MATHEMATICS
6 Probability
Let’s Recall
Sanvi and Aditya are playing a game with their father. Their father rolls the dice. What are the possible numbers that can come up? Is it possible that he gets the number 7?
A dice has only 6 numbers, so it is not possible to get the number 7 on rolling the dice.
Probability is a way of guessing how likely something is to happen. It helps us figure out if something is sure, impossible, likely, unlikely or equally likely to happen compared to other events.
Let us look at the chance of the numbers coming up on a dice when rolled.
Getting a number from 1 to 6:
Sure, since there are only 6 numbers on a dice.
Let’s Warm-up
Getting an even number:
Equally likely, since 3 out of 6 numbers are even on a dice.
Getting the number 7:
Impossible, since the number does not appear on a dice.
Write the chance of the given events happening using words like sure, impossible, likely, unlikely and equally likely.
1 A dog flying.
2 The cycle of day and night.
3 Getting heads when a coin is tossed.
4 Drawing a blue marble from a bag containing 6 blue and 2 green marbles.
5 Drawing a green marble from a bag containing 10 yellow and 3 green marbles.
I scored out of 5.
Mean, Median and Mode
Understanding Probability
Naina and her brother Sam are visiting the local village fair. They go to the Wheel of Fortune stall. They are excited to play the game since they will get a prize each time the wheel stops at an even number. The wheel has numbers from 1 to 10. Naina says 8 and spins the wheel. The wheel stops at 8.
Sam: Wow! You guessed right! Now, it’s my turn. I want the number 12.
Naina laughs upon hearing this and says, ‘No chance! You lose a point.’
In other words, we can say that probability helps us make smart guesses, similar to predicting outcomes when playing games, spinning wheels, flipping coins or rolling dice.
For example, the chance of the wheel stopping at a number greater than 10 is an impossible event. The chance of the wheel stopping at numbers from 1 to 10 is a sure event. Let us recall some terms.
The chance of an event happening is its probability.
An outcome is the result of some activity.
Event: The collection of some or all possible outcomes of an activity is called an Event
Total Outcomes: The total number of possible outcomes is called total outcomes. E.g., in a wheel showing 5 colours, the total outcomes are 5.
Favourable Outcomes: The number of favourable outcomes help us predict the chances of an event happening.
The chance of the occurrence of an event can be measured mathematically, and it is called the probability of the event. Events which have no chance of occurring have a probability of 0.
In general, the probability of a sure event is 1 and the probability of an event that has no chance of occurring or is impossible is 0. For any other event, probability varies between 0 to 1.
Total numbers on the wheel = Total Outcomes = Possible Outcomes = 10
Sure: The wheel stopping at a number from 1 to 10.
Favourable outcomes = 10 (Same as possible outcomes)
Likely: The wheel stopping at a number greater than 4.
Favourable outcomes = 6
(Numbers 5, 6, 7, 8, 9 and 10)
Favourable outcomes > 5 (More than half of the possible outcomes)
Impossible: The wheel stopping at the number 12.
Favourable outcomes = 0
Unlikely: The wheel stopping at a number less than 3.
Favourable outcomes = 2
(Numbers 1 and 2)
Favourable outcomes < 5 (Less than half of the possible outcomes)
Equally Likely: The wheel stopping at a number from 1 to 5.
Favourable outcomes = 5 (Numbers 1, 2, 3, 4 and 5)
Favourable outcomes = Same has half the possible outcomes
Theoretical Probability
Since the wheel shows numbers from 1 to 10, anytime the wheel is spun, it will stop at a number from 1 to 10. In other words, the possible outcomes of the event are the wheel stopping at the numbers 1 to 10.
The total outcomes will be 10. Let us see how.
Probability of the Occurrence of an Event
The term ‘chance’ when used in real life is measured mathematically and is called the probability of the event.
Let E be an event. Then the probability of an event E is the ratio of the number of outcomes favourable to the event E to the number of all possible outcomes of the experiment.
P(E) = Number of outcomes favourable to the event E Number of all possible outcomes of the experiment
Where, P(E) = probability of an event E and P(E) lies between 0 and 1.
In the above case, the probability of the wheel stopping at an even number can be calculated using the formula.
Here, E = the event of the wheel stopping at an even number.
Total outcomes = 10 (total numbers on the wheel)
Favourable outcomes = 5
Applying the formula,
P(E) = Number of outcomes favourable to the event E
Number of all possible outcomes of the experiment
P(E) = Total number of even numbers
Total numbers on the wheel
P(E) = 5 10 = 1 2
So, the theoretical probability of the wheel stopping at an even number is 5 10 or 1 2.
Theoretical probability gives exact probabilities based on mathematical principles in ideal conditions. It’s like predicting the chance of something based on what we know and the rules.
For example, if we roll a fair six-sided dice, we know each number has a 1 in 6 chance of showing up because there are 6 sides.
Example 1: Rahul tosses a coin before a cricket match.
What is the probability that he gets:
1 heads?
2 tails?
Think and Tell
Is the probability of the wheel stopping at an odd number, the same as the probability of the wheel stopping at an even number?
The possible outcomes when a coin is tossed are heads or tails.
Total possible outcomes of the experiment = 2 (either heads or tails).
1 He gets heads.
E = Event of getting a head
P(E) = Numbers of heads
Total number of outcomes = 1 2
Therefore, the possibility of getting a head is 1 2 .
2 He gets tails.
E = Event of getting a tail
P(E) = Number of tails
Total number of outcomes = 1 2
Therefore, the possibility of getting a tail is 1 2 .
Did You Know?
Anil Kumar Gain, an Indian mathematician, studied stochastic processes and queueing theory, which are ways of understanding and predicting random events and waiting times.
Example 2: A fruit box contains 4 oranges and 8 apples. If a fruit is taken out randomly, what is the probability of it being an apple?
E = Event of getting an apple
P(E) = Number of apples
Total number of fruits = 8 12 = 2 3
Therefore, the possibility of getting an apple is 2 3 .
