Math Class 6th Chapter 1 - PDF

Knowing our Numbers Chapter 1 1.1 Introduction Counting things is easy for us now. We can count objects in large numbers, for example, the number of students in the school, and represent them through numerals. We can...

Knowing our Numbers Chapter 1 1.1 Introduction Counting things is easy for us now. We can count objects in large numbers, for example, the number of students in the school, and represent them through numerals. We can also communicate large numbers using suitable number names. It is not as if we always knew how to convey large quantities in conversation or through symbols. Many thousands years ago, people knew only small numbers. Gradually, they learnt how to handle larger numbers. They also learnt how to express large numbers in symbols. All this came through collective efforts of human beings. Their path was not easy, they struggled all along the way. In fact, the development of whole of Mathematics can be understood this way. As human beings progressed, there was greater need for development of Mathematics and as a result Mathematics grew further and faster. We use numbers and know many things about them. Numbers help us count concrete objects. They help us to say which collection of objects is bigger and arrange them in order e.g., first, second, etc. Numbers are used in many different contexts and in many ways. Think about various situations where we use numbers. List five distinct situations in which numbers are used. We enjoyed working with numbers in our previous classes. We have added, subtracted, multiplied and divided them. We also looked for patterns in number sequences and done many other interesting things with numbers. In this chapter, we shall move forward on such interesting things with a bit of review and revision as well. Rationalised 2023-24 MATHEMATICS 1.2 Comparing Numbers As we have done quite a lot of this earlier, let us see if we remember which is the greatest among these : (i) 92, 392, 4456, 89742 (ii) 1902, 1920, 9201, 9021, 9210 So, we know the answers. Discuss with your friends, how you find the number that is the greatest. Can you instantly find the greatest and the smallest numbers in each row? 1. 382, 4972, 18, 59785, 750. Ans. 59785 is the greatest and 18 is the smallest. 2. 1473, 89423, 100, 5000, 310. Ans. ____________________ 3. 1834, 75284, 111, 2333, 450. Ans. ____________________ 4. 2853, 7691, 9999, 12002, 124. Ans. ____________________ Was that easy? Why was it easy? We just looked at the number of digits and found the answer. The greatest number has the most thousands and the smallest is only in hundreds or in tens. Make five more problems of this kind and give to your friends to solve. Now, how do we compare 4875 and 3542? This is also not very difficult.These two numbers have the same number of digits. They are both in thousands. But the digit at the thousands place in 4875 is greater than that in 3542. Therefore, 4875 is greater than 3542. Next tell which is greater, 4875 or 4542? Here too the numbers have the Find the greatest and the smallest same number of digits. Further, the digits numbers. at the thousands place are same in both. (a) 4536, 4892, 4370, 4452. What do we do then? We move to the (b) 15623, 15073, 15189, 15800. next digit, that is to the digit at the (c) 25286, 25245, 25270, 25210. hundreds place. The digit at the hundreds (d) 6895, 23787, 24569, 24659. place is greater in 4875 than in 4542. Therefore, 4875 is greater than 4542. 2 Rationalised 2023-24  K NOWING OUR N UMBERS If the digits at hundreds place are also same in the two numbers, then what do we do? Compare 4875 and 4889 ; Also compare 4875 and 4879. 1.2.1 How many numbers can you make? Suppose, we have four digits 7, 8, 3, 5. Using these digits we want to make different 4-digit numbers in such a way that no digit is repeated in them. Thus, 7835 is allowed, but 7735 is not. Make as many 4-digit numbers as you can. Which is the greatest number you can get? Which is the smallest number? The greatest number is 8753 and the smallest is 3578. Think about the arrangement of the digits in both. Can you say how the largest number is formed? Write down your procedure. 1. Use the given digits without repetition and make the greatest and smallest 4-digit numbers. (a) 2, 8, 7, 4 (b) 9, 7, 4, 1 (c) 4, 7, 5, 0 (d) 1, 7, 6, 2 (e) 5, 4, 0, 3 (Hint : 0754 is a 3-digit number.) 2. Now make the greatest and the smallest 4-digit numbers by using any one digit twice. (a) 3, 8, 7 (b) 9, 0, 5 (c) 0, 4, 9 (d) 8, 5, 1 (Hint : Think in each case which digit will you use twice.) 3. Make the greatest and the smallest 4-digit numbers using any four different digits with conditions as given. (a) Digit 7 is always at Greatest 9 8 6 7 ones place Smallest 1 0 2 7 (Note, the number cannot begin with the digit 0. Why?) (b) Digit 4 is always Greatest 4 at tens place Smallest 4 (c) Digit 9 is always at Greatest 9 hundreds place Smallest 9 (d) Digit 1 is always at Greatest 1 thousands place 3 Smallest 1 Rationalised 2023-24 MATHEMATICS 4. Take two digits, say 2 and 3. Make 4-digit numbers using both the digits equal number of times. Which is the greatest number? Which is the smallest number? How many different numbers can you make in all? Stand in proper order 1. Who is the tallest? 2. Who is the shortest? (a) Can you arrange them in the increasing order of their heights? (b) Can you arrange them in the decreasing order of their heights? Ramhari Dolly Mohan Shashi (160 cm) (154 cm) (158 cm) (159 cm) Which to buy? Sohan and Rita went to buy an almirah. There were many almirahs available with their price tags. ` 2635 ` 1897 ` 2854 ` 1788 ` 3975 (a) Can you arrange their prices in increasing Think of five more situations order? where you compare three or (b) Can you arrange their prices in more quantities. decreasing order? Ascending order Ascending order means arrangement from the smallest to the greatest. 4 Descending order Descending order means arrangement from the greatest to the smallest. Rationalised 2023-24  K NOWING OUR N UMBERS 1. Arrange the following numbers in ascending order : (a) 847, 9754, 8320, 571 (b) 9801, 25751, 36501, 38802 2. Arrange the following numbers in descending order : (a) 5000, 7500, 85400, 7861 (b) 1971, 45321, 88715, 92547 Make ten such examples of ascending/descending order and solve them. 1.2.2 Shifting digits Have you thought what fun it would be if the digits in a number could shift (move) from one place to the other? Think about what would happen to 182. It could become as large as 821 and as small as 128. Try this with 391 as well. Now think about this. Take any 3-digit number and exchange the digit at the hundreds place with the digit at the ones place. (a) Is the new number greater than the former one? (b) Is the new number smaller than the former number? Write the numbers formed in both ascending and descending order. Before 7 9 5 Exchanging the 1st and the 3rd tiles. After 5 9 7 If you exchange the 1st and the 3rd tiles (i.e. digits), in which case does the number become greater? In which case does it become smaller? Try this with a 4-digit number. 1.2.3 Introducing 10,000 We know that beyond 99 there is no 2-digit number. 