9th Class Mathematics Unit 1 Real Numbers Numericals with Solutions

1.3.1.The sum of three consecutive integers is forty-two, find the three integers.

Given

Consecutive integers x, (x+1), (x+2)

Working

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

3x+3=42 ⇒ 3x=39 ⇒ x=13

Result

Integers 13, 14, 15

1.3.2.The diagram shows right angled \triangle ABC in which the length of \overline{AC} is (\sqrt3+\sqrt5)\ cm. The area of \triangle ABC is (1+\sqrt{15})\ cm^2. Find the length \overline{AB} in the form (a\sqrt3+b\sqrt5)\ cm, where a and b are integers.

Given

overline{AC}=(sqrt3+sqrt5) cm,quad Area=(1+sqrt{15}) cm²

Formula

Area of right triangle Area=frac12 × overline{AC} × overline{AB}

Working

(1+sqrt{15})=frac12(sqrt3+sqrt5)overline{AB} ⇒ (2+2sqrt{15})=(sqrt3+sqrt5)overline{AB}

overline{AB}=frac{2+2sqrt{15}{sqrt3+sqrt5} × sqrt3-sqrt5sqrt3-sqrt5=2sqrt3-2sqrt5+6sqrt5-10sqrt3-2

Result

AB overline{AB}=(4sqrt3-2sqrt5) cm

1.3.3.A rectangle has sides of length 2+\sqrt{18}\ m and \left(5-\frac{4}{\sqrt2}\right)\ m. Express the area of the rectangle in the form a+b\sqrt2, where a and b are integers.

Formula

Area Area=L × W=(2+sqrt{18}) × left(5-4sqrt2right)

=10-8sqrt2+5sqrt{18}-sqrt{18}left(4sqrt2right)=10-4sqrt2+15sqrt2-12

Result

Area (11sqrt2-2) m²

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

Given

x+y=68,quad x-y=22

Adding both 2x=90 ⇒ x=45

Subtracting both 2y=46 ⇒ y=23

Result

x=45, y=23

1.3.5.The weather in Lahore was unusually warm during the summer of 2024. The TV news reported temperature as high as 48°C. By using the formula, (°F=\frac{9}{5}°C+32) find the temperature as Fahrenheit scale.

Formula

{}^circF=frac95{}^circC+32

Working

{}^circF=frac95 × 48+32

Result

{}^circF=118.4

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

Given

Son's current age=x,quad Father's current age=72-x

Working

Six years ago 66-x=2(x-6)

66-x=2x-12 ⇒ 78=3x ⇒ x=26

Result

Son's age six years ago 26-6=20 years

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

Given

CP=1500,quad SP=1520

Formula

Profit Profit=SP-CP=1520-1500=20

Working

Profit \% 201500 × 100%

Result

1.33%

1.3.8.The annual income of Tayyab is Rs. 9,60,000, while the exempted amount is Rs. 1,30,000. How much tax would he have to pay at the rate of 0.75%?

Formula

Taxable income Total Income-Exempted Amount

Working

960000-130000=830000

Formula

Tax amount Taxable Income × Tax Rate

Working

830000 × 0.0075

Result

Tax Rs. 6225

1.3.9.Find the compound markup on Rs. 3,75,000 for one year at the rate of 14% compounded annually.

Given

P=375000,quad R=0.14,quad T=1

Formula

Compound interest CI=P(1+R)^T-P

Working

CI=375000(1.14)¹-375000

Result

CI=Rs. 52500

Unit 1: Real Numbers — Exercise 1 3