Mathematics Standard VII Past Paper PDF

MATHEMATICS Part - 1 Standard VII Government of Kerala ST-...

MATHEMATICS Part - 1 Standard VII Government of Kerala ST-359-1-MATHS (E)-7-VOL-1 Department of General Education Prepared by State Council of Educational Research and Training (SCERT) Kerala 2024  THE NATIONAL ANTHEM Jana-gana-mana adhinayaka, jaya he Bharatha-bhagya-vidhata Punjab-Sindh-Gujarat-Maratha Dravida-Utkala-Banga Vindhya-Himachala-Yamuna-Ganga Uchchala-Jaladhi-taranga Tava subha name jage, Tava subha asisa mage, Gahe tava jaya gatha Jana-gana-mangala-dayaka jaya he Bharatha-bhagya-vidhata Jaya he, jaya he, jaya he, Jaya jaya jaya, jaya he. PLEDGE India is my country. All Indians are my brothers and sisters. I love my country, and I am proud of its rich and varied heritage. I shall always strive to be worthy of it. I shall give my parents, teachers and all elders, respect and treat everyone with courtesy. To my country and my people, I pledge my devotion. In their well-being and prosperity alone, lies my happiness. MATHEMATICS 7 State Council of Educational Research and Training (SCERT) Poojappura, Thiruvananthapuram 695012, Kerala Website : www.scertkerala.gov.in e-mail : [email protected], Phone : 0471 - 2341883, Typesetting and Layout : SCERT First Edition : 2024 Printed at : KBPS, Kakkanad, Kochi-30 © Department of General Education, Government of Kerala Dear children, We’ve acquired some of the basic concepts of mathematics. Such as counting numbers, fractions, decimal forms and some algebra too. We’ve also had ample opportunities to use these to solve problems logically, explain them in terms of cause and effect, and to complete patterns. We are moving ahead. To more applications of the concepts learnt, to more computational techniques, to develop more skill in recognising relations between numbers, explaining them in our own language and in the language of mathematics, to more deeper analysis of geometry, to more complex problem solving, to more possibilities of mathematical and computational thinking. Let’s march ahead together with conϐidence, thinking, enquiring and enjoying ourselves. With love and regards, Dr. Jayaprakash R.K. Director SCERT, Kerala  TEXTBOOK DEVELOPMENT TEAM Advisor Dr. E. Krishnan Head (Rtd.), Department of Mathematics, University College, Thiruvananthapuram Chairperson C. Venugopal Assistant Professor (Rtd.) IASE, Thrissur Experts Dr. T.G. Sarachandran Dr. Preenu C.S. Deputy Director (Rtd.), Assistant Professor, Department of Collegiate Education, Kerala University College, Thiruvananthapuram Members Kunhabdhulla M. Thulaseedharan Pillai K.G. Suresh Babu T. Headmaster (Rtd.), Trainer, BRC Anchal, Kollam LPST (Rtd.), U.P.G.S., Muyippoth M.U.P. School, Punukkonnoor, Kollam Kozhikkode Sajeev C.S. Achuthan C.G. D.K. Biju HSST Mathematics HSST Mathematics, GHSS UPST, Kuruvangad South UP GHSS Edathanattukara, Karakkurissi, Palakkad School, Koyilandy, Kozhikode Palakkad Fathima K. Jobin Kuruvilla Dhaneshan M.V. UPST, UPST, St. Antony’s Higher HSST English, GHSS Karuvarakundu, Secondary School, Poonjar, GFHSS Padannakadappuram, Malappuram Kottayam Kasaragod Rajeev N.T. Prakashan Pachan Harikrishnan G. GHSS Thariyod, Wayanad Paruthiyara LTA, Govt. VHSS for Girls Kottarakkara Academic Coordinator Dr. Sivakumar K.S. Research Officer, SCERT State Council of Educational Research and Training (SCERT) Kerala Vidhyabhavan, Poojappura, Thiruvananthapuram 695 012  CONTENTS 1. Parallel Lines 7 2. Fractions 23 3. Triangles 43 4. Reciprocals 61 5. Decimal Methods 69 6. Ratio 83 7. Shorthand Math 93 Certain icons are used in this textbook for convenience Let's do problems ICT possibilities  Parallel Lines 1 PARALLEL LINES Let us recall We have heard about parallel lines in class 6. Lines that don't meet, keeping the same distance between them. We have also drawn them with a scale and a set square. Two lines drawn at the same slant to a given line are parallel. We've seen this also. We know that a parallelogram is a quadrilateral in which the two pairs of opposite sides are parallel. Can you 3c draw this parallelogram with measures as given ? m 3 cm 7 Standard - VII 7  Mathematics Lines and angles When a line crosses another line, how many angles are formed between them ? If we know one of these, can we calculate the others ? This too was seen in class 6. What can we say about the relation between four such angles in general ? 8 Standard - VII 8  Parallel Lines Nothing much to say if the crossing lines are perpendicular. All angles are 90°. Draw a line AB. Mark a point C on the line and another point D outside the line. Draw a line through C and D. Find BCD. Use the Angle tool and click on B, C, D in order (see what happens if you click in a different order). What if one line is a bit tilted ? Mark the other angles also like this. What is the relation between the angle measures ? Try changing the position of D. Don't you see a change in the angles ? Does the relation change ? Two small angles and two large angles. What is the relation between them?  The two small angles are of the same measure.  The two large angles are of the same measure.  The sum of a small angle and a large angle is 180°. 9 Standard - VII 9  Mathematics Look at the earlier figure once again: What if we draw another line above, parallel to the blue line ? Draw two intersecting lines and mark the angles between them as before. Mark a point E on CD and draw a line through it parallel to AB. Mark the four angles around E (You may mark more points on the line for ease of marking the angles). Now the blue line above makes four angles with the green line. What can you say about them? Let's look only at the small angle below and the marked angle above. What is the relation between the eight angles you have marked now ? Try changing the position of D. Does the relation change when the angle measures change ? Try changing the position of E. What happens when E takes the place of C ? 10 Standard - VII 10  Parallel Lines The blue lines are parallel. So these two must be A change in angle of the same measure. Suppose there is slight change in the slant of two lines with another line. The two lines won't be parallel. For example, look at this figure: m 2c The blue lines in the figure appear to be parallel. As there is a difference of 1° in the slants, they will meet when extended What about the other angles above ? sufficiently. We can calculate how much to be extended. You've to extend them by more than a metre for them to meet! Draw two parallel lines and an intersecting line as in the earlier activity. Mark the eight angles around the points of intersection. You may hide their measures (Right click and uncheck Show Label box). Give the same colour to the four small angles of the same measure (Right Click  Object Properties  Colour) Choose the colour you want. You can change Opacity. In the same way, Suppose we start with an angle other than 50°. give another colour to all the large angles The measures of the other angles will change. of the same measure. But the relation between the angles will be the same. That is, A line intersects two parallel lines at angles of the same measure. 