4 2 25 (Dividing by 4, the HCF of 8, 100) 4. 0·15 = 15 100 15 5 100 5 3 20 (Dividing by 5, the HCF of 15, 100) 5. 0·48 = 48 100 48 4 100 4 12 25 (Dividing by 4, the HCF of 48, 100) 6. ·053 = 53 1000 7. 0·125 = 125 1000 125 125 1000 125 1 8 (Dividing by 125, the HCF of 125, 1000) 8. ·224 = 224 1000 224 8 1000 8 28 125 Ans. (Dividing by 8, the HCF of 224 and 1000) Convert each of the following as a mixed fraction : Q. 9. 6·4 Q. 10. 16·5 Q. 11. 8·36 Q. 11. 4·275 Q. 13. 25·06 Q. 14. 7·004 Q. 15. 2·052 Q. 16. 3·108 Sol. 9. 6·4 = 64 10 64 2 10 2 32 5 6 2 5 (Dividing by 2, the HCF of 64 and 10) 10. 16·5 = 165 10 165 5 10 5 33 2 16 1 2 (Dividing by 5, the HCF of 165 and 10) 11. 8·36 = 836 100 836 4 100 4 209 25 8 9 25 (Dividing by 4, the HCF of 836 and 100) 12. 4·275 = 4275 1000 4275 25 1000 25 171 40 4 11 40 (Dividing by 25) 13. 25·06 = 2506 100 2506 2 100 2 1253 50 25 3 50 (Dividing by 2) 14. 7·004 = 7004 1000 7004 4 1000 4 1751 250 7 1 250 (Dividing by 4) 15. 2·052 = 2052 1000 2052 4 1000 4 513 250 2 13 250 (Dividing by 4) 16. 3·108 = 3108 1000 3108 4 1000 4 777 250 3 27 250 (Dividing by 4) Ans. Convert each of the following into a decimal : Q. 17. 23 10 Q. 18. 167 100 Q. 19. 1589 100 Q. 20. 5413 1000 Q. 21. 21415 1000 Q. 22. 25 4 Q. 23. 3 3 5 Q. 24. 1 4 25Read More

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