Chapter 1

6 y

7 x −16 Solution : In examples 6 and 7, state whether the given statements are true or false.

14 2

3 3 4 14 2 4 14 3 Solution : × ÷ = × × 7 3 3 7 3 2 = ×7 = 4 2 −5 7 2 −2

8 × 3 24 −19 −18 −17 20 Thus, rational numbers , , ,..... lie 24 24 24 24 −5 7 between and . 6 8

The additive inverse of is −7 7 19 −19

• (a)
• (b)
• (c)
• (d)19 19 7 7

Multiplicative inverse of a negative rational number is

• (a)a positive rational number.
• (b)a negative rational number.
• (c)0
• (d)1

If x + 0 = 0 + x = x, which is rational number, then 0 is called

• (a)identity for addition of rational numbers.
• (b)additive inverse of x.
• (c)multiplicative inverse of x.
• (d)reciprocal of x.

To get the product 1, we should multiply by 8 −8 21 −21

• (a)
• (b)
• (c)
• (d)21 21 8 8

– (–x) is same as 1 −1

• (a)– x
• (b)x
• (c)
• (d)x x

The multiplicative inverse of −1 is 8 −8 7 7

• (a)
• (b)
• (c)
• (d)7 7 8 −8

If x be any rational number then x + 0 is equal to

• (a)x
• (b)0
• (c)– x
• (d)Not defined

The reciprocal of 1 is

• (a)1
• (b)–1
• (c)0
• (d)Not defined

The reciprocal of –1 is

• (a)1
• (b)–1
• (c)0
• (d)Not defined

The reciprocal of 0 is

• (a)1
• (b)–1
• (c)0
• (d)Not defined

The reciprocal of any rational number , where p and q are integers and q ≠ 0, is p q

• (a)
• (b)1
• (c)0
• (d)q p

If y be the reciprocal of rational number x, then the reciprocal of y will be x y

• (a)x
• (b)y
• (c)
• (d)y x −3 −7

The reciprocal of × is 8 13 104 −104 21 −21

• (a)
• (b)
• (c)
• (d)21 21 104 104

Which of the following is an example of distributive property of multiplication over addition for rational numbers. 1 2 − 4 1 2 1 − 4 × + = − × + − × 4 3 7 4 3 4 7

• (a)− 1 2 − 4 1 2 − 4 × + = − 4 3 7 4 3 7
• (b)− × 1 2 − 4 2 1 − 4
• (c)− × + = + − × 4 3 7 3 4 7 1 2 − 4 2 − 4 1
• (d)− × + = + − 4 3 7 3 7 4

Between two given rational numbers, we can find

• (a)one and only one rational number.
• (b)only two rational numbers.
• (c)only ten rational numbers.
• (d)infinitely many rational numbers. Plan a strategy • Some problems contain a lot of information. Read the entire problem carefully to be sure you understand all the facts. You may need to read it over several times, perhaps aloud so that you can hear yourself and understand it well. • Then decide which information is most important (prioritise). Is there any information that is absolutely necessary to solve the problem? This information is most important. • Finally, put the information in order (sequence). Use comparison words like before, after, longer, shorter, and so on to help you. Write down the sequence before you try to solve the problem. Read the problem given below, and then answer the questions that follow • Five friends are standing in line for the opening of a show. They are in line according to their arrival. Shreya arrived 3 minutes after Sachin. Roy took his place in line at 9:01 P.M. He was 1 minute behind Reena and 7 minutes ahead of Shreya. The first person arrived at 9:00 P.M. Babu showed up 6 minutes after the first person. List the time of each person’s arrival. (a) Whose arrival information helped you determine each person’s arrival time? (b) Can you determine the order without the time? (c) List the friends’ order of arrival from the earliest to the last. x +y

is a rational number.

• (a)Between x and y
• (b)Less than x and y both.
• (c)Greater than x and y both.
• (d)Less than x but greater than y.

Which of the following statements is always true? x −y

• (a)is a rational number between x and y. x +y
• (b)is a rational number between x and y. x ×y
• (c)is a rational number between x and y. x ÷y
• (d)is a rational number between x and y. In questions 26 to 47, fill in the blanks to make the statements true.

The equivalent of , whose numerator is 45 is ___________.

The equivalent rational number of , whose denominator is 45 is ___________. 15 35

Between the numbers and , the greater number is __________. 20 40

The reciprocal of a positive rational number is ___________.

