9th Class Mathematics - Chapter 1

Question 1 Linear Equations

The sum of three consecutive integers is forty-two. Find the three integers.

Solution

Let the three consecutive integers be represented as:

First Integer = x Second Integer = x + 1 Third Integer = x + 2

According to the given condition, their sum is 42:

x + (x + 1) + (x + 2) = 42

x + x + 1 + x + 2 = 42

3 x + 3 = 42

3 x = 42 - 3

3 x = 39

x = 393 = 13

Now, calculate each integer:

1 st integer = x = 132 nd integer = x + 1 = 13 + 1 = 143 rd integer = x + 2 = 13 + 2 = 15

Conclusion: The three consecutive integers are 13, 14, and 15.

Question 2 Rationalization & Geometry

The diagram shows a right-angled triangle

ΔABC

in which the length of

AC

is

(\sqrt 3 + \sqrt 5)

cm. The area of

ΔABC

is

(1 + \sqrt 15)

cm². Find the length

AB

in the form

(a \sqrt 3 + b \sqrt 5)

cm where

a

and

b

are integers.

Solution

Given Parameters:

Base length AC = \sqrt 3 + \sqrt 5 cm Area of ΔABC = 1 + \sqrt 15 cm 2

To Find: Height length $AB$

We know that the area of a right-angled triangle is:

Area of ΔABC = 12 \times (Base \times Height)

Area of ΔABC = 12 \times (AC \times AB)

1 + \sqrt 15 = 12 \times [ (\sqrt 3 + \sqrt 5) \times AB ]

Multiply both sides by 2:

2(1 + \sqrt 15) = (\sqrt 3 + \sqrt 5) \times AB

2 + 2 \sqrt 15 = (\sqrt 3 + \sqrt 5) \times AB

AB = 2(1 + \sqrt 15) \sqrt 3 + \sqrt 5 = 2 + 2 \sqrt 15 \sqrt 3 + \sqrt 5

Rationalize the denominator by multiplying by the conjugate $\sqrt 3 - \sqrt 5$ :

AB = 2 + 2 \sqrt 15 \sqrt 3 + \sqrt 5 \times \sqrt 3 - \sqrt 5 \sqrt 3 - \sqrt 5

= (2 + 2 \sqrt 15)(\sqrt 3 - \sqrt 5) (\sqrt 3) 2 - (\sqrt 5) 2

Expand the numerator:

= 2 \sqrt 3 - 2 \sqrt 5 + 2 \sqrt 15 \times 3 - 2 \sqrt 15 \times 5 3 - 5

= 2 \sqrt 3 - 2 \sqrt 5 + 2 \sqrt 45 - 2 \sqrt 75 -2

Simplify radical terms: $\sqrt 45 = \sqrt 9 \times 5 = 3 \sqrt 5$ and $\sqrt 75 = \sqrt 25 \times 3 = 5 \sqrt 3$ :

= 2(\sqrt 3 - \sqrt 5 + 3 \sqrt 5 - 5 \sqrt 3) -2

Cancel out factor 2, leaving negative sign:

= \sqrt 3 - 5 \sqrt 3 - \sqrt 5 + 3 \sqrt 5 -1

= -4 \sqrt 3 + 2 \sqrt 5 -1

= 4 \sqrt 3 - 2 \sqrt 5

Conclusion: The length of side

AB

is 4√3 - 2√5 cm (where

a = 4

and

b = -2

).

Question 3 Geometry & Exponents

A rectangle has sides of length

(2 + \sqrt 18)

m and

(5 - 4 \sqrt 2)

m. Express the area of the rectangle in the form

a + b \sqrt 2

where

a

and

b

are integers.

Solution

Given Parameters:

Length (L) = 2 + \sqrt 18 m Width (W) = 5 - 4 \sqrt 2 m

Simplifying components first:

\sqrt 18 = \sqrt 9 \times 2 = 3 \sqrt 2

4 \sqrt 2 = 2 \times 2 \sqrt 2 = 2 \times \sqrt 2 = 2 \sqrt 2 [ ∵ x \sqrt x = \sqrt x]

Thus, we can write the simplified dimensions as:

L = 2 + 3 \sqrt 2 m and W = 5 - 2 \sqrt 2 m

Calculate Area of the Rectangle:

Area (A) = L \times W

A = (2 + 3 \sqrt 2) \times (5 - 2 \sqrt 2)

Expand using foil method:

= 2(5) - 2(2 \sqrt 2) + 3 \sqrt 2 (5) - (3 \sqrt 2)(2 \sqrt 2)

= 10 - 4 \sqrt 2 + 15 \sqrt 2 - 6(2)

= 10 + 11 \sqrt 2 - 12

= -2 + 11 \sqrt 2

Conclusion: The area of the rectangle is -2 + 11√2 m² (where

a = -2

and

b = 11

).