11 Standard - VII 11  Mathematics In each of the pictures below, can you calculate the other seven angles which the parallel blue lines make with the green line ? Matching angles We have seen the relations between the four angles made by two intersecting lines. What can we say about the relation between the eight angles formed when a line cuts two parallel lines ? Let's take another look at this figure which we saw earlier. We know the relation between the four angles below. Same is the relation between the four angles above. What if we take an angle from below and an angle from above ? If both are small angles, each is 50°. If both are large, each is 130°. If one is small and the other is large; the small one is 50°, the large one 130°; and the sum is 180°. The relations remain the same even if the angles change, right ? So we can say this in general : Of the angles made when two parallel lines are cut by a slanting line,  the small angles are of the same measure.  the large angles are of the same measure.  a small angle and a large angle add up to 180°. If the intersecting line is perpendicular to one of the parallel lines, it would be perpendicular to the other line too, and all angles would be right angles. 12 Standard - VII 12  Parallel Lines Now, look at this figure. Position and angle We may group the angles formed when two parallel lines are cut by another line based on their position. The figures below show angle pairs in the same position. top right The top and bottom lines are parallel. What is top right the measure of angle above ? To see all the angles clearly, let's suppose the lines are extended. bottom right bottom right top left top left bottom left Look at the two small angles formed when the bottom left slanting line cuts the top and bottom parallel lines. These are the angles in the first figure. So Angles in each such pair are called they are of the same measure. corresponding angles. The angles in each pair measure the same. 13 Standard - VII 13  Mathematics Opposite positions That is, the angle above is also 45°. The figures below show the angles in opposite positions when two parallel lines are cut by another line bottom left What if the figure looks like this ? top right top left This is the first figure with the angle slightly bottom right changed and turned a bit, isn't it ? What is the measure of the other angle ? Now look at this figure : bottom right top left top right The vertical lines are parallel. But there is no bottom left line cutting them. How about drawing another vertical line ? Angles in each such pair are called alternate angles. The angles in each pair measure the same. 14 Standard - VII 14  Parallel Lines Now the, middle angle is in two parts. We can In and out find the left part. The figures below show the interior and exterior angle pairs when two parallel lines are cut by another line. inner right What about the right part ? inner right Let’s try different measures for this angle: inner left inner left outer right outer right What should be this angle to get a nice figure? outer left outer left The first two pairs are called co-interior angles and the last two pairs co-exterior angles. The sum of the angles in each such pair is 180°. 15 Standard - VII 15  Mathematics Let's draw a parallelogram. Draw Another question : Can you find out the other two lines AB and AC. Through angles in the given parallelogram ? B draw a line parallel to AC and through C draw a line parallel to AB. The point of intersection of these lines is D. Draw parallelogram ABDC using Polygon tool. We can see all angles if we click in the parallelogram using Angle tool. First, take the angle above the 55° angle. To determine this, we shall look at the angles of intersection of the left side with the top and bottom parallel lines. What is the relation between these angles? Try changing the position of C. Do the angles change ? And the relation between them ? The 55° angle and the angle above it form a pair of small angle and large angle. So their sum is 180°. This means the top angle = 180° 55° = 125°. Now look at the angle to the right of the marked angle. To calculate this, look at the angles made by the left and right parallel sides with the bottom line. 16 Standard - VII 16  Parallel Lines The 55° angle and the angle on its right are a small angle and a large angle of these angles. So, this angle also is 125° as calculated earlier. Can't you Ànd the fourth angle, like this ? Now try these problems: (1) Draw the parallelogram below with the given measures. Draw two parallel lines. Mark a point on each. Mark a third point m 3c in between them. Draw lines joining the points on the lines with the third point. Mark the angles which these lines make 5 cm with the parallel lines. Also mark the Calculate the other three angles. angle between the lines. (2) The top and bottom blue lines in the figure are parallel. Find the angle between the green lines. What is the relation between the three angles? Try changing the position of the ST-359-2-MATHS (E)-7-VOL-1 points. Does the relation between the angles remain the same ? (3) In the figure, the pair of lines slanted to the left are parallel; and also the pair of lines slanted to the right. Draw this figure : 2 cm 17 Standard - VII 17  Mathematics Triangle sum Look at this figure : The top and bottom lines are parallel. So, can you calculate the angle at the top ? If this angle is drawn less than 140°, then the two lines will meet. Let’s decrease it by 60°. Now we have a triangle. What are the angles in this triangle ? The left angle is 40°. The top angle is 140° – 60° = 80°. 