The World of Numbers Class 9 Mathematics — Competency Based Questions

20 Questions | 75 Marks

A village in Rajasthan uses a traditional method of measuring land area. They divide a larger rectangular plot into smaller, equal-sized squares. If a farmer has a plot of

12 \frac{1}{2}

marlas and wants to divide it equally among his 5 children, what fraction of a marla will each child receive? 3marks

The temperature in Shimla drops from

5^{\circ}

C to

- 10^{\circ}

C overnight. A scientist wants to represent this change using integers. If the initial temperature is considered as a positive integer, what integer represents the final temperature? 1mark

Assertion: All integers are rational numbers.Reason: A rational number can be expressed in the form

\frac{p}{q}

, where

p

and

q

are integers and

q \neq 0

. [2 marks](a) Both assertion and reason are true, and the reason is the correct explanation of the assertion.(b) Both assertion and reason are true, but the reason is not the correct explanation of the assertion.(c) Assertion is true, but the reason is false.(d) Assertion is false, but the reason is true.

A baker uses a recipe that calls for

\frac{3}{4}

cup of flour. If he wants to make only

\frac{2}{3}

of the recipe, how much flour will he need? 2marks

Consider the number line. A point P is located such that it is

\frac{3}{5}

of the way from 0 to 1. Another point Q is located such that it is

\frac{2}{3}

of the way from 1 to 2. What are the values represented by points P and Q? 3marks

Assertion: The decimal expansion of

\frac{7}{8}

terminates.Reason: A rational number

\frac{p}{q}

(in its lowest terms) has a terminating decimal expansion if and only if the prime factors of the denominator

q

are only 2s and/or 5s. [2 marks](a) Both assertion and reason are true, and the reason is the correct explanation of the assertion.(b) Both assertion and reason are true, but the reason is not the correct explanation of the assertion.(c) Assertion is true, but the reason is false.(d) Assertion is false, but the reason is true.

A group of students is participating in a science fair. One student is demonstrating the concept of square roots. They construct a right-angled triangle with perpendicular sides of length 1 unit each. They then use the hypotenuse as the radius to draw an arc on a number line. What is the length represented by the hypotenuse, and what type of number is it? 3marks

Case Study: An ancient Indian astronomer is trying to calculate the circumference of a circular well. He knows the diameter is 14 meters. He uses the approximation

π \approx \frac{22}{7}

(a) What is the circumference of the well using this approximation? [2 marks]
(b) The astronomer knows that $π$ is an irrational number. What does this imply about the accuracy of his calculation? [1 mark]

Simplify:

12 \div (- \frac{3}{4}) + (- \frac{1}{2}) \times 6

. 3marks

A researcher is studying the population growth of bacteria. The initial population is 1000. It doubles every hour.(a) How many bacteria will there be after 3 hours? Express your answer using exponents. [1 mark](b) If the population after

n

hours is represented by

P (n)

, write a formula for

P (n)

. 1mark

Assertion: The number

0.1010010001...

is irrational.Reason: Irrational numbers have decimal expansions that are non-terminating and non-repeating. [2 marks](a) Both assertion and reason are true, and the reason is the correct explanation of the assertion.(b) Both assertion and reason are true, but the reason is not the correct explanation of the assertion.(c) Assertion is true, but the reason is false.(d) Assertion is false, but the reason is true.

A historian is examining ancient Indian coins. The total weight of a collection of coins is 150 grams. If each coin weighs

\frac{2}{5}

grams, how many coins are there in the collection? 2marks

Prove that

3 \sqrt{2}

is an irrational number. 3marks

A tailor has a piece of cloth of length

10 \frac{1}{4}

meters. He needs

2 \frac{1}{2}

meters for one shirt. How many shirts can he make from the cloth? 3marks

Assertion: Between any two distinct rational numbers, there exists another rational number.Reason: The set of rational numbers is dense. [2 marks](a) Both assertion and reason are true, and the reason is the correct explanation of the assertion.(b) Both assertion and reason are true, but the reason is not the correct explanation of the assertion.(c) Assertion is true, but the reason is false.(d) Assertion is false, but the reason is true.

A farmer plants mango trees in a rectangular orchard. The length of the orchard is

15 \frac{1}{3}

meters and the width is

10 \frac{2}{5}

meters. He wants to divide the orchard into smaller square plots, each of side length

\frac{1}{2}

meter. How many such square plots can he make? 4marks

Convert the decimal

0.7 \overline{142857}

into a fraction in its simplest form. 3marks

Application: A thermometer shows a temperature of

12^{\circ}

C. If the temperature drops by

15^{\circ}

C, what will be the new temperature? Represent this scenario using integers. 2marks

Consider the numbers

\frac{2}{3}

and

\frac{4}{5}

. Find three rational numbers between them. 3marks

Assertion: The number

\sqrt{9}

is irrational.Reason: A number is irrational if it cannot be expressed as a fraction

\frac{p}{q}

where

p

and

q

are integers and

q \neq 0

Answers

Each child will receive

\frac{1}{5}

12 \frac{1}{2}

marlas.