Example 3: Even numbers between 1 and 25 are written on identical slips, put in a box and mixed up. If a slip is drawn at random, what is the probability of getting:
1 a one-digit number? 2 a prime number?
3 an odd number? 4 a number greater than 1?
There are 12 even numbers between 1 and 25. They are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22 and 24.
So, the total number of possible outcomes is 12.
1 There are 4 one-digit numbers - 2, 4, 6 and 8.
So, the number of favourable outcomes = 4
P(E) = Number of 1-digit even numbers Total number of favourable outcomes
P(one-digit number ) = 4 12 or 1 3
2 2 is the only even prime number.
So, the number of favourable outcomes = 1
P(even number ) = P(E) = 1 12
3 There are no odd numbers.
So, the number of favourable outcomes = 0
P(odd number) = P(E) = 0 12 or 0.
4 All the 12 even numbers are greater than 1 - 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22 and 24.
So, the number of favourable outcomes is 12.
P(number greater than 11) = P(E) = 12 12 = 1
Example 4: At a village fair, you have a choice of spinning Spinner A or Spinner B. You win a prize if the spinner lands on a section with a star. Which spinner should you choose if you want a better chance of winning? Give a reason for your answer.
Spinner A Spinner B
The ratio for Spinner A is greater than the ratio for Spinner B. Hence, I should choose Spinner A for a better chance of winning.
Example 5: In a game, the probability of correctly guessing which box contains a tennis ball is 1 5 . How many winners would be expected if 60 contestants played the game?
Probability of correctly guessing a box that contains a tennis ball = 1 5
Number of contestants who played the game = 60
Number of contestants expected to win the game (N) = 60 × 1 5 = 12
So, about 12 contestants are expected to win the game.
Example 6: On a game show, a contestant randomly draws a chip from a bag and replaces it. Each chip denotes either a win or a loss.
The theoretical probability of drawing a winning chip is 3 10. The bag contains 9 winning chips.
1 How many chips are in the bag?
The theoretical probability of drawing a winning chip is 3 10.
This also means that the ratio of the number of winning chips to the total number of chips in the bag is 3:10. If the bag contains 9 winning chips, then the total number of chips in the bag = 9 × 10 3 = 30.
2 Out of 20 contestants, how many do you expect to draw a winning chip?
Number of contestants in the game show = 20
Number of contestants expected to draw a winning chip = 3 10 × 20 = 6.
Theoretical probability is used for simple, well-defined situations like rolling dice and tossing coins. When we find the set of all possible outcomes of more than one event, it is called the sample space. You can use tables and tree diagrams to find the sample space of two or more events.
Spinner A
Spinner B
For example, you randomly choose a crust and style of pizza.
Select Crust
Select Style
Use a tree diagram to find the sample space.
There are 8 different outcomes in the sample space. The 8 different pizzas possible are: CM, CH, CP, CV
Example 7: A black dice and a blue dice are rolled. If all numbers are equally likely, what is the probability that the sum of the numbers that appear on their faces is equal to 5?
The table below shows the sample space for rolling a particular sum on two dice.
There are 36 possible outcomes.
The event of rolling a sum of 5 on two dice has four favourable outcomes.
P(rolling a sum of 5 on two dice) = 4 36 or 1 9 .
A well-shuffled deck of animal picture cards is placed face down on a table. Animals
What is the probability that an animal card is drawn at random:
a shows a picture of a bird?
E = Event of getting a card that shows a picture of a bird
= Number of cards that show a bird picture
= Therefore, the possibility of getting a card that shows a picture of a bird is .
b does not show a picture of a bird?
E = Event of getting a card that does not show a picture of a bird
Therefore, the possibility of getting a card that does not show a picture of a bird is .
Mexican Hawaiian Pepperoni Vegetarian
Crispy Thin
Soft Pan
Crispy Thin Soft Pan
c The animal card and the bird cards are placed side by side. How many different combinations can be made if one of each card type is picked?
28 different combinations are possible. They are:
CD, CT, CG, CR
GD, GT, GG, GR
Do It Yourself 6A
1 Fill in the blanks.
a The probability of an event having no chance of occurring is (0/1).
b The probability of a sure event is (0/1).
c The possible outcomes when you toss a coin are and .
d On rolling a dice, the event of getting a prime number has possible outcomes.
2 A glass jar contains 40 marbles in different colours such as black, green, blue and yellow. The probability of drawing a single green marble is 1 5 . What does this mean?
a There are 5 green marbles in the glass jar.
b There are 8 green marbles in the glass jar.
c There are 10 green marbles in the glass jar.
d There is only one green marble in the glass jar.
3 Each letter of the alphabet is printed on an index card. What is the theoretical probability of randomly choosing any letter except Z?
4 Neeta is playing a game using a fair coin. Contestants win the game if the coin lands tail up.
a What is the theoretical probability that the coin will land tail up?
b If 250 contestants play the game, about how many of them are expected to win?
5 Amit rolls a six-sided dice labelled 1 to 6.
a Find the probability of rolling a 4.
b Find the probability of rolling an odd number.
c If Amit rolls the dice 12 times, how many times should he expect a number greater than 4 to be rolled?
6 A game board has a spinner with 10 equal-sized sections, of which 4 are green, 3 are blue, 2 are yellow and 1 is black. What is the sum of the probabilities of the pointer landing in the green, blue, yellow and black sections?
7 A 12-sided solid has equal-sized faces numbered 1 to 12.
a Find the probability of getting a number greater than 10.
b Find the probability of getting a number less than 5.
c If the 12-sided solid is rolled 200 times, how many times would you expect either a 4, 6 or 9 to be rolled?
8 Write 2 events based on the situation given in Q7. Write their theoretical probability.
9 A black dice and a blue dice are rolled. If all numbers are equally likely, what is the probability that the sum of the numbers that appear on their faces is equal to:
a 6?
b 10?
10 Draw a spinner having numbers 1 to 7 and the number 4 with a theoretical probability of 1 4
11 At a school fair, you have the choice of randomly picking a ball from Basket A or Basket B. Basket A has 5 green balls, 3 black balls and 8 yellow balls. Basket B has 7 green balls, 4 black balls and 9 yellow balls. You can win a prize if you pick a black ball. Which basket should you choose if you want a better chance of winning?