18 Standard - VII 18  Parallel Lines What about the third angle ? It is one of the small angles which the new slanted line makes with the parallel lines. Its measure is the same as that of the small angle which this line makes with the top line. Isn't the top small angle 60° ? So the bottom small angle is also 60°. Thus the 60° which we took away from the top reappears at the bottom as an angle of the triangle. The sum of this angle and the top angle of the triangle 80° + 60° = 140°. Now look at this figure: 19 Standard - VII 19  Mathematics Can you calculate the sum of the other two angles of this triangle ? In the first problem, the right side of the triangle was got by drawing a slanted line instead of the parallel line. Let's think in the reverse. A parallel line instead of the slanted right side. What is the angle which this parallel line makes with the left side ? When we drew the triangle, this angle split into two. One part is the top angle of the triangle. What about the other part ? That is, one part of the 130° angle is the top angle of the triangle and the other part is the angle on the right in the triangle. Using the Polygon tool, draw So, the sum of these two angles of the triangle is a triangle and mark the angles 130°. (click inside the triangle using the Angle tool). What is the sum of all angles ? How do we state in general what we've learnt from Change the vertices of the triangle and this problem ? check. If we subtract the measure of one angle of a triangle from 180°, we get the sum of the other two angles. For example, if one angle of a triangle is 60°, the sum of the other two angles: 180° 60° = 120° What about the sum of the three angles of the triangle ? 20 Standard - VII 20  Parallel Lines This is true for any triangle. The sum of all angles of a triangle is 180° Now try this problem : One angle of a right triangle is 40°. What is the measure of the angle other than the right angle ? Let's think this way. The sum of the angles other than the right angle is 180° 90° = 90° One of them is 40°. Then the other angle is 90° 40° = 50° We can think in another way as well. The sum of the three angles is 180°. The sum of two of them 90° + 40° = 130° So the third angle 180°  130° = 50° Another problem : One angle of a triangle is 72°. The other two angles are of equal measure. What are their measures ? What is the sum of the other two angles ? 180° 72° = 108° Since the other two angles are equal, each is half the sum, isn't it ? So each is 108c 2 = 54c Now try these problems : (1) Draw the triangle with the given measures. 5 cm (2) The figure shows a triangle drawn in a rectangle. Calculate the angles of the triangle. 21 Standard - VII 21  Mathematics (3) The top and bottom lines in the figure are Draw two parallel lines and mark parallel. two points on each. Join them as in the figure and mark the point of intersection. Use the Polygon tool to draw two triangles and mark their angles. Calculate the third angle of the bottom triangle and all angles of the top triangle. (4) The left and right sides of the large triangle What is the relation between the angles of are parallel to the left and right sides of the the triangle ? small triangle. Try changing the positions of the points. Draw a triangle and mark a point on one of the sides. Draw a line through this point parallel to another side of the triangle. Mark the point where this Calculate the other two angles of the large line meets the third side. Draw the small triangle and all angles of the small triangle. triangle with one corner of the triangle 5. A triangle is drawn inside a parallelogram. and the points on the sides as vertices. Mark the angles of the first triangle and the small triangle. What is the relation between these angles ? Try changing the cornerss of the triangle. Calculate the angles of the triangle. 22 Standard - VII 22  2 FRACTIONS Multifold multiplication A packet of sugar weighs two kilograms. How many kilograms is four such packets ? Let's do some mental math. Four times two kilograms is eight kilograms. In detail, 4×2=2+2+2+2=8 What if they are half kilogram packets ? One kilogram in two packets; so two kilograms in four packets. That is, four times half a kilogram is two kilograms. In terms of numbers alone, four times half is two. Just as we wrote four times two as 4 × 2, we can write four times half as 4 # 12. That is, 1 1 1 1 1 1 4 # 2 = 4 times 2 = 2 + 2 + 2 + 2 = 2 A bottle holds a quarter litre of water. How much water is needed to fill three such bottles ? Three times a quarter litre is three quarters of a litre. In terms of numbers alone, 3 times 14 is 43 As a product, 3 × 14 = 3 times 14 = 43  Mathematics Another problem : Five strings, each a quarter metre long are placed end to end. What is the total length ? Four quarters make one; and another quarter makes it one and a quarter. Total length is one and a quarter metres. That is, 5 times 14 is 1 14 As a product ? 5 × 14 = 1 14 Do the following problems mentally. Write each as how many times and also as a product. (1) Each piece of a pumpkin weighs a quarter kilogram. What is the weight of two pieces together ? What is the weight of four such pieces ? Six pieces ? (2) We can fill a cup with one third of a litre of milk. How much milk is needed to fill two cups ? Four cups ? (3) What is the total length of four pieces of ribbons, each of length three fourths of a metre? What about five pieces ? (4) It takes 14 hour to walk around a play ground once. (i) How much time does it take to walk 4 times around at this speed ? (ii) What about 7 times ? Let's look at the calculations in such problems: What is 2 times 13 ? 2 × 13 = 13 + 13 = 32 What about 3 times ? 3 × 13 = 13 + 13 + 13 = 1 To get 4 times of 13 , we need just add a 13 to it, isn't it ? 4 × 13 = b3 # 13 l + 13 = 1 13 24 Standard - VII 24  Fractions It can be done this way also: 4 × 13 = 13 + 13 + 13 + 13 = 1 + 1 +3 1 + 1 = 34 = 1 13 How do we calculate 4 times 23 ? 4 × 32 = 32 + 32 + 32 + 32 = 2 + 2 +3 2 + 2 = 4 #3 2 = 38 Splitting 8 as a multiple of 3 and remainder, Share and fraction If 4 litres of milk is divided equally 8 3 = 6 +3 2 = 36 + 23 = 2 23 among 3 persons, how much of milk What about 10 times 23 ? will each one get ? First give 1 litre to each one. If the 2 We need to add ten 3 's remaining 1 litre is divided among 18 + 2 three persons, each will get 13 litre 10 × 23 = 103# 2 = 20 3 = 3 = 18 2 2 3 + 3 = 63 1 more. In total 1 3 litres. Now try this problem : Here since 4 is divided into 3, we can 3 litres of milk in a bottle; how many litres write it as a division. 4 in 7 such bottles ? 4  3 = 1 13 Also as a fraction. We need to find 7 times 43 4 1 3 = 13 7 × 43 = 7 # 3 21 4 = 4 Splitting 21 as a multiple of 4 and remainder, 21 4 = 204+ 1 = 20 1 1 4 + 4 = 54 So, 5 14 litres in 7 bottles. Now try these problems. (1) The weight of an iron block is 14 kilogram (i) What is the total weight of such 15 blocks ? (ii) 16 blocks ? 