99 is the greatest 2-digit number. Similarly, the greatest 3-digit number is 999 and the greatest 4-digit number is 9999. What shall we get if we add 1 to 9999? Look at the pattern : 9 + 1 = 10 = 10 × 1 99 + 1 = 100 = 10 × 10 999 + 1 = 1000 = 10 × 100 We observe that Greatest single digit number + 1 = smallest 2-digit number Greatest 2-digit number + 1 = smallest 3-digit number Greatest 3-digit number + 1 = smallest 4-digit number 5 Rationalised 2023-24 MATHEMATICS We should then expect that on adding 1 to the greatest 4-digit number, we would get the smallest 5-digit number, that is 9999 + 1 = 10000. The new number which comes next to 9999 is 10000. It is called ten thousand. Further, 10000 = 10 × 1000. 1.2.4 Revisiting place value You have done this quite earlier, and you will certainly remember the expansion of a 2-digit number like 78 as 78 = 70 + 8 = 7 × 10 + 8 Similarly, you will remember the expansion of a 3-digit number like 278 as 278 = 200 + 70 + 8 = 2 × 100 + 7 × 10 + 8 We say, here, 8 is at ones place, 7 is at tens place and 2 at hundreds place. Later on we extended this idea to 4-digit numbers. For example, the expansion of 5278 is 5278 = 5000 + 200 + 70 + 8 = 5 × 1000 + 2 × 100 + 7 × 10 + 8 Here, 8 is at ones place, 7 is at tens place, 2 is at hundreds place and 5 is at thousands place. With the number 10000 known to us, we may extend the idea further. We may write 5-digit numbers like 45278 = 4 × 10000 + 5 × 1000 + 2 × 100 + 7 × 10 + 8 We say that here 8 is at ones place, 7 at tens place, 2 at hundreds place, 5 at thousands place and 4 at ten thousands place. The number is read as forty five thousand, two hundred seventy eight. Can you now write the smallest and the greatest 5-digit numbers? Read and expand the numbers wherever there are blanks. Number Number Name Expansion 20000 twenty thousand 2 × 10000 26000 twenty six thousand 2 × 10000 + 6 × 1000 38400 thirty eight thousand 3 × 10000 + 8 × 1000 four hundred + 4 × 100 65740 sixty five thousand 6 × 10000 + 5 × 1000 seven hundred forty + 7 × 100 + 4 × 10 6 Rationalised 2023-24  K NOWING OUR N UMBERS 89324 eighty nine thousand 8 × 10000 + 9 × 1000 three hundred twenty four + 3 × 100 + 2 × 10 + 4 × 1 50000 _______________ _______________ 41000 _______________ _______________ 47300 _______________ _______________ 57630 _______________ _______________ 29485 _______________ _______________ 29085 _______________ _______________ 20085 _______________ _______________ 20005 _______________ _______________ Write five more 5-digit numbers, read them and expand them. 1.2.5 Introducing 1,00,000 Which is the greatest 5-digit number? Adding 1 to the greatest 5-digit number, should give the smallest 6-digit number : 99,999 + 1 = 1,00,000 This number is named one lakh. One lakh comes next to 99,999. 10 × 10,000 = 1,00,000 We may now write 6-digit numbers in the expanded form as 2,46,853 = 2 × 1,00,000 + 4 × 10,000 + 6 × 1,000 + 8 × 100 + 5 × 10 +3 × 1 This number has 3 at ones place, 5 at tens place, 8 at hundreds place, 6 at thousands place, 4 at ten thousands place and 2 at lakh place. Its number name is two lakh forty six thousand eight hundred fifty three. Read and expand the numbers wherever there are blanks. Number Number Name Expansion 3,00,000 three lakh 3 × 1,00,000 3,50,000 three lakh fifty thousand 3 × 1,00,000 + 5 × 10,000 3,53,500 three lakh fifty three 3 × 1,00,000 + 5 × 10,000 thousand five hundred + 3 × 1000 + 5 × 100 4,57,928 _______________ _______________ 4,07,928 _______________ _______________ 4,00,829 _______________ _______________ 4,00,029 _______________ _______________ 7 Rationalised 2023-24 MATHEMATICS 1.2.6 Larger numbers If we add one more to the greatest 6-digit number we get the smallest 7-digit number. It is called ten lakh. Write down the greatest 6-digit number and the smallest 7-digit number. Write the greatest 7-digit number and the smallest 8-digit number. The smallest 8-digit number is called one crore. Complete the pattern : Remember 9+1 = 10 1 hundred = 10 tens 99 + 1 = 100 1 thousand = 10 hundreds 999 + 1 = _______ = 100 tens 9,999 + 1 = _______ 1 lakh = 100 thousands 99,999 + 1 = _______ = 1000 hundreds 1 crore = 100 lakhs 9,99,999 + 1 = _______ = 10,000 thousands 99,99,999 + 1 = 1,00,00,000 We come across large numbers in 1. What is 10 – 1 =? many different situations. 2. What is 100 – 1 =? For example, while the number of 3. What is 10,000 – 1 =? 4. What is 1,00,000 – 1 =? children in your class would be a 5. What is 1,00,00,000 – 1 =? 2-digit number, the number of (Hint : Use the said pattern.) children in your school would be a 3 or 4-digit number. The number of people in the nearby town would be much larger. Is it a 5 or 6 or 7-digit number? Do you know the number of people in your state? How many digits would that number have? What would be the number of grains in a sack full of wheat? A 5-digit number, a 6-digit number or more? 1. Give five examples where the number of things counted would be more than 6-digit number. 2. Starting from the greatest 6-digit number, write the previous five numbers in descending order. 3. Starting from the smallest 8-digit number, write the next five numbers in ascending order and read them. 8 Rationalised 2023-24  K NOWING OUR N UMBERS 1.2.7 An aid in reading and writing large numbers Try reading the following numbers : (a) 279453 (b) 5035472 (c) 152700375 (d) 40350894 Was it difficult? Did you find it difficult to keep track? Sometimes it helps to use indicators to read and write large numbers. Shagufta uses indicators which help her to read and write large numbers. Her indicators are also useful in writing the expansion of numbers. For example, she identifies the digits in ones place, tens place and hundreds place in 257 by writing them under the tables O, T and H as H T O Expansion 2 5 7 2 × 100 + 5 × 10 + 7 × 1 Similarly, for 2902, Th H T O Expansion 2 9 0 2 2 × 1000 + 9 × 100 + 0 × 10 + 2 × 1 One can extend this idea to numbers upto lakh as seen in the following table. (Let us call them placement boxes). Fill the entries in the blanks left. Number TLakh Lakh TTh Th H T O Number Name Expansion 7,34,543 — 7 3 4 5 4 3 Seven lakh thirty ----------------- four thousand five hundred forty three 32,75,829 3 2 7 5 8 2 9 3 × 10,00,000 --------------------- + 2 × 1,00,000 + 7 × 10,000 + 5 × 1000 + 8 × 100 + 2 × 10 + 9 Similarly, we may include numbers upto crore as shown below : Number TCr Cr TLakh Lakh TTh Th H T O Number Name 2,57,34,543 — 2 5 7 3 4 5 4 3................................... 