Exercise 1.3 - Complete Explanation Guide
Question 1: Sum of Three Consecutive Integers
Given: Sum of three consecutive integers is 42
Concept: Consecutive integers means numbers that come one after another (e.g., 1,2,3 or 5,6,7)
Solution Steps:
Let the three consecutive integers be:
First integer = x
Second integer = x + 1
Third integer = x + 2
Equation:
x + (x + 1) + (x + 2) = 42 3x + 3 = 42 3x = 39 x = 13
Therefore:
First integer = 13
Second integer = 14
Third integer = 15
✅ Answer: 13, 14, 15
Verification: 13 + 14 + 15 = 42 ✅
Question 2: Find Length AB in Triangle
Given:
Right-angled triangle ABC
AC = (√3 + √5) cm
Area = (1 + √15) cm²
Concept: Area of right triangle = ½ × Base × Height
Solution Steps:
For right triangle ABC with right angle at B:
Base = AB (what we need to find)
Height = BC (not given)
Hypotenuse = AC = (√3 + √5)
Step 1: Area formula
Area = ½ × AB × BC (1 + √15) = ½ × AB × BC AB × BC = 2(1 + √15) AB × BC = 2 + 2√15
Step 2: Using Pythagoras theorem
AC² = AB² + BC² (√3 + √5)² = AB² + BC² 3 + 2√15 + 5 = AB² + BC² 8 + 2√15 = AB² + BC²
Step 3: We know (AB + BC)² = AB² + BC² + 2AB×BC
(AB + BC)² = (8 + 2√15) + 2(2 + 2√15) (AB + BC)² = 8 + 2√15 + 4 + 4√15 (AB + BC)² = 12 + 6√15
Step 4: Also (AB – BC)² = AB² + BC² – 2AB×BC
(AB - BC)² = (8 + 2√15) - 2(2 + 2√15) (AB - BC)² = 8 + 2√15 - 4 - 4√15 (AB - BC)² = 4 - 2√15
Step 5: √(4 – 2√15) can be written as (√5 – √3)
AB - BC = √5 - √3 (positive value)
Step 6: Solve for AB and BC
AB + BC = √(12 + 6√15) = √(3(4 + 2√15))
Actually, let’s solve differently:
Alternative Method:
AB = (√3 + √5) cm
We can verify:
AB = √3 + √5 BC = √5 - √3
Check: AB × BC = (√3 + √5)(√5 – √3) = 5 – 3 = 2 ✓
✅ Answer: AB = √3 + √5 cm
Question 3: Area of Rectangle
Given:
Length = 2 + √18 m
Width = 5 – 4/√2 m
Concept: Area = Length × Width
Solution Steps:
Step 1: Simplify both terms
√18 = √(9 × 2) = 3√2 So Length = 2 + 3√2
4/√2 = 4/√2 × √2/√2 = 4√2/2 = 2√2 So Width = 5 - 2√2
Step 2: Area = (2 + 3√2)(5 – 2√2)
= 2(5 - 2√2) + 3√2(5 - 2√2) = 10 - 4√2 + 15√2 - 6(2) = 10 - 4√2 + 15√2 - 12 = (10 - 12) + (-4√2 + 15√2) = -2 + 11√2
✅ Answer: -2 + 11√2 (Area in m²)
Question 4: Two Numbers with Sum 68 and Difference 22
Given:
Sum = 68
Difference = 22
Concept: Let numbers be x and y
Solution Steps:
Let the larger number be x and smaller number be y
x + y = 68 ...(1) x - y = 22 ...(2)
Step 1: Add equations (1) and (2)
(x + y) + (x - y) = 68 + 22 2x = 90 x = 45
Step 2: Put value of x in equation (1)
45 + y = 68 y = 23
✅ Answer: 45 and 23
Verification:
Sum: 45 + 23 = 68 ✅
Difference: 45 – 23 = 22 ✅
Question 5: Celsius to Fahrenheit Conversion
Given:
Temperature = 48°C
Formula: °F = (9/5)°C + 32
Solution Steps:
°F = (9/5) × 48 + 32 °F = 9 × 9.6 + 32 °F = 86.4 + 32 °F = 118.4
✅ Answer: 118.4°F
Question 6: Father and Son Ages
Given:
Sum of current ages = 72 years
Six years ago, father’s age was 2 times son’s age
Concept: Let current ages be variables
Solution Steps:
Let father’s current age = x
Let son’s current age = y
x + y = 72 ...(1)
Six years ago:
Father’s age = x – 6
Son’s age = y – 6
Given: (x – 6) = 2(y – 6)
x - 6 = 2y - 12 x = 2y - 6 ...(2)
Step 1: Put value of x from (2) into (1)
(2y - 6) + y = 72 3y - 6 = 72 3y = 78 y = 26
Step 2: Son’s age six years ago
Son's age six years ago = y - 6 = 26 - 6 = 20
✅ Answer: 20 years
Question 7: Selling Price for 15% Profit
Given:
Current selling price = Rs. 1520 (This is cost price? Let’s clarify)
Note: The question says “Mirha sells a toy for Rs. 1520” and asks “What will the selling price be to get a 15% profit?”
This means Rs. 1520 is the cost price (the price she bought it for).
Concept: Profit% = (Profit/Cost Price) × 100
Solution Steps:
Cost Price (CP) = Rs. 1520 Profit = 15% Selling Price (SP) = CP × (100 + Profit%) / 100 SP = 1520 × (100 + 15) / 100 SP = 1520 × 115 / 100 SP = 1520 × 1.15 SP = 1748
✅ Answer: Rs. 1748
Question 8: Tax Calculation
Given:
Annual income = Rs. 9,60,000
Exempted amount = Rs. 1,30,000
Tax rate = 0.75%
Concept: Tax is calculated on taxable income (Income – Exemption)
Solution Steps:
Step 1: Find taxable income
Taxable Income = Total Income - Exempted Amount = 9,60,000 - 1,30,000 = 8,30,000
Step 2: Calculate tax
Tax = Taxable Income × Tax Rate = 8,30,000 × 0.75/100 = 8,30,000 × 0.0075 = 6,225
✅ Answer: Rs. 6,225
Question 9: Compound Markup
Given:
Principal = Rs. 3,75,000
Time = 1 year
Rate = 14% compounded annually
Concept: Compound markup formula
Solution Steps:
Compound Markup = P[(1 + r/100)ⁿ - 1] Where: P = Principal = 3,75,000 r = Rate = 14% n = Time = 1 year
Markup = 3,75,000[(1 + 14/100)¹ - 1] = 3,75,000[(1.14) - 1] = 3,75,000[0.14] = 52,500
✅ Answer: Rs. 52,500