The reciprocal of a negative rational number is ___________.

Zero has ___________ reciprocal.

The numbers ___________ and ___________ are their own reciprocal.

If y be the reciprocal of x, then the reciprocal of y2 in terms of x will be ___________. 2 –4

The reciprocal of × is ___________. 5 9

(213 × 657)–1 = 213–1 × ___________.

The negative of 1 is ___________. Writing Strategy: any Write a Convincing Argument Your ability to write a convincing 10 For d2 . 2 an lly a argument proves that you have p ar e 10 w h a t u s u e r, understanding of the concept. An Com rs, mb o n u m b e e a t e r n u er as t w r b effective argument should include the the g r num nent? following four parts: g i v e s he greate p o t e ex (1) A goal using e or as th exception. as one the b least (2) A response to the goal Give (3) Evidence to support the response (4) A summary statement Step 1 : Identify the goal For any two numbers, explain whether using the greater number as the base will generally result in a greater number or using it as the exponent. Find one exception. Step 2 : Provide a response to the goal Using the greater number as the exponent usually gives the greater number. Step 3 : Provide evidence to support your response he n umbe r for t reate 2 x ce ption sing the g ent E on 1 0 and r, d 3. e exp mb e r be 2 an r, 3, as th a greater o r t he nu eater num w i l l e numb t result in F r the g t o n e n ber. n o Using t h e e x p will s r num numb er. 1 0 , a n a greate resul 32 = 9 10 = 23 = 8 1024 21 = 9>8 1024 100 < 0 32 > 2 10 < Step 4 : Summarise your argument Generally, for any two numbers, using the greater number as the exponent instead of as the base will result in a greater number. a c e a c e

For rational numbers , and we have × + = _________ + b d f b d f ________. −5

is ________ than –3.

There are ________ rational numbers between any two rational numbers. 1 −1

The rational numbers and are on the ________ sides of zero on 3 3 the number line.

The negative of a negative rational number is always a ________ rational number.

Rational numbers can be added or multiplied in any __________. −5

The reciprocal of is ________.

The multiplicative inverse of is _________.

The rational number 10.11 in the from is _________. 1 2 3 1 2

× + = × + _________. 5 7 8 5 7

The two rational numbers lying between –2 and –5 with denominator as 1 are _________ and _________. In each of the following, state whether the statements are true (T) or false (F).

If is a rational number, then y is always a whole number.

If is a rational number, then p cannot be equal to zero.

If is a rational number, then s cannot be equal to zero. 5 2

lies between and 1. 6 3 5 1

lies between and 1. 10 2 −7

lies between –3 and –4.

lies between 1 and 2. a b

If a ≠ 0, the multiplicative inverse of is . b a −3 5

The multiplicative inverse of is . 5 3

The additive inverse of is –2. x c x c

If is the additive inverse of , then + = 0. y d y d

For every rational number x, x + 1 = x. x c x c

If is the additive inverse of , then − = 0 y d y d

The reciprocal of a non-zero rational number is the rational number .

If x + y = 0, then –y is known as the negative of x, where x and y are rational numbers.

The negative of the negative of any rational number is the number itself.

The negative of 0 does not exist.

The negative of 1 is 1 itself.

For all rational numbers x and y, x – y = y – x.

For all rational numbers x and y, x × y = y × x.

For every rational number x, x × 0 = x.

For every rational numbers x, y and z, x + (y × z) = (x + y) × (x + z).

For all rational numbers a, b and c, a (b + c) = ab + bc.

1 is the only number which is its own reciprocal.

–1 is not the reciprocal of any rational number.

For any rational number x, x + (–1) = –x.

For rational numbers x and y, if x < y then x – y is a positive rational number.

If x and y are negative rational numbers, then so is x + y.

Between any two rational numbers there are exactly ten rational numbers.

Rational numbers are closed under addition and multiplication but not under subtraction.

Subtraction of rational number is commutative.

− is smaller than –2.

0 is a rational number.

All positive rational numbers lie between 0 and 1000.

The population of India in 2004 - 05 is a rational number. 5 8

There are countless rational numbers between and . 6 9

The reciprocal of x –1 is .

The rational number lies to the left of zero on the number line.

The rational number lies to the right of zero on the number line. −4 −8

The rational number lies neither to the right nor to the left of −3 zero on the number line.

The rational numbers and –1 are on the opposite sides of zero on the number line.