65,32,75,829 6 5 3 2 7 5 8 2 9 Sixty five crore thirty two lakh seventy five thousand eight hundred twenty nine 9 You can make other formats of tables for writing the numbers in expanded form. Rationalised 2023-24 MATHEMATICS Use of commas You must have noticed that in writing large numbers in the While writing sections above, we have often used commas. Commas help us number names, in reading and writing large numbers. In our Indian System we do not use of Numeration we use ones, tens, hundreds, thousands and commas. then lakhs and crores. Commas are used to mark thousands, lakhs and crores. The first comma comes after hundreds place (three digits from the right) and marks thousands. The second comma comes two digits later (five digits from the right). It comes after ten thousands place and marks lakh. The third comma comes after another two digits (seven digits from the right). It comes after ten lakh place and marks crore. For example, 5, 08, 01, 592 3, 32, 40, 781 7, 27, 05, 062 Try reading the numbers given above. Write five more numbers in this form and read them. International System of Numeration In the International System of Numeration, as it is being used we have ones, tens, hundreds, thousands and then millions. One million is a thousand thousands. Commas are used to mark thousands and millions. It comes after every three digits from the right. The first comma marks thousands and the next comma marks millions. For example, the number 50,801,592 is read in the International System as fifty million eight hundred one thousand five hundred ninety two. In the Indian System, it is five crore eight lakh one thousand five hundred ninety two. How many lakhs make a million? How many millions make a crore? Take three large numbers. Express them in both Indian and International Numeration systems. Interesting fact : To express numbers larger than a million, a billion is used in the International System of Numeration: 1 billion = 1000 million. 10 Rationalised 2023-24  K NOWING OUR N UMBERS How much was the increase in population Do you know? during 1991-2001? Try to find out. India’s population increased by about Do you know what is India’s population 27 million during 1921-1931; today? Try to find this too. 37 million during 1931-1941; 44 million during 1941-1951; 78 million during 1951-1961! 1. Read these numbers. Write them using placement boxes and then write their expanded forms. (i) 475320 (ii) 9847215 (iii) 97645310 (iv) 30458094 (a) Which is the smallest number? (b) Which is the greatest number? (c) Arrange these numbers in ascending and descending orders. 2. Read these numbers. (i) 527864 (ii) 95432 (iii) 18950049 (iv) 70002509 (a) Write these numbers using placement boxes and then using commas in Indian as well as International System of Numeration.. (b) Arrange these in ascending and descending order. 3. Take three more groups of large numbers and do the exercise given above. Can you help me write the numeral? To write the numeral for a number you can follow the boxes again. (a) Forty two lakh seventy thousand eight. (b) Two crore ninety lakh fifty five thousand eight hundred. (c) Seven crore sixty thousand fifty five. 1. You have the following digits 4, 5, 6, 0, 7 and 8. Using them, make five numbers each with 6 digits. (a) Put commas for easy reading. (b) Arrange them in ascending and descending order. 2. Take the digits 4, 5, 6, 7, 8 and 9. Make any three numbers each with 8 digits. Put commas for easy reading. 3. From the digits 3, 0 and 4, make five numbers each with 6 digits. Use commas. 11 Rationalised 2023-24 MATHEMATICS EXERCISE 1.1 1. Fill in the blanks: (a) 1 lakh = _______ ten thousand. (b) 1 million = _______ hundred thousand. (c) 1 crore = _______ ten lakh. (d) 1 crore = _______ million. (e) 1 million = _______ lakh. 2. Place commas correctly and write the numerals: (a) Seventy three lakh seventy five thousand three hundred seven. (b) Nine crore five lakh forty one. (c) Seven crore fifty two lakh twenty one thousand three hundred two. (d) Fifty eight million four hundred twenty three thousand two hundred two. (e) Twenty three lakh thirty thousand ten. 3. Insert commas suitably and write the names according to Indian System of Numeration : (a) 87595762 (b) 8546283 (c) 99900046 (d) 98432701 4. Insert commas suitably and write the names according to International System of Numeration : (a) 78921092 (b) 7452283 (c) 99985102 (d) 48049831 1.3 Large Numbers in Practice In earlier classes, we have learnt that we use centimetre (cm) as a unit of length. For measuring the length of a pencil, the width of a book or notebooks etc., we use centimetres. Our ruler has marks on each centimetre. For measuring the thickness of a pencil, however, we find centimetre too big. We use millimetre (mm) to show the thickness of a pencil. (a) 10 millimetres = 1 centimetre To measure the length of the classroom or 1. How many the school building, we shall find centimetres make a kilometre? centimetre too small. We use metre for the 2. Name five large cities purpose. in India. Find their (b) 1 metre = 100 centimetres population. Also, find = 1000 millimetres the distance in Even metre is too small, when we have to kilometres between each pair of these cities. state distances between cities, say, Delhi and Mumbai, or Chennai and Kolkata. For 12 this we need kilometres (km). Rationalised 2023-24  K NOWING OUR N UMBERS (c) 1 kilometre = 1000 metres How many millimetres make 1 kilometre? Since 1 m = 1000 mm 1 km = 1000 m = 1000 × 1000 mm = 10,00,000 mm We go to the market to buy rice or wheat; we buy it in kilograms (kg). But items like ginger or chillies which we do not need in large quantities, we buy in grams (g). We know 1 kilogram = 1000 grams. Have you noticed the weight of the medicine tablets given to the sick? It is very small. It is in milligrams 1. How many milligrams (mg). make one 1 gram = 1000 milligrams. kilogram? What is the capacity of a bucket for holding water? It 2. A box contains is usually 20 litres (l). Capacity is given in litres. But 2,00,000 sometimes we need a smaller unit, the millilitres. medicine tablets A bottle of hair oil, a cleaning liquid or a soft drink each weighing have labels which give the quantity of liquid inside in 20 mg. What is millilitres (ml). the total weight 1 litre = 1000 millilitres. of all the Note that in all these units we have some words tablets in the common like kilo, milli and centi. You should remember box in grams that among these kilo is the greatest and milli is the and in smallest; kilo shows 1000 times greater, milli shows kilograms? 1000 times smaller, i.e. 1 kilogram = 1000 grams, 1 gram = 1000 milligrams. Similarly, centi shows 100 times smaller, i.e. 1 metre = 100 centimetres. 1. A bus started its journey and reached different places with a speed of 60 km/hour. The journey is shown on page 14. (i) Find the total distance covered by the bus from A to D. (ii) Find the total distance covered by the bus from D to G. (iii) Find the total distance covered by the bus, if it starts from A and returns back to A. (iv) Can you find the difference of distances from C to D and D to E? 13 Rationalised 2023-24 MATHEMATICS (v) Find out the time taken by the bus to reach (a) A to B (b) C to D (c) E to G (d) Total journey 2. Raman’s shop Things Price Apples ` 40 per kg Oranges ` 30 per kg Combs ` 3 for one Tooth brushes ` 10 for one Pencils ` 1 for one Note books ` 6 for one Soap cakes ` 8 for one The sales during the last year Apples 2457 kg Oranges 3004 kg Combs 22760 Tooth brushes 25367 Pencils 38530 Note books 40002 Soap cakes 20005 (a) Can you find the total weight of apples and oranges Raman sold last year? Weight of apples = __________ kg Weight of oranges = _________ kg Therefore, total weight = _____ kg + _____ kg = _____ kg Answer – The total weight of oranges and apples = _________ kg. (b) Can you find the total money Raman got by selling apples? (c) Can you find the total money Raman got by selling apples and oranges together? (d) Make a table showing how much money Raman received from selling each item. Arrange the entries of amount of money received in descending order. Find the item which brought him the highest amount. How much is this amount? 