12 Navin is getting dressed. He considers two different shirts, three pairs of pants and three pairs of shoes. He chooses one of each of the articles at random.
Shirts
Collared shirt
T-shirt
Pants Shoes
Khakis
Jeans
Capris
a How many different outfits are possible?
b What is the probability that he will wear his jeans but not his sneakers?
Experimental Probability
Sneakers
Flip-flops
Sandals
The wheel of fortune that Naina and Sam spin records the number of times the wheel lands on an even number. The picture shows a digital scorecard to record the number of people who have won a prize.
Sam is excited to know that 60 people out of a total of 100 people who have spun the wheel got lucky! The wheel has 10 sections, numbered 1 to 10. 5 sections are odd, and the remaining 5 are even.
Even though the theoretical probability of spinning an even number is equal to that of an odd number, the actual results of the experiment have turned out to be different.
Thus, the experimental probability of spinning an even number in this case is 60 100 or 60%. This is more than the theoretical probability of spinning an even number which is 50%.
Such an experiment where we cannot predict the exact result is called a random experiment
Here, each outcome has an equal chance of occurring.
A Random Experiment is one whose outcome cannot be predicted exactly in advance. That is, a random experiment is an experiment, which, when repeated under identical conditions, does not produce the same outcome every time.
Experimental probability gives the probability based on observed data. We can find the experimental probability by physically doing tests or experiments. The accuracy of experimental probability depends on the number of trials conducted. That is, as the number of trials increases, we get more accurate results.
The experimental probability of an event E can be calculated using the formula,
P(event) ꞊ Number of times the event occurs
Total number of trials
For example, if we toss a coin 100 times and it comes up heads 47 times, the experimental probability of getting heads is 47 out of 100.
Experimental probability is used when theoretical probability is difficult to calculate.
Think and Tell
What will be the approximate experimental probability of getting heads if we toss a coin 1000 times?
Example 8: A coin is tossed 50 times to see how often tails shows up. If tails appears 20 times, what is the experimental probability of getting tails? Write as fraction, decimal and percentage.
E = Event of getting tails
Experimental Probability, P(E) = Number of times tails appears Total number of trials = 20 50 = 2 5
Therefore, the experimental probability of getting tails is 2 5, 0.40, or 40%.
Example 9: Amit randomly pulls a coloured block from a bag. He records the colour and then puts the block back into the bag. The table shows the results of his experiment. If he does the experiment 50 times, predict the number of times he will pull a black block from the bag.
The total number of trials in Amit’s experiment = 3 + 4 + 6 + 7 = 20
P(black) = Number of black blocks
Total number of trials = 4 20
In 20 trials, Amit pulled a black block 4 times.
In 50 trials, Amit would pull a black block, 50 × 4 20 ꞊ 10 times
So, Amit will pull a black block about 10 times.
Example 10: Which of the following is an example of a random experiment?
1 Taking a marble out of a bag containing coloured marbles.
Taking a marble out of a bag containing coloured marbles is an example of a random experiment.
2 Measuring the length of a table.
Measuring the length of a table is not an example of a random experiment.
Example 11: In tennis, Govind serves an ace - a ball that can’t be returned, he serves 4 out of the 10 times.
1 What is the experimental probability that Govind will serve an ace in the first match of the next game?
P(E) = Number of aces Total number of serves = 4 10 = 2 5
Therefore, the experimental probability of Govind serving an ace in the first match of the next game is 2 5 or 0.40 or 40%.
2 Make a prediction about how many aces Govind will have for the next 40 serves.
In 10 serves, Govind serves an ace 4 times.
In 40 serves, Govind would serve an ace, 40 × 4 10 = 16 times.
So, Govind will serve an ace about 16 times.
Example 12: A bag contains 50 marbles. You randomly draw a marble from the bag, record its colour and then replace it. The table shows the results after 30 draws.
1 Find the experimental probability of drawing a red marble.
The total number of trials in the experiment = 30
P(red) = P(E) = 9 30 or 3 10
2 Predict the total number of red marbles in the bag.
In 30 trials, a red marble was drawn 9 times.
In 50 trials, a red marble would be drawn about, 50 × 9 30 = 15 times
Example 13: We held a survey of all the students in our school to find out how many likes to dance. The results are shown in the given table.
If we choose a student at random, what is the probability that the student:
1 likes to dance?
55 students like to dance.
Probability that a student likes to dance = 55 110 = 1 2
2 do not like to dance?
55 students do not like to dance.
Probability that a student does not don’t like to dance = 55 110 = 1 2
3 is a boy who likes to dance?
There are 25 boys who likes to dance.
Probability that a student is a boy who likes to dance = 25 110 or 5 22
4 is a girl who does not like to dance?
There are 20 girls who does not like to dance.
Probability that a student is a girl who does not like to dance = 20 110 or 2 11
The bar graph shows the results of rolling a dice 300 times. How does the experimental probability of rolling an odd number compare with the theoretical probability?
We first find the experimental probability of rolling an odd number. The bar graph shows ones, threes and fives.
So, an odd number was rolled + + = times in a total of 300 rolls.
P(odd) = Number of times an odd number was rolled Total number of rolls �
Next, we find the theoretical probability of rolling an odd number.
P(odd) = Number of favourable outcomes
Number of possible outcomes �
The experimental probability of rolling an odd number is %, which is close to the theoretical probability of %.
Do It Yourself 6B
1 Which of the following are examples of a random experiment?
a Tossing a coin
c Throwing a ball
b Drawing a black card from a deck of playing cards
d Rolling two dice together
2 To play a game, you spin the spinner shown. You win if the arrow lands in one of the areas marked ‘WIN’. Leena played this game many times and recorded her results. She won 8 times and lost 40 times. Use Leena’s data to find the experimental probability of winning this game.
3 During a 24-hour period, the ratio of pop songs to rap songs played on a radio station is 60:75.
a What is the experimental probability that the next song played will be a rap song?
b Out of the next 90 songs, how many would you expect to be pop songs?