12 \frac{1}{2} = \frac{25}{2}

Amount for each child = $\frac{1}{5} \times \frac{25}{2} = \frac{5}{2}$ marlas.
This is not the question asked. The question is "what fraction of a marla will each child receive?".
The total amount is $12 \frac{1}{2} = \frac{25}{2}$ marlas.
If divided among 5 children, each child gets $\frac{25 / 2}{5} = \frac{25}{2 \times 5} = \frac{5}{2}$ marlas.
The question asks for the fraction of a marla each child will receive. This phrasing is slightly ambiguous. Assuming it means what fraction of one marla each child gets:
Each child gets $\frac{5}{2}$ marlas. This means each child gets $2.5$ marlas.
If it means the portion of the total plot, each child gets $\frac{1}{5}$ of the plot.

Re-interpreting: "what fraction of a marla will each child receive?" might be asking for the value of $\frac{p}{q}$ if each child gets $\frac{p}{q}$ marlas. In this case, each child gets $\frac{5}{2}$ marlas. So the fraction is $\frac{5}{2}$ .

Let's assume the question implies finding the value of each share in marlas, and then expressing that value as a fraction.
Each child receives $\frac{1}{5}$ of $12 \frac{1}{2}$ marlas.
$12 \frac{1}{2} = \frac{25}{2}$ marlas.
Each child receives $\frac{1}{5} \times \frac{25}{2} = \frac{5}{2}$ marlas.
So, each child receives $\frac{5}{2}$ marlas. The fraction is $\frac{5}{2}$ . [3 marks]

The initial temperature is

+ 5^{\circ}

C. A drop of

15^{\circ}

C means subtracting 15.Final temperature =

5 + (- 15) = 5 - 15 = - 10^{\circ}

C.The integer representing the final temperature is -10. 1mark

Assertion: All integers are rational numbers. (True, because any integer

n

can be written as

\frac{n}{1}

)Reason: A rational number can be expressed in the form

\frac{p}{q}

, where

p

and

q

are integers and

q \neq 0

. (True, by definition)The reason correctly explains why integers are rational numbers. [2 marks](a) Both assertion and reason are true, and the reason is the correct explanation of the assertion.

The recipe calls for

\frac{3}{4}

cup of flour. The baker wants to make

\frac{2}{3}

of the recipe.Amount of flour needed =

\frac{2}{3} \times \frac{3}{4} = \frac{2 \times 3}{3 \times 4} = \frac{6}{12} = \frac{1}{2}

cup. 2marks

Point P is

\frac{3}{5}

of the way from 0 to 1. So, P represents

\frac{3}{5} \times 1 = \frac{3}{5}

.Point Q is

\frac{2}{3}

of the way from 1 to 2. The interval between 1 and 2 is 1 unit.The position of Q from 0 is

1 + (\frac{2}{3} \times 1) = 1 + \frac{2}{3} = \frac{3}{3} + \frac{2}{3} = \frac{5}{3}

.So, P represents

\frac{3}{5}

and Q represents

\frac{5}{3}

. 3marks

Assertion: The decimal expansion of

\frac{7}{8}

terminates. (True, since

8 = 2^{3}

, which has only prime factor 2)Reason: A rational number

\frac{p}{q}

(in its lowest terms) has a terminating decimal expansion if and only if the prime factors of the denominator

q

are only 2s and/or 5s. (True, this is the condition for terminating decimals)The reason correctly explains the assertion. [2 marks](a) Both assertion and reason are true, and the reason is the correct explanation of the assertion.

The lengths of the perpendicular sides are 1 unit each. By the Pythagorean theorem, the square of the hypotenuse (

d^{2}

) is

1^{2} + 1^{2} = 1 + 1 = 2

.So, the length of the hypotenuse is

d = \sqrt{2}

units.