25 Standard - VII 25  Mathematics (2) Some 2 metre long rods are cut into 5 Just see how many pieces of equal length. problems in my Why so much milk ? (i) What is the length of each piece ? math book are about dividing milk ! (ii) What is the total length of 4 pieces ? (iii) Of 10 pieces? (3) 5 litres of milk is filled in 6 bottles of the same size. (i) How many litres of milk does each bottle hold ? (ii) How many litres in 3 bottles together ? (iii) In 4 bottles ? Part and multiplication A six metre long ribbon is cut into two equal pieces. What is the length of each piece? Half of six metres, that is three metres. Half means one of two equal parts, that is 12. So 12 of 6 metres is 3 metres. In terms of numbers alone, 1 2 of 6 is 3 Just as in the case of multiples, parts can also be written as products. That is, 1 2 × 6 = 12 of 6 = 3 Suppose we divide a six metre long ribbon into three equal pieces ? The length of each piece is a third of six metres; that is two metres 1 3 of 6 metres is 2 metres In terms of numbers alone, 1 of 6 is 2 3 26 Standard - VII 26  Fractions Writing it as a product, 1 3 ×6=2 What if we divide a two metre long string into three equal pieces ? We can divide each metre into three equal parts and then take two parts together. 1 1 1 metre 1 metre 3 metre 3 metre 2 metres 2 3 metre Then, 1 3 of 2 metres is 23 metre In terms of numbers alone, 1 Parts and times 3 of 2 is 23 If three litres of milk is divided between four people, how much will each get ? Writing it as a product, One fourth of three litres; that is three 1 of 2 = 1 ×2 = 2 quarters of a litre. There's another way of 3 3 3 thinking. When one litre of milk is divided Another problem : among four people, each gets a quarter of a litre. Because it is three litres, this can be What is a quarter of five kilogram ? done three times. So one gets three times a quarter of a litre which is three quarters A quarter of four kilograms is one kilogram; and of a litre. then a quarter of the remaining one kilogram. That is, one fourth of three litres and three So one and a quarter kilogram in all. In terms of times a quarter of a litre are both three quarters of a litre. numbers, As products of numbers, 1 4 of 5 is 1 14 1 1 1 4 ×3 = 3 × 14 4 of 5 = 4 ×5 = 1 14 27 Standard - VII 27  Mathematics Do these problems in head. Then write each as a part and also as a product of numbers. (1) Nine litres of milk is divided equally among three children. How many litres will each get ? Sharing must be fair ! What if there are four children ? Each must get his fill (2) Six kilograms of rice was packed in five bags of the same size. How many kilograms of rice in each bag ? What if it is packed in four bags ? (3) A seven metre long string is divided into six equal pieces. What is the length of each piece ? What if it is divided into three equal pieces ? Let's look at the calculations to find parts of numbers. What is 14 of 8 ? When 8 is divided into 4 equal parts, each part is 2. Writing as a product, 1 4 × 8 = 48 = 2 What about 43 of 8 ? 8 divided into 4 equal parts, and 3 of these parts taken together. In other words, 3 times what is got by dividing 8 into 4 equal parts. That is, 3 times 2, which is 6. Writing as a product, 3 4× 8 = 3 × 48 = 3 × 2 = 6 It can be done this way also. 3 4 × 8 = 3 × 48 = 24 4 =6 What is 43 of 9 ? 3 4 × 9 = 3 × 49 Here, instead of writing 49 = 2 14 and proceeding, it is easier to do like this: 3 4 × 9 = 3 × 49 = 3 #4 9 = 27 4 28 Standard - VII 28  Fractions To see what this actually means, we write 27 as a multiple of 4, and remainder : 27 24 + 3 4 = 4 = 24 3 3 4 + 4 = 64 Now see this problem: We have to cut off 35 of a 7 metre long string. How long is this piece ? We need to calculate 53 of 7. How we do this ? 3 20 + 1 20 5 × 7 = 3 #5 7 = 21 1 1 5 = 5 = 5 + 5 = 45 We need to cut off 4 15 metres (It is easy to cut it if we say this as 4 metres, 20 centimetres). Now try these problems: (1) There are 35 children in a class. 53 of them are girls. How many girls are there in the class ? (2) 10 kilograms of rice is filled equally in 8 bags. If the rice in 3 such bags are taken together, how many kilograms would that be ? (3) The area of the rectangle in the figure is 27 square centimetres. It is divided into 9 equal parts. What is the area of the darker part in square centimetres ? Part of part Look at this figure: 1 1 2 2 29 Standard - VII 29  Mathematics A rectangle is divided into two equal parts. Each part is half of the large rectangle, that is, 12 of the large rectangle. Now, suppose we divide one of these parts into three equal parts. Each of these three small rectangles is 13 of 12 of the large rectangle, isn't it ? That is, 1 1 3 × 2 of it. 1 #1 3 2 To look at it another way, suppose we extend the horizontal lines : What part of the large rectangle is each of these 6 small rectangles ? 1 6 Now, what do we get ? 1 3 × 12 = 16 30 Standard - VII 30  Fractions Can we calculate 15 of 13 like this ? That is 15 × 13 ? Let's draw a rectangle and divide it into three equal parts : 1 3 Now we draw horizontal lines to divide one of these rectangles into 5 equal parts. Each of these small rectangles is 15 of 13 of the largest rectangle; that is, 15 × 13 of it. 1 5 × 13 Suppose we extend the horizontal lines. So, 15 × 13 = 15 1 1 15 31 Standard - VII 31  Mathematics Now try these problems: (1) Draw rectangles and find these products. (i) 12 × 14 (ii) 13 × 16 (iii) 15 × 18 (2) A one metre long string is divided into five equal parts. How long is half of each part in metres ? In centimetres? (3) One litre of milk is filled in two bottles of equal size. A quarter of the milk in one bottle was used to make tea. How many litres of milk were used for tea ? In millilitres ? Another type of problem: what is 15 of 23 ? Let's think this way :  2 3 means 13 of 2  1 5 of 23 is 15 of 13 of 2 1 * 5 of 13 is 15 1 1 * 5 of 13 of 2 is 15 1 of 2  1 5 of 23 is 15 2 As a product, 1 5 × 23 = 15 × 13 × 2 = 15 1 2 × 2 = 15 32 Standard - VII 32  Fractions We can draw a rectangle to find 15 of 32 directly : 2 3 1 #2 1 1 2 5 3 15 15 15 What about 54 of 23 , that is 54 × 23 ? We first find 15 of 23 and then find 4 times that. 4 5 × 23 = 4 × 15 × 23 = 4 × 15 × 13 × 2 2 = 4 × 15 8 = 15 We can do it like this also: 4 5 × 23 = 4 × 15 × 13 × 2 ST-359-3-MATHS (E)-7-VOL-1 1 = 4 × 5# 3 ×2 1 = 4 × 15 ×2 2 = 4 × 15 8 = 15 33 Standard - VII 33  Mathematics Again we can draw a picture to show this : 2 3 1 #2 5 3 4#2 8 5 3 15 1 15 Now try these problems : (1) A rope 2 metres long is cut into 5 equal pieces. What is the length of three quarters of one of the pieces in metres ? In centimetres ? (2) 4 bottles of the same size were filled with 3 litres of water. One of these