Hard Level Worksheet Questions

Part A: Subjective Questions

(1) Find a fraction greater than 34 but less than 45.

Awesome! 1924 lies between 34 and 4/5.

(2) Write a rational number which lies between −23 and −12.

Great job! −712 is between −23 and -1/2.

(3) Which is greater: 0.7 or 79? Justify without converting to decimals.

Perfect! 79 > 0.7 by cross multiplication: 7×10 > 7×9.

(4) If 34 of a number is 6, what is the number?

Excellent! If 34 × x = 6, then x = 6 × 43 = 8.

(5) Which decimal has 5 in the hundredths place and 3 in the tenths place?

Super! 0.35 has 3 in tenths and 5 in hundredths.

(6) Express the reciprocal of 59 as a decimal.

That's correct! Reciprocal of 59 is 9/5 = 1.8.

(7) Identify the property: (ab) + (−ab) = 0 additive inverseDistributive

Well done! This demonstrates the additive inverse property.

(8) What rational number should be added to −38 to get 0?

Brilliant! 38 is the additive inverse of −38.

(9) How many rational numbers lie between 13 and 12 (list at least 2)?

You nailed it! There are infinitely many rational numbers between any two rationals.

(10) If 0.1 is multiplied 3 times, what is the final product?

Perfect! 0.1 × 0.1 × 0.1 = 0.001.

Drag each expression to its correct operation type:

(1) A person walks 34 km in 15 hour. What is his speed in km/hr?

Speed = DistanceTime

Speed = 3415

Speed = km/hr = km/hr (in decimal form)

Excellent! The speed is 154 or 3.75 km/hr.

(2) Evaluate and simplify:

(a) (56 – 23) × 34 =

Great! The answer is 18.

(b) (−35) + 2 – (110) =

Perfect! The result is 1310.

(3) A chocolate bar is divided into 8 equal parts. Rani ate 34 of it.

(a) How many pieces did she eat?

(b) What fraction of the bar is left?

Excellent! She ate 6 pieces and 1/4 is left.

(4) Find the value of:

(a) (310 ÷ 25) + (43 × 16) =

Great work! The answer is 35/36.

(b) (1.2 × 2.5) – (0.4 ÷ 2) =

Perfect! The final answer is 2.8.

(1) A bag contains coins worth ₹7.75 in the form of 50 paisa and 25 paisa coins.

(a) If there are 10 coins of 50 paisa, how many 25 paisa coins are there?

Number of 50 paisa coins =

Value of 50 paisa coins = ₹

Remaining value = ₹7.75 - ₹5.00 = ₹

Number of 25 paisa coins = ₹2.75 ÷ ₹ =

Total coins =

(b) Fraction of 25 paisa coins

The Fraction of 25 paisa coins =

Excellent! There are 11 coins of 25 paisa, making 1121 of total.

(2) A tank is 25 full of water. After adding 18 litres, it becomes 34 full.

(a) What is the total capacity of the tank?

(b) How much water was already in the tank?

Perfect! Tank capacity is 54 litres with 21.6 litres initially.

(3) Simplify and express as a rational number:

(a) (12 + 23 – 56) ÷ (34 – 12) =

Great! The answer is 43.

(b) [(0.5 × 0.2) + 0.75] ÷ 0.25 =

Excellent! The decimal result is 3.4.

(4) A student scored 9.25, 7.75, and 8.5 in three tests.

(a) What is the average score?

(b) How much more should they score in the fourth test to get an average of 8.5?

Outstanding! Current average is 8.5 and needs 8.5 in fourth test.

Part B: Objective Questions

Test your understanding with these challenging multiple choice questions:

1. If a number is multiplied by 0, the result is:

(a) The number itself (b) 1 (c) 0 (d) Not defined

Super job! Any number multiplied by 0 equals 0.

2. Which of these is irrational?

(a) 1/3 (b) 2/5 (c) √5 (d) –3/4

Well done! √5 cannot be expressed as p/q, so it's irrational.

3. 0.75 of 80 is:

(a) 60 (b) 65 (c) 55 (d) 50

That's right! 0.75 × 80 = 60.

4. The fraction 7/9 lies between:

(a) 0.6 and 0.7 (b) 0.7 and 0.8 (c) 0.8 and 0.9 (d) 0.9 and 1

Correct! 7/9 ≈ 0.778, which is between 0.7 and 0.8.

10. Which of the following is closest to –2/3?

(a) –0.5 (b) –0.67 (c) –0.25 (d) –0.9

Fantastic! –2/3 ≈ –0.667, so –0.67 is closest.

🎉 Outstanding! You've Mastered Hard-Level Fractions & Decimals!

Here's what you conquered:

• Advanced fraction comparisons without decimal conversion
• Complex multi-step operations with mixed numbers
• Real-world applications involving money, speed, and capacity problems
• Understanding infinite density of rational numbers
• Advanced decimal operations and place value concepts
• Distinguishing between rational and irrational numbers
• Complex word problems requiring multiple operations

Your advanced skills in fractions and decimals prepare you for algebra, advanced arithmetic, and real-world mathematical problem solving!