14 Rationalised 2023-24  K NOWING OUR N UMBERS We have done a lot of problems that have addition, subtraction, multiplication and division. We will try solving some more here. Before starting, look at these examples and follow the methods used. Example 1 : Population of Sundarnagar was 2,35,471 in the year 1991. In the year 2001 it was found to be increased by 72,958. What was the population of the city in 2001? Solution : Population of the city in 2001 = Population of the city in 1991 + Increase in population = 2,35,471 + 72,958 Now, 235471 + 72958 308429 Salma added them by writing 235471 as 200000 + 35000 + 471 and 72958 as 72000 + 958. She got the addition as 200000 + 107000 + 1429 = 308429. Mary added it as 200000 + 35000 + 400 + 71 + 72000 + 900 + 58 = 308429 Answer : Population of the city in 2001 was 3,08,429. All three methods are correct. Example 2 : In one state, the number of bicycles sold in the year 2002-2003 was 7,43,000. In the year 2003-2004, the number of bicycles sold was 8,00,100. In which year were more bicycles sold? and how many more? Solution : Clearly, 8,00,100 is more than 7,43,000. So, in that state, more bicycles were sold in the year 2003-2004 than in 2002-2003. Now, 800100 Check the answer by adding – 743000 743000 + 57100 057100 800100 (the answer is right) Can you think of alternative ways of solving this problem? Answer : 57,100 more bicycles were sold in the year 2003-2004. Example 3 : The town newspaper is published every day. One copy has 12 pages. Everyday 11,980 copies are printed. How many total pages are 15 printed everyday? Rationalised 2023-24 MATHEMATICS Solution : Each copy has 12 pages. Hence, 11,980 copies will have 12 × 11,980 pages. What would this number be? More than 1,00,000 or lesser. Try to estimate. Now, 11980 × 12 23960 + 119800 143760 Answer:Everyday 1,43,760 pages are printed. Example 4 : The number of sheets of paper available for making notebooks is 75,000. Each sheet makes 8 pages of a notebook. Each notebook contains 200 pages. How many notebooks can be made from the paper available? Solution : Each sheet makes 8 pages. Hence, 75,000 sheets make 8 × 75,000 pages, Now, 75000 ×8 600000 Thus, 6,00,000 pages are available for making notebooks. Now, 200 pages make 1 notebook. Hence, 6,00,000 pages make 6,00,000 ÷ 200 notebooks. 3000 Now, 200 )600000 – 600 0000 The answer is 3,000 notebooks. EXERCISE 1.2 1. A book exhibition was held for four days in a school. The number of tickets sold at the counter on the first, second, third and final day was respectively 1094, 1812, 2050 and 2751. Find the total number of tickets sold on all the four days. 2. Shekhar is a famous cricket player. He has so far scored 6980 runs in test matches. He wishes to complete 10,000 runs. How many more runs does he need? 3. In an election, the successful candidate registered 5,77,500 votes and his nearest rival secured 3,48,700 votes. By what margin did the successful candidate win the 16 election? Rationalised 2023-24  K NOWING OUR N UMBERS 4. Kirti bookstore sold books worth ` 2,85,891 in the first week of June and books worth ` 4,00,768 in the second week of the month. How much was the sale for the two weeks together? In which week was the sale greater and by how much? 5. Find the difference between the greatest and the least 5-digit number that can be written using the digits 6, 2, 7, 4, 3 each only once. 6. A machine, on an average, manufactures 2,825 screws a day. How many screws did it produce in the month of January 2006? 7. A merchant had ` 78,592 with her. She placed an order for purchasing 40 radio sets at ` 1200 each. How much money will remain with her after the purchase? 8. A student multiplied 7236 by 65 instead of multiplying by 56. By how much was his answer greater than the correct answer? (Hint: Do you need to do both the multiplications?) 9. To stitch a shirt, 2 m 15 cm cloth is needed. Out of 40 m cloth, how many shirts can be stitched and how much cloth will remain? (Hint: convert data in cm.) 10. Medicine is packed in boxes, each weighing 4 kg 500g. How many such boxes can be loaded in a van which cannot carry beyond 800 kg? 11. The distance between the school and a student’s house is 1 km 875 m. Everyday she walks both ways. Find the total distance covered by her in six days. 12. A vessel has 4 litres and 500 ml of curd. In how many glasses, each of 25 ml capacity, can it be filled? What have we discussed? 1. Given two numbers, one with more digits is the greater number. If the number of digits in two given numbers is the same, that number is larger, which has a greater leftmost digit. If this digit also happens to be the same, we look at the next digit and so on. 2. In forming numbers from given digits, we should be careful to see if the conditions under which the numbers are to be formed are satisfied. Thus, to form the greatest four digit number from 7, 8, 3, 5 without repeating a single digit, we need to use all four digits, the greatest number can have only 8 as the leftmost digit. 3. The smallest four digit number is 1000 (one thousand). It follows the largest three digit number 999. Similarly, the smallest five digit number is 10,000. It is ten thousand and follows the largest four digit number 9999. Further, the smallest six digit number is 100,000. It is one lakh and follows the largest five digit number 99,999. This carries on for higher digit numbers in a similar manner. 4. Use of commas helps in reading and writing large numbers. In the Indian system of numeration we have commas after 3 digits starting from the right and thereafter every 2 digits. The commas after 3, 5 and 7 digits separate thousand, lakh and crore 17 Rationalised 2023-24 MATHEMATICS respectively. In the International system of numeration commas are placed after every 3 digits starting from the right. The commas after 3 and 6 digits separate thousand and million respectively. 5. Large numbers are needed in many places in daily life. For example, for giving number of students in a school, number of people in a village or town, money paid or received in large transactions (paying and selling), in measuring large distances say betwen various cities in a country or in the world and so on. 6. Remember kilo shows 1000 times larger, Centi shows 100 times smaller and milli shows 1000 times smaller, thus, 1 kilometre = 1000 metres, 1 metre = 100 centimetres or 1000 millimetres etc. 18 Rationalised 2023-24  Whole Numbers Chapter 2 2.1 Introduction As we know, we use 1, 2, 3, 4,... when we begin to count. They come naturally when we start counting. Hence, mathematicians call the counting numbers as Natural numbers. Predecessor and successor Given any natural number, you can add 1 to that number and get the next number i.e. you 1. Write the predecessor get its successor. and successor of The successor of 16 is 16 + 1 = 17, 19; 1997; 12000; that of 19 is 19 +1 = 20 and so on. 49; 100000. The number 16 comes before 17, we 2. Is there any natural say that the predecessor of 17 is 17–1=16, number that has no the predecessor of 20 is 20 – 1 = 19, and predecessor? so on. 3. Is there any natural The number 3 has a predecessor and a number which has no successor. What about 2? The successor is successor? Is there a last natural number? 