was used to fill 5 cups of the same size. How much water is there in one such cup, in litres ? And in millilitres ? (3) A watermelon weighing four kilograms was cut into five equal pieces. One piece was again halved. What is the weight of each of these two pieces in kilograms ? And in grams ? (4) A vessel full of milk is used to fill three bottles of the same size. Then the milk in each bottle was used to fill four cups of the same size. What fraction of the milk in the first vessel does each cup contain ? (5) Draw a line AB of length 12 centimetres. Mark AC as 23 of AB. Mark AD as 14 of AC. What part of AB is AD ? (6) Calculate the following using multiplication : 3 (i) 7 of 25 (ii) 3 5 of 27 (iii) 23 of 43 (iv) 5 6 3 of 10 34 Standard - VII 34  Fractions More on multiplication A bottle can contain one and a half litres of water. Four such bottles of water was poured into a vessel. How much water is there in the vessel ? When we pour two bottles it is three litres; four bottles makes it six litres, isn't it ? Here we've found 4 times 1 12. Writing this as a product, 4 × 1 12 = 6 Suppose we poured the water in 3 bottles, each containing 2 14 litres, into the vessel ? If it is 2 litre bottles, then 6 litres. Here we have 14 litre more in each bottle. So we must add 43 litre also. That is 6 43 litres. Writing this as a product, 3 × 2 14 = 3 × c2 + 14 m = (3 × 2) + c3 # 14 m = 6 + 43 = 6 43 There is another way to calculate this. We can write 2 14 litres as 49 litres. Then, 3 × 2 14 = 3 × 49 = 3 × 14 × 9 = 43 × 9 = 27 4 = 6 43 35 Standard - VII 35  Mathematics Like this, in what all ways can you calculate 5 times 3 12 ? We can do it this way : 5 × 3 12 = 5 × c3 + 12 m = (5 × 3) + c5 # 12 m = 15 + 2 12 = 17 12 And this is another way : 5 × 3 12 = 5 × 27 = 5 #2 7 = 35 2 = 17 12 Let's look at another thing : Six metres is thrice two metres. What about seven metres ? It's thrice two metres and a metre more. In other words, thrice two metres and half of two metres. So we may say that seven metres is three and a half of two metres. As a product, 3 12 × 2 = 7 How did we get this ? 3 12 × 2 = c3 + 12 m × 2 = (3 × 2) + c 1 # 2m 2 = 6+1 = 7 36 Standard - VII 36  Fractions In the same way, two and a quarter times five means, twice five together with a quarter of five. That is, ten and one and a quarter; makes eleven and a quarter. As a product, 2 14 × 5 = b2 + 14 l × 5 = (2 × 5) + b 14 # 5 l = 10 + 1 14 = 11 14 Another way to do this is : 2 14 × 5 = 49 × 5 = 9# 4 5 = 45 4 You're selling okra so = 11 14 cheap! why don't you sell at 30 rupees a kilo ? How do we calculate 3 12 times 2 14 ? Writing 2 14 as 49 Then I can get and 3 12 as 27 , the price for two and a half kilos ! 3 12  2 14 = 27  49 = 27 # 9 63 #4 = 8 = 56 + 7 8 = 7 87 Now try these problems yourself : (1) One and a half metres of cloth is needed for a shirt. How much cloth is required for five such shirts ? (2) The price of one kilogram of okra is thirty rupees. What is the price of two and a half kilograms ? (3) A person walks two and a half kilometres in an hour. At the same speed, how far will he walk in one and a half hours ? 37 Standard - VII 37  Mathematics (4) Roni has 36 stamps with her. Sahira says she has 2 12 times this. How many stamps does Sahira have ? (5) Joji works 4 12 hours each day. How many hours does he work in 6 days ? (6) Calculate the following : (i) 4 times 5 13 (ii) 4 13 times 5 (iii) 1 12 times 23 (iv) 25 times 2 12 (v) 2 12 times 5 12 Fractional area We've studied areas of rectangles in class 5. A rectangle is 5 centimetres long and 3 centimetres high. What is its area in square centimetres ? We calculated this area by filling this rectangle with squares of side one centimetre. 3 cm 5 cm 5  3 = 15 such squares fill the rectangle. So, area is 15 square centimetres. What happens if the sides are of length 5 centimetres and 1 12 centimetres ? 1 2 cm 1 5 cm 38 Standard - VII 38  Fractions Isn't this half the first rectangle ? 3 cm 1 12 cm 5 cm 5 cm So the area is half of 15, that is 7 12 square centimetres. This may be done in another way. What is the area of each small rectangle in the top row of the picture on the right ? Each one is half the area of the square with area 1 square centimetre, isn't it ? So we can say that the area of each rectangle is 12 square centimetre. Taking five of them together, the total area is 5 # 12 = 2 12 square centimetres. So the area of the large rectangle is 5 + 2 12 = 7 12 square centimetres. How do we calculate the area of a rectangle with One and half sides of length 12 centimetre and 13 centimetre ? We halved a rectangle of length To draw such a rectangle, we first draw a square 5 centimetres and height 3 centimetres. of side 1 centimetre. Then we draw one vertical This is the result. line and two horizontal lines to split it into 6 1 2 cm equal parts. 1 1 1 2 2 1 cm 5 cm Suppose we rearrange the half squares iin 1 cm 1 cm 1 3 the top row this way ? 1 cm 1 2 1 2 1 Each small rectangle has sides 12 centimetre and 2 1 1 3 centimetre. What about its area ? Area = 5 + 2 12 = 7 12 sq.cm. 39 Standard - VII 39  Mathematics Each of them is 16 of the large square. The area of the square is 1 square centimetre. So the area of the small rectangle is 16 square centimetre. What about the area of a rectangle with sides of length 13 centimetre and 1 5 centimetre ? Dividing the square of side 1 centimetre in another way, we find this area to be 1 #1 1 3 5 = 15 What is the area of a rectangle of length 5 12 centimetres and breadth 3 13 centimetres ? 3 13 cm 5 12 cm Into how many parts of length 12 centimetre each, can we divide the bottom side? 10 lines of length 12 centimetre, each make 5 centimetres. To make 5 12 centimetres, we need one more such line 5 12 = 11  12 11 parts 1 2 5 12 cm Now let's divide the left side of the rectangle into parts of length 13 centimetre. 9 lines of length 13 centimetre each, make 3 centimetres. To get 3 13 centimetres, we need one more line. 3 13 = 10  13 40 Standard - VII 40  Fractions 10 parts 3 13 cm 11 parts 5 12 cm We now fill part of the rectangle with small Rectangle Division rectangles of length 12 centimetre and breadth 1 We can divide a rectangle of length 5 2 1 3 centimetre. 