\sqrt{2}

is an irrational number. 3marks

(a) Circumference =

π \times diameter

.Using

π \approx \frac{22}{7}

and diameter = 14 m.Circumference =

\frac{22}{7} \times 14 = 22 \times 2 = 44

meters. [2 marks](b) Since

π

is an irrational number, its decimal expansion is non-terminating and non-repeating. This means any calculation using an approximation of

π

(like

\frac{22}{7}

or 3.14) will not be the exact circumference. It will be an approximation. 1mark

12 \div (- \frac{3}{4}) + (- \frac{1}{2}) \times 6

= 12 \times (- \frac{4}{3}) + (- \frac{6}{2})

= - \frac{48}{3} + (- 3)

= - 16 - 3

$= - 19$ . [3 marks]

(a) Initial population = 1000.After 1 hour:

1000 \times 2 = 1000 \times 2^{1}

After 2 hours:

1000 \times 2 \times 2 = 1000 \times 2^{2}

After 3 hours:

1000 \times 2 \times 2 \times 2 = 1000 \times 2^{3} = 1000 \times 8 = 8000

bacteria. [1 mark](b) The formula for the population after

n

hours is

P (n) = 1000 \times 2^{n}

. 1mark

Assertion: The number

0.1010010001...

is irrational. (True, as the decimal is non-terminating and the pattern of zeros between 1s is increasing, so it's non-repeating)Reason: Irrational numbers have decimal expansions that are non-terminating and non-repeating. (True, this is the definition of irrational numbers)The reason correctly explains the assertion. [2 marks](a) Both assertion and reason are true, and the reason is the correct explanation of the assertion.

Total weight of coins = 150 grams.Weight of each coin =

\frac{2}{5}

grams.Number of coins = Total weight / Weight of each coinNumber of coins =

150 \div \frac{2}{5} = 150 \times \frac{5}{2} = \frac{150 \times 5}{2} = 75 \times 5 = 375

coins. 2marks

Assume

3 \sqrt{2}

is rational. Then

3 \sqrt{2} = \frac{p}{q}

for some integers

p, q

with

q \neq 0

and gcd(p, q) = 1.Squaring both sides:

(3 \sqrt{2})^{2} = {(\frac{p}{q})}^{2}

9 \times 2 = \frac{p^{2}}{q^{2}}

18 = \frac{p^{2}}{q^{2}}

18 q^{2} = p^{2}

This shows that $p^{2}$ is divisible by 18, which means $p^{2}$ is divisible by 2 and 9. If $p^{2}$ is divisible by 2, then $p$ must be divisible by 2. Let $p = 2 k$ .

18 q^{2} = (2 k)^{2} = 4 k^{2}

9 q^{2} = 2 k^{2}

This shows that $2 k^{2}$ is divisible by 9. Since 2 and 9 are coprime, $k^{2}$ must be divisible by 9, which means $k$ is divisible by 3. Let $k = 3 m$ .
Then $p = 2 k = 2 (3 m) = 6 m$ .
Now substitute $p = 6 m$ back into $18 q^{2} = p^{2}$ :

18 q^{2} = (6 m)^{2} = 36 m^{2}

q^{2} = \frac{36 m^{2}}{18} = 2 m^{2}

This shows that $q^{2}$ is divisible by 2, which implies $q$ is divisible by 2.
So, both $p$ and $q$ are divisible by 2. This contradicts our assumption that $p$ and $q$ have no common factors other than 1.
Therefore, $3 \sqrt{2}$ must be irrational. [3 marks]

Total length of cloth =

10 \frac{1}{4}

meters =

\frac{41}{4}

meters.Length needed for one shirt =

2 \frac{1}{2}

meters =

\frac{5}{2}

meters.Number of shirts = Total length / Length per shirtNumber of shirts =

\frac{41 / 4}{5 / 2} = \frac{41}{4} \times \frac{2}{5} = \frac{41 \times 2}{4 \times 5} = \frac{82}{20} = \frac{41}{10} = 4.1

.Since the tailor can only make whole shirts, he can make 4 shirts. 3marks

Assertion: Between any two distinct rational numbers, there exists another rational number. (True)Reason: The set of rational numbers is dense. (True, density is the property that between any two distinct rational numbers there is another rational number.)The reason correctly explains the assertion. [2 marks](a) Both assertion and reason are true, and the reason is the correct explanation of the assertion.

Length of orchard =

15 \frac{1}{3}

meters =

\frac{46}{3}

meters.Width of orchard =

10 \frac{2}{5}

meters =

\frac{52}{5}

meters.Side length of square plot =

\frac{1}{2}

meter.Number of plots along the length =

\frac{Length of orchard}{Side length of plot} = \frac{46 / 3}{1 / 2} = \frac{46}{3} \times 2 = \frac{92}{3}

.Number of plots along the width =

\frac{Width of orchard}{Side length of plot} = \frac{52 / 5}{1 / 2} = \frac{52}{5} \times 2 = \frac{104}{5}

.Total number of plots = (Number of plots along length)

\times

(Number of plots along width)Total number of plots =

\frac{92}{3} \times \frac{104}{5} = \frac{9568}{15}

.This result assumes we can have fractional plots, which is not practical for making distinct plots. However, the question implies division into smaller squares.