3 and the predecessor is 1. Does 1 have both a successor and a predecessor? We can count the number of children in our school; we can also count the number of people in a city; we can count the number of people in India. The number of people in the whole world can also be counted. We may not be able to count the number of stars in the sky or the number of hair on our heads but if we are able, there would be a number for them also. We can then add one more to such a number and Rationalised 2023-24 MATHEMATICS get a larger number. In that case we can even write the number of hair on two heads taken together. It is now perhaps obvious that there is no largest number. Apart from these questions shared above, there are many others that can come to our mind when we work with natural numbers. You can think of a few such questions and discuss them with your friends. You may not clearly know the answers to many of them ! 2.2 Whole Numbers We have seen that the number 1 has no predecessor in natural numbers. To the collection of natural numbers we add zero as the predecessor for 1. The natural numbers along with zero form the collection of whole numbers. In your previous classes you have learnt to perform all the basic operations like addition, 1. Are all natural numbers subtraction, multiplication and division on also whole numbers? 2. Are all whole numbers numbers. You also know how to apply them to also natural numbers? problems. Let us try them on a number line. 3. Which is the greatest Before we proceed, let us find out what a whole number? number line is! 2.3 The Number Line Draw a line. Mark a point on it. Label it 0. Mark a second point to the right of 0. Label it 1. The distance between these points labelled as 0 and 1 is called unit distance. On this line, mark a point to the right of 1 and at unit distance from 1 and label it 2. In this way go on labelling points at unit distances as 3, 4, 5,... on the line. You can go to any whole number on the right in this manner. This is a number line for the whole numbers. What is the distance between the points 2 and 4? Certainly, it is 2 units. Can you tell the distance between the points 2 and 6, between 2 and 7? 20 On the number line you will see that the number 7 is on the right of 4. This number 7 is greater than 4, i.e. 7 > 4. The number 8 lies on the right of 6 Rationalised 2023-24  W HOLE N UMBERS and 8 > 6. These observations help us to say that, out of any two whole numbers, the number on the right of the other number is the greater number. We can also say that whole number on left is the smaller number. For example, 4 < 9; 4 is on the left of 9. Similarly, 12 > 5; 12 is to the right of 5. What can you say about 10 and 20? Mark 30, 12, 18 on the number line. Which number is at the farthest left? Can you say from 1005 and 9756, which number would be on the right relative to the other number. Place the successor of 12 and the predecessor of 7 on the number line. Addition on the number line Addition of whole numbers can be shown on the number line. Let us see the addition of 3 and 4. Start from 3. Since we add 4 to this number so we make 4 jumps to the right; from 3 to 4, 4 to 5, 5 to 6 and 6 Find 4 + 5; to 7 as shown above. The tip of the last arrow in the fourth 2 + 6; 3 + 5 jump is at 7. and 1+6 using the The sum of 3 and 4 is 7, i.e. 3 + 4 = 7. number line. Subtraction on the number line The subtraction of two whole numbers can also be shown on the number line. Let us find 7 – 5. Start from 7. Since 5 is being subtracted, so move towards left with 1 jump of 1 unit. Make 5 such jumps. We Find 8 – 3; reach the point 2. We get 7 – 5 = 2. 6 – 2; 9 – 6 Multiplication on the number line using the number line. We now see the multiplication of whole numbers on the number line. Let us find 4 × 3. 21 Rationalised 2023-24 MATHEMATICS Start from 0, move 3 units at a time to the right, make such 4 moves. Where do you reach? You will reach 12. Find 2 × 6; So, we say, 3 × 4 = 12. 3 × 3; 4 × 2 using the EXERCISE 2.1 number line. 1. Write the next three natural numbers after 10999. 2. Write the three whole numbers occurring just before 10001. 3. Which is the smallest whole number? 4. How many whole numbers are there between 32 and 53? 5. Write the successor of : (a) 2440701 (b) 100199 (c) 1099999 (d) 2345670 6. Write the predecessor of : (a) 94 (b) 10000 (c) 208090 (d) 7654321 7. In each of the following pairs of numbers, state which whole number is on the left of the other number on the number line. Also write them with the appropriate sign (>, and < signs. 0 –1 – 100 –101 – 50 – 70 50 –51 – 53 –5 –7 1 118  Let us once again observe the integers which are represented on the Fract I NTEGERS number line. Intege Integ We know that 7 > 4 and from the number line shown above, we observe that 7 is to the right of 4 (Fig 6.3). Similarly, 4 > 0 and 4 is to the right of 0. Now, since 0 is to the right of –3 so, 0 > – 3. Again, – 3 is to the right of – 8 so, – 3 > – 8. Thus, we see that on a number line the number increases as we move to the right and decreases as we move to the left. Therefore, – 3 < – 2, – 2 < – 1, – 1 < 0, 0 < 1, 1 < 2, 2 < 3 so on. Hence, the collection of integers can be written as..., –5, –4, – 3, – 2, – 1, 0, 1, 2, 3, 4, 5... Compare the following pairs of numbers using > or , < or = sign. (a) (– 3) + (– 6) ______ (– 3) – (– 6) (b) (– 21) – (– 10) _____ (– 31) + (– 11) (c) 45 – (– 11) ______ 57 + (– 4) (d) (– 25) – (– 42) _____ (– 42) – (– 25) 3. Fill in the blanks. (a) (– 8) + _____ = 0 (b) 13 + _____ = 0 (c) 12 + (– 12) = ____ (d) (– 4) + ____ = – 12 (e) ____ – 15 = – 10 4. Find (a) (– 7) – 8 – (– 25) (b) (– 13) + 32 – 8 – 1 (c) (– 7) + (– 8) + (– 90) (d) 50 – (– 40) – (– 2) What have we discussed? 1. We have seen that there are times when we need to use numbers with a negative sign. This is when we want to go below zero on the number line. These are called negative numbers. Some examples of their use can be in temperature scale, water level in lake or river, level of oil in tank etc. They are also used to denote debit account or outstanding dues. 131 MATHEMATICS 2. The collection of numbers..., – 4, – 3, – 2, – 1, 0, 1, 2, 3, 4,... is called integers. So, – 1, – 2, – 3, – 4,... called negative numbers are negative integers and 1, 2, 3, 4,... called positive numbers are the positive integers. 3. We have also seen how one more than given number gives a successor and one less than given number gives predecessor. 4. We observe that (a) When we have the same sign, add and put the same sign. (i) When two positive integers are added, we get a positive integer [e.g. (+ 3) + ( + 2) = + 5]. (ii) When two negative integers are added, we get a negative integer [e.g. (–2) + ( – 1) = – 3]. (b) When one positive and one negative integers are added we subtract them as whole numbers by considering the numbers without their sign and then put the sign of the bigger number with the subtraction obtained. The bigger integer is decided by ignoring the signs of the integers [e.g. (+4) + (–3) = + 1 and (–4) + ( + 3) = – 1]. (c) The subtraction of an integer is the same as the addition of its additive inverse. 