4
A board game uses a bag of 105 lettered tiles. You randomly choose a tile and then return it to the bag. The table shows the number of vowels and the number of consonants after 50 draws.
Predict the number of vowels in the bag.
5 The Venn diagram shows two sports played by 90 Grade 8 students. What is the probability that a randomly chosen student plays:
a both tennis and squash?
c tennis?
b neither tennis nor squash?
d tennis or squash, but not both?
6 We held a survey of all the students in our school to find out how many own a mobile phone. The results are shown in the two-way table. Owns a mobile phone Does not own a mobile phone
If we choose a student at random, what is the probability that the student:
a owns a mobile phone?
c is a boy who owns a mobile phone?
b does not own a mobile phone?
d is a girl who does not own a mobile phone?
7 You randomly select 200 pairs of jeans and find 5 defective pairs. About how many pairs of jeans do you expect to be defective in a shipment of 5000?
Challenge
1 The table shows the result of flipping two coins. Read the statements and choose the correct option.
Assertion (A): The experimental probability of flipping two tails is 11 50 .
Reason (R): Experimental probability of an event = Number of times the event occurs Total number of trials
a Both A and R are true, and the R is the correct explanation of A.
b Both A and R are true, and the R is not the correct explanation of A.
c A is true, but the R is false.
d A is false, but the R is true.
Points to Remember
• Probability is the concept which numerically measures the degree of certainty of the occurrence of an event.
• An operation which can produce some well-defined outcomes is called an experiment.
• An experiment in which all possible outcomes are known and the exact outcome cannot be predicted in advance is called a random experiment.
• The collection of all or some of the possible outcomes is called an event.
• Probability of an event P(E) = Favourable outcomes
Total number of outcomes
• The probability of an event always lies between 0 and 1.
• The probability of a sure event is 1 and that of an impossible event is 0.
• Probability cannot be negative.
• The sum of the probabilities of all possible outcomes of an event is 1.
Math Lab
Tossing Coins
Aim: To find the experimental probability of an event.
Collaboration & Experiential Learning
Materials Required: A coin of ₹1, paper and pencil
Setting: In groups of 3
Method:
1 Toss the coin 10 times consecutively and record the result in a table.
2 Count the number of times a head appears and the number of times a tail appears.
3 Calculate the probability of getting heads by dividing the number of times heads appeared by the total number of tosses. Similarly, calculate the probability of getting tails.
4 Increase the number of tosses to 20 and take the cumulative result of the previous tosses. Again, calculate the probabilities.
5 As the number of tosses increases, the probabilities tend to get closer to 1 2.
Chapter Checkup
1 Fill in the blanks.
a The probability of a sure event is
b If an event cannot occur, then its probability is .
c The probability of selecting P from the word ‘SPECIAL’ is .
d The probability of an event can’t be more than
e If a dice is rolled once, the probability of getting an even prime number is
2 Write all the possible outcomes of the following events.
a Flipping a fair coin
b Rolling a standard six-sided dice
c Selecting a single letter at random from the English alphabet
d Flipping two fair coins simultaneously
e Rolling two standard six-sided dice simultaneously
3 Sneha draws a card at random from the cards given. What is the probability of drawing a card that shows a shape with:
a three sides?
b four sides?
c more than four sides?
d no sides?
4 Find the probability of getting an A on a random selection of a letter from the word ‘APPRECIATION’.
5 A box contains 100 bolts, two-fifths of which are rusted. One bolt is drawn at random from the box. Find the probability that it is:
a a rusted bolt.
b not a rusted bolt.
6 200 apples are packed in a carton out of which 30 apples are rotten. If one apple is taken out from the carton at random, find the probability of getting a:
a good apple.
b rotten apple.
7 A college student has to select a stream with the given subject combinations:
a Physics, Chemistry, Mathematics
c Accountancy, Mathematics, Economics
b Physics, Chemistry, Biology
d English, Hindi, Social Sciences
Find the probability of getting mathematics as a subject on a random selection of the stream.
8 The bar graph shows the result of spinning a wheel 150 times. Compare the theoretical and experimental probabilities of the events.
a Wheel stopping at green
b Wheel stopping at red
c Wheel stopping at blue or purple
d Wheel stopping at a colour
9 Prime numbers between 1 and 25 are written on identical slips, put in a box and mixed up. If a slip is drawn randomly, what is the probability of getting:
a a one-digit number?
c an odd number?
b an even number?
d a number greater than 11?
10 The keypad of a mobile phone has digits from 0 to 9 on it. If a person enters a digit at random, what is the probability of typing:
a an even number?
b a multiple of 5?
c a factor of 9?
11 In a store, 50 pairs of bangles are on display. The probability that a customer will choose a pair of pearl bangles is 6 25 . How many pearl bangles are on display?
1 Amita has a ₹1 coin, a ₹2 coin, a ₹5 coin and a ₹10 coin in her purse. She needs to pay for a pencil that costs ₹4. Namita says the probability of her using the ₹5 coin is 25%, while Ranita insists that it is 50%. Who is correct - Namita or Ranita?
2 Statement: 300 marbles are placed in a jar, of which 50 are blue. One marble is drawn from the jar at random.
Conclusion I: The probability of drawing a blue marble is one-sixth.
Conclusion II: The probability of drawing a non-blue marble is 80%.
Which of the following conclusions follow?
a Only conclusion I follows
c Either I or II follows
Case Study
b Only conclusion II follows
d Neither I nor II follows
Cross Curricular & Value Development
Observing the Biodiversity!
A biodiversity park is a special area of wild nature where groups of native plants, animals and birds live together. Rashmi was visiting the biodiversity park with her classmates. The children decided to observe the types of birds present there. Over a week, they observed 100 birds, of which 40 were sparrows, 30 were pigeons, 20 were crows and 10 were some other species. Answer the following questions.
1 What is the probability of randomly spotting a sparrow in the park?
2 If Rashmi sees a bird, what is the probability that it is either a pigeon or a crow?
3 The probability of not seeing a sparrow is
4 If Rashmi sees a bird in the park, it is more likely to be a pigeon than a crow. True or False?
5 Have you ever been to a biodiversity park? What did you see there? What should you keep in mind while visiting the park?