Let's re-read: "divide the orchard into smaller square plots, each of side length $\frac{1}{2}$ meter." This means we should find how many $\frac{1}{2}$ m by $\frac{1}{2}$ m squares fit.

Area of orchard = Length $\times$ Width = $\frac{46}{3} \times \frac{52}{5} = \frac{2392}{15}$ square meters.
Area of each square plot = Side $\times$ Side = $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$ square meters.
Number of plots = $\frac{Area of orchard}{Area of each plot} = \frac{2392 / 15}{1 / 4} = \frac{2392}{15} \times 4 = \frac{9568}{15}$ .
As a mixed number, $\frac{9568}{15} = 637 \frac{13}{15}$ .
This means 637 full square plots can be made, with some leftover area. Since the question asks "how many such square plots can he make?", it implies the maximum number of whole plots.
So, 637 plots. [4 marks]

Let

x = 0.7 \overline{142857}

.The repeating block has 6 digits.Multiply by

10^{6}

1000000 x = 714285.7142857...

Original equation:

x = 0.7142857...

Subtracting the original equation from the multiplied one:

1000000 x - x = 714285.7142857... - 0.7142857...

999999 x = 714285

x = \frac{714285}{999999}

Now simplify this fraction.
Sum of digits of numerator: $7 + 1 + 4 + 2 + 8 + 5 = 27$ (divisible by 9).
Sum of digits of denominator: $9 \times 6 = 54$ (divisible by 9).
Divide both by 9:

714285 \div 9 = 79365

999999 \div 9 = 111111

So, $x = \frac{79365}{111111}$ .
Sum of digits of numerator: $7 + 9 + 3 + 6 + 5 = 30$ (divisible by 3).
Sum of digits of denominator: $1 + 1 + 1 + 1 + 1 + 1 = 6$ (divisible by 3).
Divide both by 3:

79365 \div 3 = 26455

111111 \div 3 = 37037

So, $x = \frac{26455}{37037}$ .
The number 142857 is related to $1 / 7$ . $1 / 7 = 0.142857...$ . $1 / 7 \times 5 = 5 / 7 = 0.7142857...$
So, the fraction should be $\frac{5}{7}$ .
Let's check if $\frac{714285}{999999} = \frac{5}{7}$ .

714285 \times 7 = 4999995

$999999 \times 5 = 4999995$ .
So the fraction is indeed $\frac{5}{7}$ . [3 marks]

Initial temperature =

12^{\circ}

C.Drop in temperature =

15^{\circ}

C.New temperature =

12 - 15 = - 3^{\circ}

C.Representing this using integers: The initial temperature is represented by the integer +12. The drop is represented by subtracting 15. The final temperature is represented by the integer -3. 2marks

Let's find the average of

\frac{2}{3}

and

\frac{4}{5}

.Average =

\frac{1}{2} (\frac{2}{3} + \frac{4}{5}) = \frac{1}{2} (\frac{10 + 12}{15}) = \frac{1}{2} \times \frac{22}{15} = \frac{11}{15}

.So,

\frac{11}{15}

is one rational number between them.

Now, find the average of $\frac{2}{3}$ and $\frac{11}{15}$ :
Average = $\frac{1}{2} (\frac{2}{3} + \frac{11}{15}) = \frac{1}{2} (\frac{10 + 11}{15}) = \frac{1}{2} \times \frac{21}{15} = \frac{21}{30} = \frac{7}{10}$ .
So, $\frac{7}{10}$ is another rational number.

Now, find the average of $\frac{11}{15}$ and $\frac{4}{5}$ :
Average = $\frac{1}{2} (\frac{11}{15} + \frac{4}{5}) = \frac{1}{2} (\frac{11 + 12}{15}) = \frac{1}{2} \times \frac{23}{15} = \frac{23}{30}$ .
So, $\frac{23}{30}$ is a third rational number.
Three rational numbers between $\frac{2}{3}$ and $\frac{4}{5}$ are $\frac{11}{15}$ , $\frac{7}{10}$ , and $\frac{23}{30}$ . [3 marks]

Assertion: The number

\sqrt{9}

is irrational. (False,

\sqrt{9} = 3

, which is a rational number as it can be written as

\frac{3}{1}

)Reason: A number is irrational if it cannot be expressed as a fraction

\frac{p}{q}

where

p

and

q

are integers and

q \neq 0

. (True, this is the definition of an irrational number)The assertion is false, but the reason is true. [2 marks](d) Assertion is false, but the reason is true.

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The World of Numbers — Competency Based Questions

Competency based questions

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