1 cm and breadth 3 3 cm this way also : 10 rectangles 3 13 cm 11 rectangles 5 12 cm How many such small rectangles are needed to The area can be calculated in four steps : fill the rectangle completely ? Blue rectangle 11 rectangles 5 × 3 = 15 Green rectangle 10 rectangles 5 × 13 = 1 23 3 13 cm Yellow rectangle 1 2 × 3 = 1 12 1 Red rectangle 5 2 cm 11  10 = 110 small rectangles in all; each of 1 2 × 13 = 16 area 16 square centimetre. Total area 15 + 1 23 + 1 12 + 16 = 18 13 sq.cm 41 Standard - VII 41  Mathematics Total area. 110 × 16 = 18 13 square centimetres Let's take another look at the operations involved : 5 12 = 11 × 12 3 13 = 10 × 13 11 × 10 × 12 × 13 = 110 × 16 The last multiplication may also be written as, 11 × 10 × 12 × 13 = c11 # 12 m × c10 # 13 m = 5 12 3 13 Thus we see that even if the lengths are in fractions, the area of a rectangle is still the product of the lengths of sides. Now try these problems : (1) The length and breadth of some rectangles are given below. Find the area of each: (i) 3 14 centimetres, 4 12 centimetres (ii) 5 13 metres, 6 43 metres (iii) 1 13 metres, 43 metres (2) What is the area of a square of side 1 12 metres ? (3) The perimeter of a square is 14 metres. What is its area ? 42 Standard - VII 42  3 TRIANGLES Star Picture Let's see how to draw stars using geogebra. Look at this picture: Draw a regular hexagon using the Regular Polygon tool. (For this, choose the tool and click on the two points. In the pop-up window give the number of corners as 6). Join alternate corners to get an equilateral triangle. Joining the other three corners, we can draw the next triangle also. To draw the circles, use the Circle through 3 points tool and click on three corners. Now we may hide the hexagon. Like this we can draw a regular polygon of 9 sides and draw this star: How do we draw this figure ? What all do we need to draw ? Two triangles and a circle. What is special about the triangles ? All sides are equal. Let's draw such a triangle first. Let's take the sides to be 3 cm long. Here we have three equilateral triangles. Try First we draw the bottom line. drawing star pictures with more equilateral 3 cm triangles. Where do we mark the third corner of the triangle? The left and right sides need to be 3 centimetres each. So the third corner must be 3 centimetres away from the two ends of the first line. How many points can you mark 3 centimetres away from one end ?  Mathematics Imagine such points : 3c m 3 cm If we draw this circle, the third corner must be somewhere on it. 3c m 3 cm Like this, if we draw a circle of radius 3 centimetres, centred on the right end, the third corner must be somewhere on this circle also. 3c m m 3c 3 cm 44 Standard - VII 44  Triangles These circles intersect at two points at the top and bottom, don't they ? What can you say about those points ? Since they are on the first circle, they are 3 centimetres away from the left end. Also as they are on the second circle, they are 3 centimetres away from the right end. Any one of these could be the third corner of our triangle. Let's draw the triangle using the top point. m 3c 3c m 3 cm Now we can delete the circles. We have one triangle now. How do we draw the star ? For this, mark points on the three sides 1 centimetre apart. I have to find some triangular shapes. Can you give a hand? But you've one within your arm's reach! ST-359-4-MATHS (E)-7-VOL-1 Joining these points, we can draw the second triangle. 45 Standard - VII 45  Mathematics If we draw a circle around it, we have the first figure. Where do we fix the compass to draw it ? Find it out (Wild guesses won't do, it should be exact). When the picture is completed, you can colour it as you wish and make it pretty. By drawing two triangles like this, adding some lines, removing some and We have drawn a star using the regular with a change of colour, you can get the hexagon. We can draw other beautiful pictures below: pictures like this. Using the Intersect tool, mark the points where the sides of the triangle meet. Use the midpoint of centre tool to find the centre of the circle. Now hide the triangle and use the polygon tool to join the dots properly to draw various figures. Right click on the polygon and choose the appropriate colour from Object properties → colour. Use Opacity to change the clarity of the picture. Try drawing this picture. Try it out. 46 Standard - VII 46  Triangles Also try to draw the pictures below : Lines and Math We've drawn triangles with all sides equal. Such triangles are called equilateral triangles. Can't we draw triangles with unequal sides also this way ? For example, a triangle with sides 3 centimetres, 4 centimetres and 6 centimetres. 6c m How do we draw this ? 4c m Draw a line 3 centimetres long. Mark the third corner, 4 centimetres from one end and 6 centimetres from the other end. 3 cm For this, isn't it enough to draw a circle from each end ? 6c m 4c m 4 cm 3 cm 6 cm 47 Standard - VII 47  Mathematics For the third corner, we need only the position above. So we draw only the top half of the circles (Even that is not needed, in fact. It's enough to draw two small pieces of the circles). Two Sides We can also draw like this, interchanging the sides. Draw triangle ABC, with AB = 6, AC = 5. Use Segment with Given Length tool to draw a line AB of length 6. Draw a circle of radius 5 centimetres centred at cm m 6c 6 A and mark a point C. Draw triangle ABC. m 4c Try changing the position of C along the m circle. Do the lengths of the sides AB; AC 4c change? What about the length of BC ? 3 cm What is the maximum length ? What is the minimum length ? What is the relation Is there any difference between these two between that of the possible length of this triangles ? side with the other two sides. Just flipped right to left, isn't it ? We can see this immediately if we draw them together. 6c m m 6c 4c m 4c m 3 cm Without changing the lengths of the sides, we can draw the triangles changing the position of the sides. 6 cm m 6c 3c m 3c m 4 cm 4 cm 48 Standard - VII 48  Triangles m 4c 3c m m m 3c 4c 6 cm 6 cm Apart from flips and turns, is there any real difference between these six triangles ? If you have any doubt, cut out one of these figures from a thick piece of paper and place it on each in different ways and see. Look at one of those figures now. 4 cm Draw a triangle and m 3c mark the measures of angles and sides. Compare 6 cm the size of the angles and the Which is the largest angle ? length of the sides. Change The smallest ? the vertices of the triangle and Can't you see it without actually measuring? see what happens. Look at the sides opposite them : rtest largest med ium sho ium smalles med t longest Draw several triangles with different lengths for the sides. In all these, is the longest side is opposite the largest angle, and the shortest side opposite the smallest angle ? m 5c 4.5 c 4.5 m 3.5 cm m 5c cm 4 cm 7 cm 2.5 cm 3.5 cm 49 Standard - VII 49  Mathematics Why is it so ? As the angle becomes larger, its sides spread