14 Exponents and Powers
Let's Recall
We have learnt about numbers that are perfect squares. A number that can be arranged in the form of a square using unit squares is called a perfect square. We have also learnt about perfect cubes. A number that can be arranged in the form of a cube using unit cubes is called a perfect cube. For example, look at the set of unit squares and unit cubes in 64.
The unit squares are arranged in such a way that they form a bigger square. So, 64 is a perfect square.
The unit cubes are arranged in such a way that they form a bigger cube. So, 64 is a perfect cube.
82 = 8 × 8 = 64
43 = 4 × 4 × 4 = 64
We can find the last digit of the square and the cube of any number by looking at the digit in that number's ones place. For example, look at the square numbers and the cube numbers.
Both squares end in 6
142 = 196 and 162 = 256
If a number has 4 or 6 in its ones place, the number's square ends in 6.
Letʼs Warm-up
Fill in the blanks.
1 The square of 11 is and its cube is .
This cube ends in 8 This cube ends in 2
123 = 1728 and 183 = 5832
If a number has 2 in its ones place, its cube ends in 8 and vice versa.
2 If the units digit of a number is 3, then the units digit of its square is and that of its cube is .
3 The prime factorisation of 48 is .
a Is it a perfect square? (Yes/No)
b The smallest number that 48 should be divided by to get a perfect square is .
4 Reeti says, “125 is a perfect square” whereas Meeti says, “125 is a perfect cube”. Who is correct?
Exponents
Bacteria are very small microorganisms that reproduce very quickly. Under favourable environmental conditions, the bacterial growth shows an increase in the number of bacteria in a population. During binary fission, with each generation, the bacteria split into identical copies every time.
The table given below shows how quickly bacteria grows. Based on the table, let us see how can we find the population of bacteria in its 10th generation.
Exponents of a Rational Number
We saw that for generation 5, the population of bacteria was 32. This was expressed as: 32 = 2 × 2 × 2 × 2 × 2 = 25
It is read as 2 raised to the power 5 or the 5th power of 2.
So, we can say that for generation n, the population of bacteria will be 2
Therefore, for generation 10, the population of bacteria will be 210, that is: 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1024.
Here, 2 is a rational number. What if the numbers are other types of rational numbers, such as fractions or decimals? Let us see how can we write 0.7 multiplied 10 times or 4 7 multiplied 5 times.
To write such numbers, we use exponents. This helps us read such numbers easily and in less time.
7 × 4 7 × 4 7 × 4 7 × 4 7 = 4 7 5
Did You Know?
The Indian mathematician, Aryabhata, in the 5th century CE, used the concept of powers with base 10 to express large numbers. He developed a place value system where powers of 10 were implied by position, which is fundamental in our understanding of exponents today.
A number that can be expressed in exponential form can be either a positive or a negative rational number.
For example, a positive rational number 6 7 when multiplied by itself 6 times is written as:
6 7 6 = 6 7 × 6 7 × 6 7 × 6 7 × 6 7 × 6 7 = 46,656 1,17,649
It is read as 6 7 raised to the power 6 or the 6th power of 6 7.
Similarly, a negative rational number, say −7 when multiplied by itself 7 times is written as: (−7)7 = (−7) × (−7) × (−7) × (−7) × (−7) × (−7) × (−7) = −8,23,543
It is read as −7 raised to the power 7 or the 7th power of −7.
Therefore, if any rational number, say a, is multiplied by itself n times, then a × a × a × a … n times = an
It is read as a raised to the power n or the nth power of a.
Positive Integral Exponent of a Rational Number
A number that can be expressed in exponential form can also be either a positive or a negative rational number. For example, a positive rational number 5 6 when multiplied by itself 6 times is written as:
5 6 6 = 5 6 × 5 6 × 5 6 × 5 6 × 5 6 × 5
Similarly, a negative rational number, say 6 7 when multiplied by itself 7 times is written as:
We know that positive and negative integers, fractions and decimal numbers all together constitute rational numbers. So, if any rational number, say a b is multiplied n times, then a b n = a b × a b × a b … n times = an bn, where a b is the base and n is the exponent.
Example 1: Find the value of:
2: Evaluate:
14 • Exponents and Powers
Negative Integral Exponent of a Rational Number
We have already learnt about how to solve numbers with positive exponents. Now, let us understand how to solve numbers with negative exponents.
Look at the given pattern.
23 = 2 × 2 × 2 = 8 = 16 × 1 2
22 = 2 × 2 = 4 = 8 × 1 2
21 = 2 = 4 × 1 2
20 = 1 = 2 × 1 2
Think and Tell
As the exponent decreases by 1, what do you think about the rule of this pattern? Can we extend this pattern further?
We know that the set of integers is represented by {…, −3, −2, −1, 0, 1, 2, 3, …}. So, if using integers, continuing this pattern with negative exponents, this is what we get as their values.
2 1 = 1 × 1 2 =
2 3 = 1 22 × 1 2 =
4 × 1 2 = 1 8 = 1 23 and so on.
On observing the pattern, we can say that in each case, exponential numbers are reciprocals of each other.
Remember!
The reciprocal (multiplicative inverse) of a number is 1 divided by that number. For example, the reciprocal of 3 is 1 3 .
So, if a is any number and n is any positive integer, then its reciprocal is written as a n. It is read as a raised to the power −n.
a n = 1 a
If a b be any rational number and n be a positive integer, then
Thus,
Example 3: Express the numbers as negative exponents. 1
Example 4: Evaluate the numbers.
1 (−4)−4
3 (1.2)−2
(1.2)−2 =
Example 5: By what number should
Let the required number be a. Then,
6
Think and Tell Is 2 3 2 = 3 2 2 ? Why?
be multiplied so that the product is
Remember!
When a number has a negative exponent, it represents repeated division. Negative exponents tell us the number of times a number is divided by itself.
So, the required number is 45 49 .