Every fraction is a rational number.

Every integer is a rational number.

The rational numbers can be represented on the number line.

The negative of a negative rational number is a positive rational number.

If x and y are two rational numbers such that x > y, then x – y is always a positive rational number.

0 is the smallest rational number.

Every whole number is an integer.

Every whole number is a rational number.

0 is whole number but it is not a rational number. 1 5

The rational numbers and − are on the opposite sides of 0 on 2 2 the number line.

Rational numbers can be added (or multiplied) in any order −4 −6 −6 −4 × = × 5 5 5 5

Solve the following: Select the rational numbers from the list which are also the integers. 9 8 7 6 9 8 7 6 5 4 3 3 1 0 –1 –2 −3 −4 −5 −6 , , , , , , , , , , , , , , , , , , , , 4 4 4 4 3 3 3 3 2 2 1 2 1 1 1 1 2 2 2 2

Select those which can be written as a rational number with denominator 4 in their lowest form: 7 64 36 − 16 5 140 , , , , , 8 16 − 12 17 − 4 28

Using suitable rearrangement and find the sum: 4 − 4 3 −13

• (a)+ + + 7 9 7 9 7 3 5 −4
• (b)−5 + + + ( −3) + + 10 7 14 5

Verify – (– x) = x for 3 −7 13

• (i)x =
• (ii)x =
• (iii)x = 5 9 −15

Give one example each to show that the rational numbers are closed under addition, subtraction and multiplication. Are rational numbers closed under division? Give two examples in support of your answer.

Verify the property x + y = y + x of rational numbers by taking 1 1 −2 −5

• (a)x = , y=
• (b)x = , y= 2 2 3 6 −3 20 −2 −9
• (c)x = , y=
• (d)x = , y= 7 21 5 10

Simplify each of the following by using suitable property. Also name the property. 1 1 1 1 2 1 2 −3 3 −5

• (a)× + × 6
• (b)× × + 5 15 5 5 − ×
• (c)2 4 2 5 7 6

Tell which property allows you to compute 1 5 7 1 5 7 × × as × × 5 6 9 5 6 9

Verify the property x × y = y × z of rational numbers by using 1 2 9

• (a)x = 7 and y =
• (b)x = and y = 2 3 4 −5 14 −3 −4
• (c)x = and y =
• (d)x = and y = 7 15 8 9

Verify the property x × (y × z) = (x × y) × z of rational numbers by using −1 1 2 −3 1

• (a)x = 1, y = and z =
• (b)x = , y= and z = 2 4 3 7 2 −2 −5 1 1
• (c)x = , y= and z =
• (d)x = 0, y = 7 6 4 2 and What is the name of this property?

Verify the property x × (y + z) = x × y + x × z of rational numbers by taking. −1 3 1

• (a)x = , y= , z= 2 4 4 −1 2 3
• (b)x = , y= , z= 2 3 4 −2 −4 −7
• (c)x = , y= , z= 3 6 9 −1 2 −3
• (d)x = , y= , z= 5 15 10

Use the distributivity of multiplication of rational numbers over addition to simplify 3 35 10 −5 8 16 5 24 1 4 5 15

• (a)× +
• (b)× + 2 7 21 3 8 × − 40 7 16 4
• (c)× −
• (d)4 9

Simplify 32 23 22 3 28 14

• (a)+ ×
• (b)× ÷ 5 11 15 7 15 5 3 −2 −5 7 1 1
• (c)+ ×
• (d)+ − 7 21 6 8 16 12

Identify the rational number that does not belong with the other three. Explain your reasoning −5 −1 −4 −7 , , , 11 2 9 3 19 171

The cost of metres of wire is Rs. . Find the cost of one metre 4 2 of the wire. 1445 17

A train travels km in hours. Find the speed of the train in 2 2 km/h.

If 16 shirts of equal size can be made out of 24m of cloth, how much cloth is needed for making one shirt?

of all the money in Hamid’s bank account is Rs. 77,000. How much money does Hamid have in his bank account? 1 1

A 117 m long rope is cut into equal pieces measuring 7 m each. 3 3 How many such small pieces are these? 1 1

of the class students are above average, are average and rest 6 4 are below average. If there are 48 students in all, how many students are below average in the class? 2 1

of total number of students of a school come by car while of 5 4 students come by bus to school. All the other students walk to school of which walk on their own and the rest are escorted by their parents. If 224 students come to school walking on their own, how many students study in that school?