5. We have shown how addition and subtraction of integers can also be shown on a number line. 132  Fractions Chapter 7 7.1 Introduction Subhash had learnt about fractions in Classes IV and V, so whenever possible he would try to use fractions. One occasion was when he forgot his lunch at home. His friend Farida invited him to share her lunch. She had five pooris in her lunch box. So, Subhash and Farida took two pooris each. Then Farida made two equal halves of the fifth poori and gave one-half to Subhash 2 pooris + half-poori–Subhash and took the other half herself. Thus, 2 pooris + half-poori–Farida both Subhash and Farida had 2 full pooris and one-half poori. Where do you come across situations with fractions in your life? 1 Subhash knew that one-half is written as. While Fig 7.1 2 eating he further divided his half poori into two equal parts and asked Farida what fraction of the whole poori was that piece? (Fig 7.1) Without answering, Farida also divided her portion of the half puri into two equal parts and kept them beside Subhash’s shares. She said that these four equal parts together make Fig 7.2 2015-16 (11-11-2014) MATHEMATICS one whole (Fig 7.2). So, each equal part is one-fourth of one whole poori and 4 4 parts together will be or 1 whole poori. 4 When they ate, they discussed what they had learnt earlier. Three 3 parts out of 4 equal parts is. 4 3 Similarly, is obtained when we Fig 7.3 Fig 7.4 7 divide a whole into seven equal parts 1 and take three parts (Fig 7.3). For , we divide a whole into eight equal parts 8 and take one part out of it (Fig 7.4). Farida said that we have learnt that a fraction is a number representing part of a whole. The whole may be a single object or a group of objects. Subhash observed that the parts have to be equal. 7.2 A Fraction Let us recapitulate the discussion. A fraction means a part of a group or of a region. 5 is a fraction. We read it as “five-twelfths”. 12 What does “12” stand for? It is the number of equal parts into which the whole has been divided. What does “5” stand for? It is the number of equal parts which have been taken out. Here 5 is called the numerator and 12 is called the denominator. 3 4 Name the numerator of and the denominator of. 7 15 2 Play this Game 3 You can play this game with your friends. Take many copies of the grid as shown here. 1 Consider any fraction, say. 2 1 Each one of you should shade of the grid. 2 134 2015-16 (11-11-2014)  EXERCISE 7.1 Fracti FRACTIONS 1. Write the fraction representing the shaded portion. Intege (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) 2. Colour the part according to the given fraction. 1 1 1 6 4 3 3 4 4 9 135 2015-16 (11-11-2014) MATHEMATICS 3. Identify the error, if any. 1 1 3 This is This is This is 2 4 4 4. What fraction of a day is 8 hours? 5. What fraction of an hour is 40 minutes? 6. Arya, Abhimanyu, and Vivek shared lunch. Arya has brought two sandwiches, one made of vegetable and one of jam. The other two boys forgot to bring their lunch. Arya agreed to share his sandwiches so that each person will have an equal share of each sandwich. (a) How can Arya divide his sandwiches so that each person has an equal share? (b) What part of a sandwich will each boy receive? 7. Kanchan dyes dresses. She had to dye 30 dresses. She has so far finished 20 dresses. What fraction of dresses has she finished? 8. Write the natural numbers from 2 to 12. What fraction of them are prime numbers? 9. Write the natural numbers from 102 to 113. What fraction of them are prime numbers? 10. What fraction of these circles have X’s in them? 11. Kristin received a CD player for her birthday. She bought 3 CDs and received 5 others as gifts. What fraction of her total CDs did she buy and what fraction did she receive as gifts? 7.3 Fraction on the Number Line You have learnt to show whole numbers like 0,1,2... on a number line. We can also show fractions on a number line. Let us draw a number line 1 and try to mark on it? 2 1 We know that is greater than 0 and less than 1, so it should lie between 2 0 and 1. 1 Since we have to show , we divide the gap between 0 and 1 into two 2 1 equal parts and show 1 part as (as shown in the Fig 7.5). 2 136 2015-16 (11-11-2014)  Fracti FRACTIONS ×1 Intege 0 1 2 Fig 7.5 1 Suppose we want to show on a number line. Into how many equal parts 3 should the length between 0 and 1 be divided? We divide the length between 1 0 and 1 into 3 equal parts and show one part as (as shown in the Fig 7.6) 3 0 ×1 1 3 Fig 7.6 2 2 Can we show on this number line? means 2 parts out of 3 parts as 3 3 shown (Fig 7.7). Fig 7.7 0 Similarly, how would you show 3 3 3 and on this number line? 1. Show on a number line. 3 5 0 3 1 0 5 10 is the point zero whereas since is 2. Show , , and on 3 3 10 10 10 10 1 whole, it can be shown by the point a number line. 1 (as shown in Fig 7.7) 3. Can you show any other fraction 3 between 0 and 1? So if we have to show on a 7 Write five more fractions that number line, then, into how many you can show and depict them equal parts should the length between on the number line. 3 4. How many fractions lie between 0 and 1 be divided? If P shows then 0 and 1? Think, discuss and 7 how many equal divisions lie between write your answer? 0 7 0 and P? Where do and lie? 7 7 137 2015-16 (11-11-2014) MATHEMATICS 7.4 Proper Fractions You have now learnt how to locate fractions on a number line. Locate the fractions 3 1 9 0 5 , , , , on separate number lines. 4 2 10 3 8 Does any one of the fractions lie beyond 1? All these fractions lie to the left of 1as they are less than 1. In fact, all the fractions we have learnt so far are less than 1. These are proper fractions. A proper fraction as Farida said (Sec. 7.1), is a number representing part of a whole. In a proper fraction the denominator shows the number of parts into which the whole is divided and the numerator shows the number of parts which have been considered. Therefore, in a proper fraction the numerator is always less than the denominator. 1. Give a proper fraction : (a) whose numerator is 5 and denominator is 7. (b) whose denominator is 9 and numerator is 5. (c) whose numerator and denominator add up to 10. How many fractions of this kind can you make? (d) whose denominator is 4 more than the numerator. (Give any five. How many more can you make?) 2. A fraction is given. How will you decide, by just looking at it, whether, the fraction is (a) less than 1? (b) equal to 1? 3. Fill up using one of these : ‘>’, ‘. Note the number of the parts taken is 8 8 given by the numerator. It is, therefore, clear that for two fractions with the same denominator, the fraction with the greater numerator is greater. Between 4 3 4 11 13 13 and , is greater. Between and , is greater and so on. 5 5 5 20 20 20 1. Which is the larger fraction? 7 8 11 13 17 12 (i) or (ii) or (iii) or 10 10 24 24 102 102 Why are these comparisons easy to make? 2. Write these in ascending and also in descending order. 1 5 3 1 11 4 3 7 1 3 13 11 7 (a) , , (b) , , , , (c) , , , , 8 8 8 5 5 5 5 5 7 7 7 7 7 7.9.2 Comparing unlike fractions Two fractions are unlike if they have different denominators. For example, 1 1 2 3 and are unlike fractions. So are and. 3 5 3 5 Unlike fractions with the same numerator : 1 1 Consider a pair of unlike fractions and , in which the numerator is the 3 5 same. 1 1 Which is greater or ? 