Example 6: Write the expanded form of 345.28 as negative exponents. 345.28 = 300 + 40 + 5 + 0.2 + 0.08 = 300 + 40 + 5 + 2 10 + 8 100 = (3 × 100) + (4 × 10) + (5 × 1) + 1 10 2 × + 1 10 8 × = 3 × 10² + 4 × 10¹ + 5 × 100 + 2 × 10 ¹ + 8 × 10 ²
Write the expanded form of 23.456 as negative exponents. 23.456 = 20 + 3 + 0.4 + 0.05 + 0.006 = 20 + 3 + 4 10 + + = 2 × 10 + 3 × 1 + 4 × 1 10 + × 1 + × 1 = 2 × 101 + 3 × 100 + 4 × 10−1 + × 10 + × 10
14 • Exponents and Powers
Do It Yourself 14A
1 Expand and find the value of the numbers.
2 Express numbers as exponents.
3 Express the numbers with negative exponents. a
4 Write the expanded form of the numbers using exponents.
5 Which one is larger? a 2⁵ or 5² b 2⁷ or 7²
6 Solve and express your answer with negative exponents.
7 By what number should 3 3 8 be completely divided so that the quotient is 16?
8 By what number should 3 5 6 be multiplied so that the product is (0.8)4?
9 An insect weighs only 5–3 g. But each day, it can eat 26 times its body weight. How much food can the insect eat on the first day?
1 Vivan’s school had two branches. After the school’s annual day function, the videos for the events were uploaded on the Youtube channels of both the branches. Since the time it was uploaded, the online traffic has been monitored. The table shows the number of views the video got each week.
The pattern for the number of views for both the branches continued for few weeks.
a What is the ratio of the growth rate of the number of views for both the branches?
b What is the percentage increase in the number of views for the first branch every week?
c What will be the ratio of the number of views for the second branch in week 2 and week 6?
Application of Exponents and Laws of Exponents
We know that the table given below shows how quickly bacteria grow. Each generation increases its population to 2 times.
Exponential Value
What will be bacteria population after 4 more generations?
We already know that each generation increases its population to 2 times.
Population after 4 more generations = Population after 6 generations × 2⁴ = 2⁶ × 2⁴
We know that, am × an = a(m + n)
Population after 4 more generations = 2(6 + 4) = 1024
What if the powers are negative? Let us see!
We know that, 2(
So, am × an = am + n holds true for any negative integers m and n and rational number a also.
Apart from this, we have some other laws of exponents.
Applying Laws of Exponents
We can apply the laws of exponents in various forms.
For example, let us find the value of
Here, like the laws for positive exponents, we have to find the quotient of two rational numbers when the bases are the same but the powers are different. So, in this case, we apply the law:
Similarly, we can apply other laws as well.
Example 7: Find the value of x for which 7x ÷ 7 5 = 78 . 7x ÷ 7−5 = 78 ⇒ 7x−(−5) = 78 ⇒ 7x + 5 = 78 ⇒ x + 5 = 8 ⇒ x = 8 – 5 ⇒ x = 3
Example 8: Evaluate:
Example 9: Find the value of:
Example 10: Simplify: 64 × 5−3 × 34 243 × 22 × 10
64 × 5−3 × 34 243 × 22 × 10 = 43 × 5−3 × 34 35 × 22 × 2 × 5 = 43 × 5−3 − 1 × 34 − 5 22 + 1 = (22)3 × 5−4 × 3−1 23 = 26 × 5−4 × 3−1 23 = 26 − 3 × 5−4 × 3−1 = 23 × 5−4 × 3−1 = 23 × 1
Use of Exponents
In our daily lives, we come across numbers like 2 cm, 766 km, 3,59,000 km, 18,76,53,00,000 m, 0.000000045 mm, etc. in the fields of science and engineering. Among these, numbers like 3,59,000 km and 18,76,53,00,000 m are very large and a number like 0.000000045 mm is very small. Reading these numbers in the usual form is very difficult and time-consuming. So, to read and write these numbers easily in less time, we express them in exponential form using the standard form or scientific notation.
Scientific notation is a way of expressing very large or very small numbers as a product of a decimal number (ranging from 1.0 to 10) and an exponent of 10, where the exponent tells us the number of places the decimal point is moved. For example, 2.9 × 102, 3.6587 × 10−3, etc. 2 4
Therefore, a number can be expressed in the standard form (or using scientific notation) as: k × 10n, where 0 < k < 10 and n is an integer
Now, let us see how we can express 3,59,000 km and 0.000000045 mm using scientific notation.
359000 km
The decimal is moved 5 places to the left.
359000. = 359 × 1000
= 3.59 × 100 × 103 = 3.59 × 102 × 103 = 3.59 × 102 + 3 = 3.59 × 105 km
Example 11: Write the numbers in standard form.
0.000000045 mm
The decimal is moved 8 places to the right.
0.000000045 = 45 1000000000 = 45 109 = 4.5 × 10 109 = 4.5 × 101 − 9 = 4.5 × 10−8 mm
1 548900000 = 5489 × 100000 = 5.489 × 103 × 105 = 5.489 × 108
2 0.0000000003465 = 3465 10000000000000 = 3465 1013 = 3.465 × 103 1013 = 3.465 × 103 – 13 = 3.465 × 10−10
Example 12: Write the numbers in the usual form.
1 5.67 × 105 = 567 100 × 105 = 567 102 × 105 = 567 × 105 – 2 = 567 × 103 = 5,67,000
2 8.912 × 10−8 = 8912 1000 × 1 108 = 8912 103 × 1 108 = 8912 103 + 8 = 8912 1011 = 8912 100000000000 = 0.00000008912
Example 13: Which is greater, 7.42 × 104 or 5.6 × 107?
7.42 × 104 = 742 100 × 104 = 742 × 104 – 2 = 742 × 102 = 742 × 100 = 74,200
5.6 × 107 = 56 10 × 107 = 56 × 107 – 1 = 56 × 106 = 56 × 10,00,000 = 5,60,00,000
Here, 5,60,00,000 > 74,200.
So, 5.6 × 107 > 7.42 × 104.
