ICSE Class 8 Maths Number Systems Unit Test Paper

Unit Test Paper

Numbers

1. Given that \(a = 3\sqrt{7}\), \(b = 2\sqrt{7}\), and \(c = \sqrt{7}\), verify that

(i) \(a + b = b + a\)

(ii) \(a \times c = c \times a\)

(iii) \(a + (b + c) = (a + b) + c\)

(iv) \(a \times (b + c) = (a \times b) + (a \times c)\)

(v) \(a \times (b - c) = (a \times b) - (a \times c)\)

(vi) \(a - b \neq b - a\)

(vii) \(a(b - c) \neq b(a - c)\)

2. Simplify the following expressions.

(i) \((+3) \times (-5) - (-2) \text{ of } (+6) \div (-4) + (-7) \text{ of } (-2)\)

(ii) \((-2) + (+3) - [7 - 2(9 - (8 - 6 - 2 - 5));) + 4]\)

(iii) \(\frac{1\frac{3}{1} - \frac{1}{2}}{1\frac{2}{1} - \frac{3}{7}}\)

(iv) \(\left[\frac{1}{2} - \frac{1}{5}\left|\frac{1}{1} + \frac{1}{7}\right|\left(\frac{6}{14} - \frac{1}{6}\left(\frac{9}{1} - \frac{6}{7}\right)\right)\right] + 1\frac{2}{7}\)

(v) \(\frac{0.89 \times 1.66}{\frac{4}{5}}\) and \(\frac{4.15 \times 1.78}{5}\)

(vi) \(1 - 0.5 \text{ of } [1.8 - \{1.5 - 0.7 \text{ of } (2.75 - 2.63 - 1.28)\} + 1.3]\)

(vii) \(\frac{3.375^2 - 2.125^2}{6.75^2 - 2 \times 6.75 \times 4.25 + 4.25^2}\)

Teacher's Note

Simplifying complex mathematical expressions helps develop logical thinking skills used in everyday problem-solving and decision-making.

3. Find the HCF of the following numbers.

(i) 546, 637, 728, and 819

(ii) 792, 1512, 1728, and 1368

(iii) 6273, 6642, 7011, and 7380

(iv) 17175, 17862, 18549, and 19236

(v) \(4\frac{2}{3}, 10\frac{1}{2}, \text{ and } \frac{21}{24}\)

(vi) \(1\frac{5}{1}, 1\frac{1}{7}, \text{ and } 4\frac{4}{5}\)

(vii) 7.2, 12.96, and 10.08

(viii) 2.73, 2.31, and 3.15

Teacher's Note

Finding the highest common factor is essential in real-world applications like dividing items equally among groups or simplifying recipes.

4. Find the LCM of the following numbers.

(i) 570, 228, and 380

(ii) 1008, 756, and 864

(iii) 221, 153, and 117

(iv) 2376, 3960 and 3168

(v) \(\frac{5}{6}, \frac{5}{18}, \text{ and } 2\frac{1}{2}\)

(vi) \(\frac{2}{5}, \frac{9}{10}, \text{ and } \frac{3}{10}\)

(vii) 0.78, 0.66, and 1.43

(viii) 0.21, 1.155, and 0.77

Teacher's Note

Finding the least common multiple is useful in planning schedules, such as determining when two events will coincide or when to reorder supplies.

5. Find the greatest number that divides 24307, 25091, and 25875 leaving exactly 3 as remainder in each case.

6. Find the greatest 7-digit number that can be divided by 10368, 6912 as well as 8640 exactly.

7. Convert the following common fractions into decimals, correct up to 3 decimal places.

(i) \(\frac{8}{9}\)

(ii) \(\frac{7}{13}\)

(iii) \(1\frac{4}{9}\)

(iv) \(2\frac{7}{15}\)

(v) \(4\frac{1}{17}\)

Teacher's Note

Converting fractions to decimals is practical for everyday measurements in cooking, shopping, and financial calculations.

8. Convert the following decimals into common fractions.

(i) 0.0875

(ii) 0.15625

(iii) 0.5

(iv) 6.45

(v) 3.252

9. A man wins Rs 34,00,000 in a lottery and donates \(\frac{11}{17}\) of the money to a charitable institution. The charitable institution uses \(\frac{2}{5}\) part of the donation to buy an ambulance and \(\frac{1}{6}\) part of the remaining money to build a dispensary. How much did it cost to build the dispensary?

Teacher's Note

Understanding fractions of amounts helps in managing charitable giving and understanding how donations are allocated to different causes.

10. \(\frac{2}{3}\) books in a library are works of fiction, out of which \(\frac{1}{5}\) are in Hindi. If there are 4640 works of fiction that are not in Hindi, how many books are there in the library?

Teacher's Note

This type of problem demonstrates how fractions are used in library management and inventory systems in real institutions.

11. If 15 mg of baking soda makes up 0.0000125 part of the weight of a cake, how much does the cake weigh?

12. Divide Rs 13878 between A, B, and C such that A's share is 0.65 part of B's share and C's share is 0.92 part of B's share.

13. Write the following decimal numbers in expanded form as multiples of powers of 10.

(i) 1.1

(ii) 10.01

(iii) 100.001

(iv) 1000.0001

(v) 1000000

(vi) 0.000001

Teacher's Note

Understanding decimal expansion helps in reading and interpreting precise measurements in scientific and technical applications.

14. Write the scientific notation for the following numbers.

(i) 38000

(ii) 67001

(iii) 4800000

(iv) 720003

(v) 621000000

(vi) 6210000400

15. Write the scientific notation for the following numbers.

(i) 0.02

(ii) 0.00043

(iii) 0.78

(iv) 0.0000008

(v) 0.000000700

(vi) 0.000070003

Teacher's Note

Scientific notation simplifies working with very large and very small numbers, commonly used in astronomy, chemistry, and physics.

16. Round off the following decimals to 2 significant digits.

(i) 6.36

(ii) 0.2237

(iii) 5.019003

(iv) 0.007091

(v) 0.001003005

17. Light travels at a speed of 299792500 m/s. Write the speed of light in scientific notation.

18. If the distance between Earth and Mars is 7800000 km, write this distance in scientific notation.

19. The weight of a proton in kilograms is written in scientific notation as \(1.67252 \times 10^{-27}\) kg. Write the scientific notation for the weight of a proton in grams.

Teacher's Note

Scientific notation is essential in physics and astronomy for expressing astronomical distances and subatomic particle measurements.

20. Find the square roots of the following by the division method.

(i) 47089

(ii) 60516

(iii) 61009

(iv) 776161

(v) 1572516

(vi) \(\frac{529}{1369}\)

(vii) \(\frac{3721}{5041}\)

(viii) 73.96

(ix) 22.8484

(x) 1.030225

21. Find the smallest and the greatest 8-digit numbers that are perfect squares.

Teacher's Note

Understanding square roots and perfect squares is fundamental in geometry, construction, and land measurement applications.

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