3 5 1 1 3 5 149 2015-16 (11-11-2014) MATHEMATICS 1 1 In , we divide the whole into 3 equal parts and take one. In , we divide the 3 5 1 whole into 5 equal parts and take one. Note that in , the whole is divided into 3 1 1 a smaller number of parts than in. The equal part that we get in is, therefore, 5 3 1 larger than the equal part we get in. Since in both cases we take the same 5 1 number of parts (i.e. one), the portion of the whole showing is larger than the 3 1 1 1 portion showing , and therfore >. 5 3 5 2 2 In the same way we can say >. In this case, the situation is the same as in 3 5 the case above, except that the common numerator is 2, not 1. The whole is 2 2 divided into a large number of equal parts for than for. Therefore, each 5 3 2 2 equal part of the whole in case of is larger than that in case of. Therefore, 3 5 2 2 the portion of the whole showing is larger than the portion showing and 3 5 2 2 hence, >. 3 5 We can see from the above example that if the numerator is the same in two fractions, the fraction with the smaller denominator is greater of the two. 1 1 3 3 4 4 Thus, > , > , > and so on. 8 10 5 7 9 11 2 2 2 2 2 Let us arrange , , , , in increasing order. All these fractions are 1 13 9 5 7 unlike, but their numerator is the same. Hence, in such case, the larger the 2 denominator, the smaller is the fraction. The smallest is , as it has the 13 2 2 2 largest denominator. The next three fractions in order are , ,. The greatest 9 7 5 2 fraction is (It is with the smallest denominator). The arrangement in 1 2 2 2 2 2 increasing order, therefore, is , , , ,. 13 9 7 5 1 150 2015-16 (11-11-2014)  Fracti FRACTIONS 1. Arrange the following in ascending and descending order : (a) 1 12 , 1 , 1 1 , , 23 5 7 50 9 17 1 , 1 , 1 Intege 3 3 3 3 3 3 3 (b) , , , , , , 7 11 5 2 13 4 17 (c) Write 3 more similar examples and arrange them in ascending and descending order. 2 3 Suppose we want to compare and. Their numerators are different 3 4 and so are their denominators. We know how to compare like fractions, i.e. fractions with the same denominator. We should, therefore, try to change the denominators of the given fractions, so that they become equal. For this purpose, we can use the method of equivalent fractions which we already know. Using this method we can change the denominator of a fraction without changing its value. 2 3 Let us find equivalent fractions of both and. 3 4 2 4 6 8 10 3 6 9 12 = = = = =.... Similarly, = = = =.... 3 6 9 12 15 4 8 12 16 2 3 The equivalent fractions of and with the same denominator 12 are 3 4 8 9 and repectively. 12 12 2 8 3 9 9 8 3 2 i.e. = and =. Since, > we have, >. 3 12 4 12 12 12 4 3 4 5 Example 6 : Compare and. 5 6 Solution : The fractions are unlike fractions. Their numerators are different too. Let us write their equivalent fractions. 4 8 12 16 20 24 28 = = = = = = =........... 5 10 15 20 25 30 35 5 10 15 20 25 30 and = = = = = =........... 6 12 18 24 30 36 151 2015-16 (11-11-2014) MATHEMATICS The equivalent fractions with the same denominator are : 4 24 5 25 = and = 5 30 6 30 25 24 5 4 Since, > so, > 30 30 6 5 Note that the common denominator of the equivalent fractions is 30 which is 5 × 6. It is a common multiple of both 5 and 6. So, when we compare two unlike fractions, we first get their equivalent fractions with a denominator which is a common multiple of the denominators of both the fractions. 5 13 Example 7 : Compare and. 6 15 Solution : The fractions are unlike. We should first get their equivalent fractions with a denominator which is a common multiple of 6 and 15. 5 × 5 25 13 × 2 26 Now, = , = 6 × 5 30 15 × 2 30 26 25 13 5 Since > we have >. 30 30 15 6 Why LCM? The product of 6 and 15 is 90; obviously 90 is also a common multiple of 6 and 15. We may use 90 instead of 30; it will not be wrong. But we know that it is easier and more convenient to work with smaller numbers. So the common multiple that we take is as small as possible. This is why the LCM of the denominators of the fractions is preferred as the common denominator. EXERCISE 7.4 1. Write shaded portion as fraction. Arrange them in ascending and descending order using correct sign ‘’ between the fractions: (a) (b) 152 2015-16 (11-11-2014)  2 4 8 Fracti FRACTIONS and 6 on the number line. Put appropriate signs between Intege (c) Show , , 6 6 6 6 the fractions given. 5 2 3 0 11 66 , 88 55 , 0, 6 6 6 6 66 66 66 66 2. Compare the fractions and put an appropriate sign. 3 5 1 1 4 5 3 3 (a) (b) (c) (d) 6 6 7 4 5 5 5 7 3. Make five more such pairs and put appropriate signs. 4. Look at the figures and write ‘’, ‘=’ between the given pairs of fractions. 1 1 3 2 2 2 6 3 5 5 (a) (b) (c) (d) (e) 6 3 4 6 3 4 6 3 6 5 Make five more such problems and solve them with your friends. 5. How quickly can you do this? Fill appropriate sign. ( ‘’) 11 11 22 33 33 2 (a) (b) (c) 22 55 44 66 55 3 33 22 33 66 77 3 (d) (e) (f) 44 88 55 55 99 99 153 2015-16 (11-11-2014) MATHEMATICS 11 22 66 44 3 27 (g) (h) (i) 44 88 10 10 55 4 88 66 44 55 115 5 ( j) (k) 10 10 55 77 2211 6. The following fractions represent just three different numbers. Separate them into three groups of equivalent fractions, by changing each one to its simplest form. 2 3 8 16 10 15 (a) (b) (c) (d) (e) (f ) 12 15 50 100 60 75 12 16 12 12 3 4 (g) (h) (i) ( j) (k) (l) 60 96 75 72 18 25 7. Find answers to the following. Write and indicate how you solved them. 5 4 9 5 (a) Is equal to ? (b) Is equal to ? 9 5 16 9 4 16 1 4 (c) Is equal to ? (d) Is equal to ? 5 20 15 30 2 8. Ila read 25 pages of a book containing 100 pages. Lalita read of the same 5 book. Who read less? 3 3 9. Rafiq exercised for of an hour, while Rohit exercised for of an hour. 6 4 Who exercised for a longer time? 10. In a class A of 25 students, 20 passed in first class; in another class B of 30 students, 24 passed in first class. In which class was a greater fraction of students getting first class? 7.10 Addition and Subtraction of Fractions So far in our study we have learnt about natural numbers, whole numbers and then integers. In the present chapter, we are learning about fractions, a different type of numbers. Whenever we come across new type of numbers, we want to know how to operate with them. Can we combine and add them? If so, how? Can we take away some number from another? i.e., can we subtract one from the other? and so on. Which of the properties learnt earlier about the numbers hold now? Which are the new properties? We also see how these help us deal with our daily life situations. 154 2015-16 (11-11-2014)  Look at the following Fracti FRACTIONS 1. My mother divided an apple into 4 equal parts. She gave me two parts and my brother one part. How much apple did she give to both of us together? example. A tea stall owner consumes in her shop 2 litres of milk in the morning 1 2 Intege 2. Mother asked Neelu and her brother to 1 pick stones from the wheat. Neelu picked and 1 litres of milk in the 2 one fourth of the total stones in it and her evening. What is the total brother also picked up one fourth of the amount of milk she uses in stones. What fraction of the stones did both the stall? pick up together? Or Shekhar ate 2 chapatis 3. Sohan was putting covers on his note books. 