Example 14: Find the value of m if 5² × m = 125.
5² × m = 5³
m = 5³ 5²
m = 5³ ²
m = 5
Did You Know?
The distance from the Earth to Proxima Centauri (the star nearest to us after the Sun) is about 4.0 x 1013 km.
Example 15: Find the value of a if 2−5 × 4a + 1 8² × 2−1 = 4²
2−5 × 4a + 1 8² × 2−1 = 2−5 × (22)a + 1 (23)2 × 2−1 = (22)2
2−5 × 22a + 2 26 × 2−1 = 24
2−5 + 2a + 2 26−1 = 24
22a − 3 25 = 24 22a − 3 − 5 = 24
a − 8 = 24 On comparing, 2a − 8 = 4 2a = 12 a = 6
Example 16: The mass of the Earth is 5.97 × 1024 kg and that of the moon is 7.34 × 1022 kg. What is the combined mass of the two? Express your answer using scientific notation.
Total mass = 5.97 × 1024 + 7.34 × 1022 kg = 5.97 × 1022 × 102 + 7.34 × 1022 kg = (5.97 × 102 + 7.34) × 1022 kg = (597 + 7.34) × 1022 kg = 604.34 × 1022 kg = 6.0434 × 1024 kg
Example 17: The distance from the Sun to the Canis Major Dwarf Galaxy is around 1.47 × 1017 miles. Light travels at a speed of 5.88 × 1012 miles per year. How long does it take (in years) for light to travel from the Sun to the Canis Major Dwarf Galaxy?
Distance from the Sun to the Canis Major Dwarf Galaxy = 1.47 × 1017 miles
Speed of light = 5.88 × 1012 miles per year
The time light takes to travel from the Sun to the galaxy = 1.47 × 1017 5.88 × 1012 =
= 1.47 × 105 5.88
= 1.47 5.88 × 105 = 0.25 × 105
= 2.5 × 10−1 × 105
= 2.5 × 10−1 + 5 = 2.5 × 104 years
Which is smaller, 5.09 × 10 6 or 2.13 × 10 8 and by how much?
5.09 × 10−6 = 509 100 × 1 106
= 509 10 × 1 106 = ____________
2.13 × 10−8 = 213 × 1 10
= 213 × 1 108 = 213 10 =
In scientific notation, ‘× 10’ NEVER means multiplication. It represents the exponent, that is, the number of times we need to move the decimal point towards the left or right to obtain the actual value.
2.5 × 104 = 2.5 × 10 = 25 2.5 × 104 = 2.5 × 10,000 = 25,000
Error Alert!
Do It Yourself 14B
1 Write the numbers in standard form.
a 10980000 b 2300000000000000
c 10.446678 d 453.98785
2 Evaluate:
a 4 9 4 × 4 9 –2 b 7 13 4 ÷ 7 13 2
c 2 3 2 2 d 1 2 × 4 7
e 5 6 ÷ 2 3 2 f (3–2+ 8–3+ (–2)–4)0
3 Write the numbers in the usual form.
a 4.21 × 10⁸ b 1.03 × 10⁸
c 40.857 × 10–2 d 3.467 × 10–5
4 Compare the numbers.
a 2.11 × 10², 1.67 × 10–2 b 7.3 × 10–5, 0.7 × 10–8
5 Solve as directed. Give your answers in standard form.
a 6.757869 × 10⁵ + 980.436 × 10³ b (3.87 × 10–7) + (1.02 × 10–7)
c 3.6548 × 10⁸ + 154.142 × 10⁶ d 46157.22 × 10² – 23.1572 × 10⁵
6 Find the value of: a (5−9 × 5³) × 15⁵
2 5 –4 × 3 8 –4 × 3 5 4 c 3 4 –5 ÷ 3 4 –2 d ((−2) 2 + (1.6) 3) × (0.8)³ e 2 5 –3 ÷ (2.4) 3 × (0.5)0
7 Simplify using laws of exponents.
a 72 × 3−6 × 4² 81 × 16 × 27−2
20 × 3−3 × 40 36 × 5 5 × 45
c 9 × t −3 18 × 3 5 × t 5 d 16 × 2m + 1 − 4 × 2m 16 × 2m + 1 − 2 × 2m + 2
8 Write the missing number in the equation: –1 5 2 × –1 –2 × 511 = 55
9 Find the value of x.
a 1 2 = 1 8 b 2x = 32 –1 x
10 An ant carries a small amount of food that is 2.78 × 10–3 inches long. Another ant carries an amount of food that is 4.56 × 10−5 inches long. How long are the pieces of food carried by the ants altogether?
11 In the population of a country, 7.7 × 103 people spoke French as their first language and 3.2 × 104 people spoke German. Which language had more speakers and by how much?
12 A swarm of mosquitoes contains as many as 80 million mosquitos per square mile on a 1500 square mile area of land. How many mosquitoes should the swarm have to cover the entire area of land?
13 Scientists discovered that the size of the Antarctic Ocean is 20,330,000 sq. km. How big is the Arctic Ocean, if it is 3 4 th of the size of the Antarctic Ocean? Express your answer in standard form.
Challenge
1 What is the value of 2x + y?
Critical Thinking
Read the statements and choose the option that correctly answers the question above.
Statement 1: 2x = 8
Statement 2: 2y = 16
a Statement 1 alone is sufficient to answer while statement 2 alone is not sufficient.
b Statement 2 alone is sufficient to answer while statement 1 alone is not sufficient.
c Both statements together are sufficient to answer.
d Both statements together are not sufficient to answer.
e Either of the statement alone is sufficient to answer.
Points to Remember
• This representation of numbers with a base and exponent is the exponential form of numbers.
• If any integer, say a, is multiplied by itself n times, then a × a × a × a … n times = an. It is read as: a raised to the power n or the nth power of a.
• If any rational number, say a b is multiplied by itself n times, then a b n = a b × a b × a b n times = a × a × a...× n times b × b × b...× n times = an bn , where a b is the base and n is the exponent.