Question 4 System of Equations

Find two numbers whose sum is 68 and whose difference is 22.

Solution

Let the two numbers be x and y.

According to the given conditions:

Sum condition: x + y = 68 — Eq. (1) Difference condition: x - y = 22 — Eq. (2)

Add Eq. (1) and Eq. (2) to eliminate y:

(x + y) + (x - y) = 68 + 22

2 x = 90

x = 902 = 45

Substitute the value of x in Eq. (1):

45 + y = 68

y = 68 - 45

y = 23

Conclusion: The required two numbers are 45 and 23.

Question 5 Fahrenheit conversion

The weather in Lahore was unusually warm during the summer of 2024. The TV news reported temperatures as high as 48°C. By using the formula

°F = 95 °C + 32

, find the temperature on the Fahrenheit scale.

Solution

Given Celsius temperature:

C = 48°C

Use the conversion formula:

°F = 95 °C + 32

°F = 95 \times (48) + 32

= 9 \times 9.6 + 32 [ ∵ 485 = 9.6 ]

= 86.4 + 32

= 118.4°F

Conclusion: The reported temperature on the Fahrenheit scale is 118.4°F.

Question 6 Age Problems

The sum of the ages of the father and son is 72. Six years ago, the father's age was 2 times the age of the son. What was the son's age six years ago?

Solution

Let current ages be:

Father's age = x Son's age = y

By given condition: The sum of current ages is 72:

x + y = 72 — Eq. (1)

Six years ago, their ages were:

Father's age = x - 6 Son's age = y - 6

Given: Six years ago, the father was twice as old as the son:

x - 6 = 2(y - 6)

x - 6 = 2 y - 12

x - 2 y = -12 + 6

x - 2 y = -6 — Eq. (2)

Subtract Eq. (2) from Eq. (1) to eliminate x:

(x + y) - (x - 2 y) = 72 - (-6)

x + y - x + 2 y = 72 + 6

3 y = 78

y = 783 = 26

Substitute y = 26 in Eq. (1) to find father's current age:

x + 26 = 72

x = 72 - 26 = 46

Calculate their ages six years ago:

Son's age six years ago = y - 6 = 26 - 6 = 20 years Father's age six years ago = x - 6 = 46 - 6 = 40 years

Conclusion: The son's age six years ago was 20 years.

Question 7 Business Math

Mirha bought a toy for Rs. 1500 and sold it for Rs. 1520. What was her profit percentage?

Solution

Given parameters:

Cost Price (CP) = Rs. 1500 Selling Price (SP) = Rs. 1520

Calculate the profit amount:

Profit = SP - CP

Profit = 1520 - 1500 = Rs. 20

Use the profit percentage formula:

Profit Percentage = Profit Cost Price \times 100%

= 201500 \times 100%

Cancel out zeros from 1500 and 100:

= 2015 %

= 43 % = 1.33%

Conclusion: Mirha's profit percentage was 1.33%.

Question 8 Tax Calculations

The annual income of Tayab is Rs. 960,000, while the exempted amount is Rs. 130,000. How much tax would he have to pay at the rate of 0.75%.

Solution

Given parameters:

Annual Income = Rs. 960,000 Exempted Amount = Rs. 130,000 Tax Rate = 0.75%

Calculate taxable income:

Taxable Income = Annual Income - Exempted Amount

Taxable Income = 960,000 - 130,000 = Rs. 830,000

Calculate tax amount:

Tax Amount = Taxable Income \times Tax Rate

= 830,000 \times 0.75%

= 830,000 \times 0.75 100

= 8,300 \times 0.75

= 8,300 \times 75100

= 83 \times 75 = Rs. 6,225

Conclusion: Tayab will have to pay a tax amount of Rs. 6,225.

Question 9 Markup calculations

Find the compound markup on Rs. 375,000 for one year at the rate of 14% compounded markup annually.

Solution

Given parameters:

Principal Amount (P) = Rs. 375,000 Time (T) = 1 year Rate (R) = 14%

Calculate markup amount:

Markup = P \times T \times R

= 375,000 \times 1 \times 14%

= 375,000 \times 14100

= 3,750 \times 14 = Rs. 52,500

Conclusion: The compound markup amount is Rs. 52,500.

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