out more, don't they ? This fact about triangles can be stated thus: In any triangle, the angles and their opposite sides have sizes in the same order. We have drawn triangles with different sides. Draw a line AB, 6 units This raises a question. Can we draw a triangle with any long. Draw a circle with three numbers as the lengths of the sides? centre A and radius 2 units. Make a slider a with Min = 0, For example, let's try, 6 centimetres, 2 centimetres, Max = 10. Choose Circle → 3 centimetres. Centre & Radius tool and click on B. Give the radius of the circle as a in the window. Use Intersect tool to mark the 6 cm point C of the intersection of 2 cm 3 cm m the circles. (The value on the 2c 3c slider can be adjusted so that m the circles intersect). Draw triangle ABC and mark the We see that all the points 2 centimetres away from lengths of the sides. Change the radius using the slider and one end of the 6 centimetre long line, are more than 3 check. For what values do you centimetres away from the other end; and the other way get a triangle ? round also. What if we try to draw the triangle with the 2 centimetre long line as the bottom side? m 6c 3c m 3 cm 6 cm 2 cm 50 Standard - VII 50  Triangles Take another look at the red lines in the two pictures. Why is it that we can't draw a triangle ? The greatest length 6 is more than the sum of the other two lengths 2 and 3. So, for three lengths to be the sides of a triangle, the greatest length must be less than the sum of the other two. Let's draw a picture as shown above. For This we state as a general principle about the this, we first draw an equilateral triangle, lengths of the sides of a triangle: and divide each side into three equal parts. In any triangle, the length of the greatest side is less than the sum of the length of the other two sides. This may be worded differently. The sum of the length of the two smaller sides of a triangle is more than the length of the greatest side. Draw equilateral triangles on the middle part of each side. From this, we can see that the sum of the lengths of any two sides of a triangle is greater than the length of the third side (Why is it so ?). This is easily seen with a little thought. Next divide each side of this star into three equal parts and draw equilateral triangles. Now our picture is ready. If the sides of the first triangle is a multiple of 9 then, things are easy. To divide a side into three equal parts draw a circle of radius one third the side, centred at a corner of The direct path from A to B is definitely shorter the triangle and mark the point where the than a detour through C, however near C is to A, circle meets the side. isn't it? 51 Standard - VII 51  Mathematics Try these problems: (1) The sides of a triangle are natural numbers. If the lengths of two sides are 5 centimetres and 8 centimetres, what are the possible numbers which can be the length of the third side ? (2) The lengths of the sides of a triangle are all natural numbers and two of the sides are 1 centimetre and 99 centimetres. What is the length of the third side? (3) Which of the following sets of three lengths can be used to draw a triangle? (i) 4 centimetres, 6 centimetres, 10 centimetres (ii) 3 centimetres, 4 centimetres, 5 centimetres (iii) 10 centimetres, 5 centimetres, 4 centimetres (4) Draw these pictures : 6 cm 6c m 4 cm 5 cm m 4c 6 cm 10 cm m 8c 12 cm Angle math We've seen how to draw triangles, with specified lengths for the sides. We also found out the relation between the lengths of the sides. What if we specify the angles? We know that the sum of the angles of a triangle is 180°(The chapter, Parallel Lines). So we can fix only two angles. For example, if we take two angles to be 40° and 60°, then the third angle has to be 80°. How do we draw the triangle with these angles? 52 Standard - VII 52  Triangles We can draw not one, but several triangles, right? 6 cm 4 cm 3 cm So, if we have one particular triangle in mind, we must specify not only two angles, but the length of the side on which they stand also. This raises a question : can't we draw several triangles with one side of length 4 centimetres and two angles of measures 40° and 60° standing on it? 4c cm m 4 4 cm We may interchange the position of the angles and draw several more (Try it!). But aren't they all, the same triangle flipped and turned, as we saw earlier? That is to say, there is in fact only one triangle with one side 4 centimetres and angles 40°, 60° on this side. It is easy to change the size of a triangle with these angles. For example, look at this figure. We want to draw another triangle adjacent to it as shown below : 6 cm 6 cm 4 cm 53 Standard - VII 53  Mathematics First we extend the bottom side by another 4 centimetres. 6 cm 4 cm We make a 50° angle at the left end of the extended side. Draw a triangle ABC using the Polygon tool. Use the Ray tool to extend AB. Mark a point D on AB. Through D, draw a line parallel to BC and another line through B parallel to AC. Mark E, their point of intersection. Draw triangle BDE. Choose the Angle tool and click inside 6 cm 4 cm the triangles to see the angle measures. And a 70° angle at the right end of the line. Change the position of D along the line and see what happens. 6 cm 4 cm Did you notice something about the figure ? The sides of the new triangle are parallel to the sides of the first triangle (Why ?). 54 Standard - VII 54  Triangles So, instead of drawing a new triangle, by measuring the angles once again, it is enough to draw lines parallel to the sides of the first triangle (This is easy in GeoGebra) Now, can't you draw a picture like the one below? Can you draw the triangle shown below ? Draw triangle ABC. Extend AB and mark D on it. Join CD. Through B, draw a line parallel to AC. Mark E, the point of intersection of this line with CD. Draw a line through E parallel to BC. Mark 6 cm F, the point of intersection of this line with the bottom line. Draw triangle BFE. Draw By joining three of them we have an equilateral other triangles similarly and colour them. triangle. Now we can hide the unnecessary lines. 6 cm 55 Standard - VII 55  Mathematics Can you draw these pictures by joining more and more small triangles like this ? Sides and angles The sides of an angle may be extended as much as we want : If we block them with a line, we have a triangle. 