• If a is any number and n is any positive integer, then its reciprocal is written as a n. It is read as: a raised to the power −n
• Some laws of exponents are:
a
where m is a positive integer
• A number can be expressed in the standard form or scientific notation as k × 10n, where 0 < k < 10 and n is an integer.
Math Lab
Settings: In groups of 4
Communication & Collaboration
Counting Exponentially!!
Materials Required: Chart paper; Small rectangular pieces of paper; Counters or rajma beans
Method: All 5 members of each group must follow these steps.
1 Take a chart paper. Place a rectangular piece of paper and paste two counters on the piece of paper. This represents 21 which is equal to 2.
2 Each rectangular piece is to be followed by two more rectangular pieces and each piece of paper has 2 counters.
3 Repeat this process multiple times.
4 The number of counters on each piece of paper shows the base and rectangular pieces of paper placed in a row show the exponent. The total number of counters shows the answer.
5 How many counters will be there in the fourth step? How many counters will be there in the sixth step?
6 Repeat the same activity by keeping 3 counters on each rectangular piece of paper.
Chapter Checkup
1 Write the expanded form of the given numbers using exponents.
2 Identify if the statements are written in standard form or usual form. If the numbers are in standard form, express them in usual form and vice versa.
a The size of a plant cell is 1.275 × 10−5 m.
b The total volume of water on Earth is 1.386 × 109 cu. km.
c The weight of an electron is 0.0000000000000000000000000000911 kg.
3 Which of the following is in usual form? Give reason.
a 8.09 × 105
4 Express the given numbers as directed.
a 256 as a power of 2
b 0.000000654879 c 9.21 × 10−21
b 59,049 as a power of 3 c 78,125 as a power of 5
5 Write the reciprocal of the given expressions using exponents.
a 7 9 × 7 9 × 7 9 × 7 9
6 Solve.
d [(2)5 × (−1.5)2] × 1 4 3
7 What power of 49 is equal to 710?
8 Arrange the numbers in ascending and descending orders.
a 210, 323, 642, 128−2, 83 b 310, 273, 7292, 18−2, 93
9 A teacher asked her students to find the multiplicative inverse of x−2. Tanya said the answer is –x2, whereas Sayani said the answer is x−3. Who do you think is right? Explain your answer.
10 Find the value of:
+
11 Simplify using laws of exponents.
128 × 36 × 4−2
× 36 × 27−1
12 Find the value of x in:
210 ÷ 26 = 2−4 × 23x −
13 Simplify: x p + l × x q + p x l + q (x p × x q × x l)2
14 A farm that has 6 × 105 m² of land produces 2.4 × 106 kg of cotton each year. What is the average number of cotton plants that are planted per metre?
15 At a warehouse, each crate weighs 2.2 × 103 kg. If there are 4 × 102 crates, what is their combined weight?
16 The diameters of the planets Jupiter and Saturn are 13.9822 × 104 km and 1.165 × 105 km, respectively. Compare the diameters of these two planets.
17 The size of a plant cell is 0.00002155 m and that of a red blood cell is 0.0000000015 m. Is the plant cell double the size of a red blood cell?
18 The area of the Pacific Ocean is 1.55 × 108 sq. km. The area of the Atlantic Ocean is 1.0646 × 108 sq. km. How many times greater is the area of Pacific Ocean than the Atlantic Ocean? Express your answer in scientific notation.
Critical Thinking
Read the statement and choose the correct option.
Assertion (A): (a³ × b³) + 2000 = –1000, where a = 5 and b = –2.
Reason (R): (a³ × b³) = (ab)³ and a0 = 1.
a Both A and R are true, and R is the correct explanation of A.
b Both A and R are true, but R is not the correct explanation of A.
c A is true, but R is false.
d A is false, but R is true.
Find the value of y if y m + 3 × (y²)m – 1 y³m−9 = 1024
Case Study
Elemental Atomic Masses: A Comparative Table
The table shows the mass of an atom for some elements (in kg). Read the data carefully and answer the questions given below.
1 Which of the following is the lightest element?
a Hydrogen
c Gold
b Oxygen
d Iron
2 Which of the following is correct?
a Carbon is the heaviest element.
b Oxygen is 0.6 times heavier than carbon.
c The mass of all the elements is written in standard form.
d The mass of gold is lighter than iron.
3 Which element is approximately 5 times heavier than carbon?
1.6735 × 10–27 kg
1.994 × 10–26 kg
2.656 × 10–26 kg
2.325 × 10–26 kg
× 10–26 kg
× 10–25 kg
4 What is the ratio of the weights (approx) of the heaviest element and the lightest element?
Cross Curricular
About the Book
Imagine Mathematics seamlessly bridges the gap between abstract mathematics and real-world relevance, offering engaging narratives, examples and illustrations that inspire young minds to explore the beauty and power of mathematical thinking. Aligned with the NEP 2020, this book is tailored to make mathematics anxiety-free, encouraging learners to envision mathematical concepts rather than memorize them. The ultimate objective is to cultivate in learners a lifelong appreciation for this vital discipline.
Key Features
• Let’s Recall: Introductory page with a quick recall of concepts learnt in previous grades
• Real Life Connect: Introduction to a new concept related to real-life situations
• Examples: Solved problems showing the correct method and complete solution
• Do It Together: Guided practice for learners with clues and hints to help solve problems
• Think and Tell: Probing questions to stimulate Higher Order Thinking Skills (HOTS)
• Error Alert: A simple tip-off to help avoid misconceptions and common mistakes
• Remember: Key points for easy recollection
• Did You Know? Interesting facts related to the application of concept
• Math Lab: Fun cross-curricular activities
• Challenge: Critical thinking questions to enhance problem-solving and analytical thinking skills
• Real-Life Maths: Scenario-based questions to apply theory to real-life situations
• QR Codes: Digital integration through the app to promote self-learning and practice
About Uolo
Uolo partners with K-12 schools to provide technology-enabled learning programs. We believe that pedagogy and technology must come together to deliver scalable learning experiences that generate measurable outcomes. Uolo is trusted by over 10,000 schools across India, Southeast Asia and the Middle East.
ISBN 978-81-979482-5-1