Huma, Hubna and Seema received a total of Rs. 2,016 as monthly allowance from their mother such that Seema gets of what Huma gets and Hubna gets 1 times Seema’s share. How much money do the three sisters get individually?

A mother and her two daughters got a room constructed for Rs. 62,000. The elder daughter contributes of her mother’s contribution while the younger daughter contributes of her mother’s share. How much do the three contribute individually?

Tell which property allows you to compare 2 3 5 2 5 3 × × and × × 3 4 7 3 7 4

Name the property used in each of the following. 7 −3 −3 −7

• (i)− × = × 11 5 5 11 2 3 −1 −2 3 −2 −1 3 4 2 3 4 3 2
• (ii)− × + = × + × 1 4 − 4 1 4 − 4 + = 3 9 3 3 9 3
• (iii)+ + + −2 −2 2
• (iv)+0 =0+ =− 7 7 7 3 3 3 (v) ×1 = 1 × = 8 8 8

Find the multiplicative inverse of 1 1

• (i)−1
• (ii)3 8 3 1 13 5

Arrange the numbers , , in the descending order. 4 16 8 −14

The product of two rational numbers is . If one of the numbers be , find the other. −15

By what numbers should we multiply so that the product may −5 be ? −8

By what number should we multiply so that the product may be 24?

The product of two rational numbers is –7. If one of the number is –5, find the other?

Can you find a rational number whose multiplicative inverse is –1?

Find five rational numbers between 0 and 1.

Find two rational numbers whose absolute value is .

From a rope 40 metres long, pieces of equal size are cut. If the length of one piece is metre, find the number of such pieces.

5 metres long rope is cut into 12 equal pieces. What is the length of each piece?

Write the following rational numbers in the descending order. 8 −9 −3 2 , , , 0, 7 8 2 5 2 1 −5 −21

Find

• (i)0÷
• (ii)× × 3 3 7 10

On a winter day the temperature at a place in Himachal Pradesh was –16°C. Convert it in degree Fahrenheit (0F) by using the formula. C F – 32 = 5 9

Find the sum of additive inverse and multiplicative inverse of 7.

Find the product of additive inverse and multiplicative inverse of – .

The diagram shows the wingspans of different species of birds. Use the diagram to answer the question given below:

• (a)How much longer is the wingspan of an Albatross than the wingspan of a Sea gull?
• (b)How much longer is the wingspan of a Golden eagle than the wingspan of a Blue jay?

Shalini has to cut out circles of diameter 1 cm from an aluminium 3 1 strip of dimensions 8 cm by 1 cm. How many full circles can 4 4 Shalini cut? Also calculate the wastage of the aluminium strip.

One fruit salad recipe requires cup of sugar. Another recipe for the same fruit salad requires 2 tablespoons of sugar. If 1 tablespoon is equivalent to cup, how much more sugar does the first recipe require?

Four friends had a competition to see how far could they hop on one foot. The table given shows the distance covered by each. Name Distance covered (km) Seema Nancy Megha Soni

• (a)How farther did Soni hop than Nancy?
• (b)What is the total distance covered by Seema and Megha?
• (c)Who walked farther, Nancy or Megha?

The table given below shows the distances, in kilometres, between four villages of a state. To find the distance between two villages, locate the square where the row for one village and the column for the other village intersect.

• (a)Compare the distance between Himgaon and Rawalpur to Sonapur and Ramgarh?
• (b)If you drove from Himgaon to Sonapur and then from Sonapur to Rawalpur, how far would you drive?

The table shows the portion of some common materials that are recycled. Material Recycled Paper Aluminium cans Glass Scrap

• (a)Is the rational number expressing the amount of paper recycled 1 1 more than or less than ? 2 2
• (b)Which items have a recycled amount less than ?
• (c)Is the quantity of aluminium cans recycled more (or less) than half of the quantity of aluminium cans?
• (d)Arrange the rate of recycling the materials from the greatest to the smallest.

The overall width in cm of several wide-screen televisions are 97.28 cm, 4 1 98 cm, 98 cm and 97.94 cm. Express these numbers as rational 9 25 numbers in the form and arrange the widths in ascending order.

Roller Coaster at an amusement park is m high. If a new roller coaster is built that is times the height of the existing coaster, what will be the height of the new roller coaster?