56 Standard - VII 56  Triangles To draw such a line, it is enough to mark a point on each of the sides, right ? In other words, once we specify an angle and the length of its sides, we have the triangle. For example, if we fix the bottom side as 6 centimetres and top side as 4.5 centimetres of the above angle, we get the earlier triangle. This can be put in another way: once if we specify the lengths of two sides of a triangle and the angle between them, we have the triangle. Now can't you draw a triangle with two sides 4 centimetres and 8 centimetres and the angle between them as 60° ? m 8c 4 cm Do you note anything special about this triangle ? Draw other triangles with one side twice the other side and the angle between them as 60°. Are all of them right triangles ? Now what is the third angle of these triangles ? From the set squares in the geometry box, choose the one with all sides different and measure the shortest and longest sides. Is the longest side twice the shortest ? Let's now look at another thing. Once we specify two sides and the angle between them, we can draw the triangle. 57 Standard - VII 57  Mathematics Suppose we specify the lengths of two sides and an angle not between these sides. 4 cm 6 cm How do we draw this ? We can draw a line 6 centimetres long and an angle of 30° at the right end: Draw a line AB, 10 centimetres long. Using the Angle with 6 cm Given Size tool, click on B and then The third corner of the triangle could be anywhere on A to get a window. Give 30 as on this line; but it should be 4 centimetres the angle measure, we get a point B′. away from the left end of the bottom line. Draw line AB′ (Use Ray tool). Make a That is to say, it should be on a circle of radius slider 'a'. Choose Circle → Centre & 4 centimetres with the centre as the left end. Let's Radius tool click on B. In the window draw this circle. give a as the radius of the circle. Mark the points of intersections C, D of the line AB′ with the circle. Draw triangles ABC, ABD. Try changing 4 cm the value of the slider. When do we get more than one triangle ? When 4 cm we get a single triangle do you 6 cm see anything special about it ? Is, there a case when we don't get a triangle at all ? 58 Standard - VII 58  Triangles We see that the circle cuts the line at two points. If we take the bottom point as the third corner, then we get triangle we have seen above. 4 cm 6 cm What if we take the top point as the third corner ? 4 cm 6 cm Thus, we can draw two triangles with these measures. Do we get two triangles, if we take the length of the left side as something other than 4 centimetres? For example, what if it is 2 centimetres ? Draw the circle and the line as below : m 2c 6 cm 59 Standard - VII 59  Mathematics Circle doesn't intersect the line; and we cannot 3 D Shapes draw a triangle. What if the length of the side is 3 centimetres ? This is the picture of a square pyramid. Can m you construct similar one? Look the shape 3c below. On each side of the square, there is an isosceles triangle drawn (triangle with two sides equal). 6 cm Circle intersects the line at a single point. And we can draw a single triangle with these Cut a similar one from a thick sheet of measures. paper. When we fold and paste them we get a square pyramid. m 3c 6 cm If we draw isosceles triangles on each side What happens if we take the side longer than of an equilateral triangle instead, as above, 6 centimetres ? Try it! then we get a pyramid with an equilateral triangle as base. Now, draw the following figures using the methods we have used in this lesson. 60 Standard - VII 60  4 RECIPROCALS Times and parts There are two pens in one packet and eight in another: We can compare these in two ways: 6 more 4 times 6 less 1 of 4 Now look at these weights: One is 5 kilograms, the other 15 kilograms. How do we say it as times and part ?  Mathematics 3 times 1 3 of Let's look at the first example using only numbers. 4 times 4 times 2 is 8 1 4 of 8 is 2 1 4 of Let's do this for the second example also: 3 times 5 is 15 3 times 1 3 of 15 is 5 1 of 3 Now try this problem : A vessel holds 10 litres of water. Another vessel holds 15 litres. In what all ways can you state the relation between them as times and part ? Let's think this way. To fill the large vessel, how many times should we fill and empty the small vessel ? When we do this once, we have 10 litres in the large vessel. Now only 5 more litres is needed. That means 12 of the small vessel. Thus, 15 litres is 1 12 times 10 litres. Let's think in the reverse. If we take the 15 litre vessel full of water, not all of it is needed to fill the 10 litre vessel. How much water will be left in the large vessel ? 62 Standard - VII 62  Reciprocals The remaining 5 litres is 13 of 15 litres, isn't it ? So the 10 litres poured into the small vessel is 23 of 15 litres. That is, 1 12 times 10 litres is 15 litres 2 3 of 15 litres is 10 litres In terms of numbers alone 3 2 times 3 2 times 10 is 15 2 3 of 15 is 10 2 3 of We can think of it in a different way. 1 If we take 10 of 10, we have 1; and 15 times 1 is 15. 15 So 15 is 10 times 10. 15 = 3 # 5 = 3 , isn't it ? 10 2 # 5 2 We can think of the reverse problem also like this 1 15 of 15 is 1; and 10 times 1 is 10. So 10 15 of 15 is 10. 10 2 # 5 2 15 = 3 # 5 = 3 Now can you do this problem ? The length of a rope is 4 metres. The length of another rope is 14 metres. How do we state the relation between them as times and part ? 14 4 times 4 is 14 4 14 of 14 is 4 63 Standard - VII 63  Mathematics Let's rewrite the fractions, 14 7 4 = 27 # 2 #2 = 2 4 14 = 27 # #2 2 2 = 7 Thus, 7 2 times 4 is 14 2 7 of 14 is 4 We can rewrite 27 in the first line as 3 12 and say this way: 3 12 times 4 is 14 2 7 of 14 is 4 Now try these problems: (1) Suma has 16 rupees with her. Safeer has 4 rupees. (i) What part of Suma's money does Safeer have ? (ii) How many times Safeer's money does Suma have ? (2) A large bag contains 9 kilograms of sugar. A small bag contains 6 kilograms. (i) The weight of sugar in the heavier bag is how much times that in the lighter bag ? (ii) The weight of sugar in the lighter bag is what part of that in the heavier bag ? (3) The weight of an iron block is 6 kilograms. The weight of another block is 26 kilograms. (i) The weight of the lighter block is what fraction of that of the heavier block ? (ii) The weight of the heavier block is how much times that of the lighter block ? (4) The length of a ribbon is 2 32 times the length of a smaller ribbon. What part of the length of the large ribbon is the length of the small ribbon ? Topsy-turvy Did you notice something in the problems we did just now ? In order to reverse times and part, we just need to turn fractions upside down. Can you say this in the first examples we discussed ? 64 Standard - VII 64  Reciprocals For example, in the pen problem, we saw 4 times 2 is 8