1 He put one fourth of the covers on Monday. for lunch and 1 chapatis for 2 He put another one fourth on Tuesday and dinner. What is the total the remaining on Wednesday. What fraction of the covers did he put on Wednesday? number of chapatis he ate? Clearly, both the situations require the fractions to be added. Some of these additions can be done orally and the sum can be found quite easily. Do This Make five such problems with your friends and solve them. 7.10.1 Adding or subtracting like fractions All fractions cannot be added orally. We need to know how they can be added in different situations and learn the procedure for it. We begin by looking at addition of like fractions. Take a 7 × 4 grid sheet (Fig 7.13). The sheet has seven boxes in each row and four boxes in each column. How many boxes are there in total? Colour five of its boxes in green. What fraction of the whole is the green region? Now colour another four of its boxes in yellow. Fig 7.13 What fraction of the whole is this yellow region? What fraction of the whole is coloured altogether? 5 4 9 Does this explain that + = ? 28 28 28 155 2015-16 (11-11-2014) MATHEMATICS Look at more examples In Fig 7.14 (i) we have 2 quarter parts of the figure shaded. This means we have 2 parts out of 4 1 shaded or of the figure shaded. 2 1 1 1 +1 2 1 Fig. 7.14 (i) Fig. 7.14 (ii) That is, + = = =. 4 4 4 4 2 Look at Fig 7.14 (ii) 1 1 1 1 +1 + 1 3 1 Fig 7.14 (ii) demonstrates + + = = =. 9 9 9 9 9 3 What do we learn from the above examples? The sum of 1. Add with the help of a diagram. two or more like fractions can 1 1 2 3 1 1 1 be obtained as follows : (i) + (ii) + (iii) + + 8 8 5 5 6 6 6 Step 1 Add the numerators. 1 1 Step 2 Retain the (common) 2. Add +. How will we show this 12 12 denominator. pictorially? Using paper folding? Step 3 Write the fraction as : 3. Make 5 more examples of problems given Result of Step 1 in 1 and 2 above. Solve them with your friends. Result of Step 2 3 1 Let us, thus, add and. 5 5 3 1 3 +1 4 We have + = = 5 5 5 5 7 3 So, what will be the sum of and ? 12 12 Finding the balance 5 2 Sharmila had of a cake. She gave out of that to her younger brother. 6 6 How much cake is left with her? A diagram can explain the situation (Fig 7.15). (Note that, here the given fractions are like fractions). 5 2 5− 2 3 1 We find that − = = or 6 6 6 6 2 (Is this not similar to the method of adding like fractions?) 156 2015-16 (11-11-2014)  Fracti FRACTIONS Intege Fig 7.15 Thus, we can say that the difference of two like fractions can be obtained as follows: Step 1 Subtract the smaller numerator from the bigger numerator. Step 2 Retain the (common) denominator. Result of Step 1 Step 3 Write the fraction as : Result of Step 2 3 8 Can we now subtract from ? 10 10 7 3 1. Find the difference between and. 8 8 2. Mother made a gud patti in a round shape. She divided it into 5 parts. Seema ate one piece from it. If I eat another piece then how much would be left? 3. My elder sister divided the watermelon into 16 parts. I ate 7 out them. My friend ate 4. How much did we eat between us? How much more of the watermelon did I eat than my friend? What portion of the watermelon remained? 4. Make five problems of this type and solve them with your friends. EXERCISE 7.5 1. Write these fractions appropriately as additions or subtractions : (a) (b) (c) 157 2015-16 (11-11-2014) MATHEMATICS 2. Solve : 1 1 8 3 7 5 1 21 12 7 (a) + (b) + (c) − (d) + (e) − 18 18 15 15 7 7 22 22 15 15 5 3 2  3 1 0 12 (f) + (g) 1 − 3 1 = 3 (h) + (i) 3 – 8 8 4 4 5 2 3. Shubham painted of the wall space in his room. His sister Madhavi helped 3 1 and painted of the wall space. How much did they paint together? 3 4. Fill in the missing fractions. 7 3 3 5 3 3 5 12 (a) − = (b) − = (c) – = (d) + = 10 10 21 21 6 6 27 27 5 5. Javed was given of a basket of oranges. What fraction of oranges was left in 7 the basket? 7.10.2 Adding and subtracting fractions We have learnt to add and subtract like fractions. It is also not very difficult to add fractions that do not have the same denominator. When we have to add or subtract fractions we first find equivalent fractions with the same denominator and then proceed. 1 1 1 1 What added to gives ? This means subtract from to get the 5 2 5 2 required number. 1 1 Since and are unlike fractions, in order to subtract them, we first find 5 2 2 5 their equivalent fractions with the same denominator. These are and 10 10 respectively. 1 1× 5 5 1 1× 2 2 This is because= = and = = 2 2× 5 10 5 5× 2 10 1 1 5 2 5–2 3 ∴ – = Therefore, – = = 2 5 10 10 10 10 Note that 10 is the least common multiple (LCM) of 2 and 5. 3 5 Example 8 : Subtract from. 4 6 3 5 Solution : We need to find equivalent fractions of and , which have the 4 6 158 2015-16 (11-11-2014) same denominator. This denominator is given by the LCM of 4 and 6. The Fracti FRACTIONS required LCM is 12. Therefore, 5 3 5 × 2 3 × 3 10 9 − = − = − = 1 6 4 6 × 2 4 × 3 12 12 12 Intege 2 1 Example 9 : Add to. 5 3 Solution : The LCM of 5 and 3 is 15. 2 1 2 × 3 1× 5 6 5 11 Therefore, + = + = + = 5 3 5 × 3 3 × 5 15 15 15 3 7 Example 10 : Simplify − 5 20 2 3 Solution : The LCM of 5 and 20 is 20. 1. Add and. 5 7 3 7 3 × 4 7 12 7 2 Therefore, − = − 5 20 5 × 4 20 20 20 = − 2. Subtract from 5. 5 7 12 − 7 55 1 = = = 20 205 4 How do we add or subtract mixed fractions? Mixed fractions can be written either as a whole part plus a proper fraction or entirely as an improper fraction. One way to add (or subtract) mixed fractions is to do the operation seperately for the whole parts and the other way is to write the mixed fractions as improper fractions and then directly add (or subtract) them. 4 5 Example 11 : Add 2 and 3 5 6 4 5 4 5 4 5 Solution : 2 + 3 = 2 + + 3 + = 5 + + 5 6 5 6 5 6 4 5 4× 6 5× 5 Now + = + (Since LCM of 5 and 6 = 30) 5 6 5× 6 6× 5 24 25 49 30 + 19 19 = + = = = 1+ 30 30 30 30 30 4 5 19 19 19 Thus, 5 + + = 5 + 1 + =6+ =6 5 6 30 30 30 4 5 19 And, therefore, 2 + 3 = 6 5 6 30 159 2015-16 (11-11-2014) MATHEMATICS Think, discuss and write Can you find the other way of doing this sum? 2 1 Example 12 : Find 4 − 2 5 5 2 1 Solution : The whole numbers 4 and 2 and the fractional numbers and 5 5 2 1 can be subtracted separately. (Note that 4 > 2 and > ) 5 5 2 1  2 1 1 1 So, 4 − 2 = (4 − 2) +  −  = 2 + = 2 5 5  5 5 5 5 1 5 Example 13 : Simplify: 8 − 2 4 6 1 5 Solution : Here 8 > 2 but <. We proceed as follows: 4 6 1 (8 4)+1 33 5 2 6+5 17 8 = = and 2 = = 4 4 4 6 6 6 33 17 33 × 3 17 × 2 Now, − = − (Since LCM of 4 and 6 = 12) 4 6 12 12 99 − 34 65 5 = = =5 12 12 12 EXERCISE 7.6 1. Solve 2 1 3 7 4 2 5 1 2 1 (a) + (b) + (c) + (d) + (e) + 3 7 10 15 9 7 7 3 5 6 4 2 3 1 5 1 2 3 1 1 1 1 (f ) + (g) (h) (i) + + (j) + + 5 3 4 3 6 3 3 4 2 2 3 6 1 2 2 1 16 7 4 1 (k) 1 + 3 (l) 4 +3 (m) (n) 3 3 3 4 5 5 3 2 2 3 2. Sarita bought metre of ribbon and Lalita metre of ribbon. What is the total 5 4 length of the ribbon they bought? 1 1 3. Naina was given 1 piece of cake and Najma was given 1 piece of cake. Find 2 3 the total amount of cake was given to both of them. 160 2015-16 (11-11-2014)  5 1 1 1 1 1 Fracti FRACTIONS Intege 4. Fill in the boxes : (a) −= (b