Hard Level Worksheet Questions

Part A: Subjective Questions

(1) What is the value of x if 2x−34 = 5?

Awesome! Multiply by 4: 2x - 3 = 20, so x = 11

(2) Find the unknown: x + x2 = 12

Great job! Combine terms: 3x2 = 12, so x = 8.

(3) If x – 3 = y and x + y = 15, find x.

Perfect! Substitute: x + (x - 3) = 15, so x = 9.

(4) Write an algebraic equation for: "One-third of a number increased by 7 gives 16."

Excellent! x3 + 7 = 16 represents the statement.

(5) Solve: 3x+12 = 12

Super! Multiply by 2: 3(x + 1) = 24, so x = 7.

(6) If 4x – 7 = 3x + 5, find the value of x.

That's correct! Subtract 3x: 4x - 3x = 5 + 7, so x = 12.

(7) Write an equation for: "Twice a number decreased by 9 equals 11."

Well done! 2x - 9 = 11 represents the statement.

(8) Solve for x: x−45 = x+23

Brilliant! Cross multiply: 3(x - 4) = 5(x + 2), so x = -22.

(9) What number multiplied by 4 and increased by 7 gives 31?

You nailed it! 4x + 7 = 31, so x = 6.

(10) Express the statement as an equation and solve: "A number divided by 6 and then subtracted by 2 gives 4." , x =

Perfect! x6 - 2 = 4 gives x = 36.

Drag each problem to its correct category:

(1) Raju bought 4 pencils and 2 pens. Each pencil costs ₹x and each pen costs ₹(x + 5). If the total cost is ₹46, find the cost of one pencil.

Cost of 4 pencils = 4x4+x

Cost of 2 pens = 2x + 102x + 5

Total cost equation: 4x+2x+10=464x-2x+10=46

Simplifying: x =

Excellent! Each pencil costs ₹6.

(2) A father's age is 3 times his son's age. In 4 years, the sum of their ages will be 64. Find their present ages.

Let son's present age be x. Then father's age = .

After 4 years: son's age = , father's age = .

Their total age after 4 years = 64.

Solving x = .

Therefore, son's age = years and father's age = years.

Perfect! Son is 14 years and father is 42 years old.

3. Two numbers differ by 7. Three times the smaller number added to the larger gives 52. Find both numbers.

Let the smaller number =

Then the larger number =

Given condition: 3xx+3 + (x+7) =

Solving x =

The numbers are and

Great work! The numbers are 11.25 and 18.25.

(4) A fruit seller sold mangoes worth ₹x per box. He sold 5 boxes and spent ₹120 on transport. If he earned a profit of ₹180, write an equation and find the cost per box.

Let selling price per box = ₹x. Revenue from 5 boxes = .

Total cost = Cost price +× = (5x – 300120) + 120 = 5x – 180.

Profit = Revenue –+ Total costCost price = 5x – 5x – 1805x + 180 = 180.

Simplifying x = .

Hence, cost per box = ₹.

Outstanding! Each box costs ₹60.

(1) A boat travels 30 km downstream and returns the same distance upstream. The speed downstream is (x + 5) km/h and upstream is (x – 3) km/h. If the total time taken is 4 hours, form and solve the equation to find x.

Time downstream = 30x+530×(x+5) hours

Time upstream = 30x−330×(x-3) hours

Total time equation: 30x+5+30x−3=430x−5-30x−3=4

Simplifying: x =

Excellent! The speed in still water is 13 km/h.

(2) A number when increased by 10% of itself and then reduced by 5 equals 42. Find the number.

Let the number be x. Increased by 10%: x+0.1xx+10

Then reduced by 5: 1.1x-5x+0.1x-5

Equation:

Simplifying: x =

Great! The number is approximately 42.73.

(3) A group of students collected ₹x each for a charity. If there were 6 students more, the collection would be ₹180 more. If each student contributed ₹30, find how many students were originally there.

Let original number of students = n. Original collection =

With 6 more students: Collection = (n+6)×30n+630

Given: (n+6) × 30 = nx + 180

Simplifying: n =

Perfect! There were originally 30 students.

(4) A father distributes ₹1000 among his two sons such that the elder gets ₹x and the younger gets ₹(x – 200). If the elder invests his share at 5% and the younger at 10%, their annual interest incomes are equal. Find how much each got.

Elder's interest = 0.05x0.5x

Younger's interest = 0.1(x-200)0.1x-200

Equal interest condition: 0.05x=0.1(x-200)0.05x+0.1(x-200)=0

Elder gets ₹

Younger gets ₹

Fantastic! Elder gets ₹400 and younger gets ₹200.

Test your understanding with these challenging multiple choice questions:

Part B: Objective Questions

1. The equation that represents "One fourth of a number is 3 less than half of that number" is:

(a) x4 = x2 – 3 (b) x2 = x4 – 3 (c) x4 + 3 = x2 (d) x4 = 32x

Excellent! "One fourth is 3 less than half" translates to x4 = x2 - 3.

2. If 2x – 3 = 7, then 4x =

(a) 20 (b) 14 (c) 12 (d) 16

Perfect! If 2x - 3 = 7, then 2x = 10, so 4x = 20.

3. The solution of 3(x + 2) = 2(x + 5) is:

(a) 4 (b) 1 (c) 0 (d) –1

Great! Expanding: 3x + 6 = 2x + 10, so x = 4.

4. A linear equation has how many solutions?

(a) 0 (b) 1 (c) 2 (d) Infinite

Excellent! A linear equation in one variable has exactly one solution.

5. If the cost of 2 pens and 3 pencils is ₹41, and a pencil costs ₹5, what is the cost of one pen?

(a) ₹12 (b) ₹13 (c) ₹14 (d) ₹11

Fantastic! 2p + 3(5) = 41, so 2p = 26, therefore p = ₹13.

🎉 Outstanding Achievement! You've Mastered Hard-Level Simple Equations!

Here's what you conquered:

• Complex equation solving with fractions and multiple steps
• Advanced word problem translation and setup
• Sophisticated age, money, and speed/distance problems
• Multi-variable relationship problems with constraints
• Advanced algebraic manipulation and cross-multiplication
• Understanding equation types and solution uniqueness
• Real-world applications requiring complex equation modeling

Your advanced simple equation skills prepare you for linear algebra, systems of equations, and complex mathematical modeling!