Here is a table which gives the information about the total rainfall for several months compared to the average monthly rains of a town. Write each decimal in the form of rational number . Month Above/Below normal (in cm) May 2.6924 June 0.6096 July – 6.9088 August – 8.636

The average life expectancies of males for several states are shown in the table. Express each decimal in the form and arrange the states from the least to the greatest male life expectancy. State-wise data are included below; more indicators can be found in the “FACTFILE” section on the homepage for each state. State Male form Lowest terms Andhra Pradesh 61.6 Assam 57.1 Bihar 60.7 Gujarat 61.9 Haryana 64.1 Himachal Pradesh 65.1 Karnataka 62.4 Kerala 70.6 Madhya Pradesh 56.5 Maharashtra 64.5 Orissa 57.6 Punjab 66.9 Rajasthan 59.8 Tamil Nadu 63.7 Uttar Pradesh 58.9 West Bengal 62.8 India 60.8 Source: Registrar General of India (2003) SRS Based Abridged Lefe Tables. SRS Analytical Studies, Report No. 3 of 2003, New Delhi: Registrar General of India. The data are for the 1995-99 period; states subsequently divided are therefore included in their pre-partition states (Chhatisgarh in MP, Uttaranchal in UP and Jharkhand in Bihar) 7 1

A skirt that is 35 cm long has a hem of 3 cm. How long will the 8 8 skirt be if the hem is let down?

Manavi and Kuber each receives an equal allowance. The table shows the fraction of their allowance each deposits into his/her saving account and the fraction each spends at the mall. If allowance of each is Rs. 1260 find the amount left with each. Fraction of allowance Where money goes Manavi Kuber 1 1 Saving Account 2 3 1 3 Spend at mall 4 5 Left over ? ? 70 – 21 25 24 1. Given below is a magic square. Place the numbers 95 , –133 , 95 , 38 in the appropriate squares so that sum in each row, column and diagonal is equal. 32 18 4 − 14 38 38 38 −38 −18 104 ? ? − 57 152 22 − 20 ? ? 38 − 95 1 −16 45 60 19 − 38 57 114 Hint: (Rewrite each rational number in its lowest term.) 2. Solve the given crossword filling up the given boxes. Clues are given below for across as well as downward filling. Also, for across and down clues, clue number is written at the corner of the boxes. Answers of clues have to be filled in their respective boxes. 2 5 Down 1: and are _______ numbers. 3 4 a –a Down 2: The _______ inverse of is . b b Down 3: The addition and multiplication of whole number integers and rational numbers is _________. Down 4: Since doesn’t exist hence 0 has no ________. Down 5: The number line extends _______ on both the sides. Down 6: The _______ of two integers may not lead to the formation of another integer. Down 7: The multiplication of a number by its reciprocal gives_______. Down 8: Rational numbers can be represented on a _______ line. Across 1: There are _______ rational numbers between two integers. Across 2: The multiplication of rational numbers is commutative and ______. Across 3: The addition and ______ of two rational numbers lead to the formation of another rational number. Across 4: All the positive integers excluding 0 are known as _______ numbers. Across 5: For any rational a ; a ÷ 0 is _______. Across 6: Reciprocal is also known as the multiplicative _____________. 3. Break the Code Solve this riddle by reducing each rational number to its lowest term. The magnitude of the numerator of rational number so obtained gives you the letter you have to encircle in the word following it. Use the encircled letters to fill in the blanks given below: S.No. Rational Number Lowest Term Word 12 (1) SPIN 24 (2) TYPE –36 (3) WITH – 48 (4) HOST (5) SHARP –34 (6) GAIN –170 (7) PROOF (8) RAIN 92 (9) AWAY (10) SWEET –42 _____ _____ _____ _____ _____ _____ _____ _____ ____ ____ 1 2 3 4 5 6 7 8 9 10 UNIT-1 1 1 × (–12 ) ÷3 × (– 2 ) 2 5 ÷ 1 1 2 ÷ (– ÷ ( × –1 ) ) æ2 1ö ¸ç - ÷ è5 2ø 2 3 5 ) 3 × 3)6× ×(–2) –2 3 1 × + ÷(–2) Its 4 12 reciprocal + Its additive identity × Its multiplicative inverse × Its multiplicative inverse ÷ –1 1 ( ( 3 ( ÷(–25 +203 × Its additive inverse ÷–3 ( 4( 2 RATIONAL NUMBERS 31 MATHEMATICS Rough Work