## A word about Hermann Weyl

“It is a crying shame that Weyl is leaving Zurich.” Thus, Albert Einstein described Hermann Weyl (1885-1955), who remains a legendary figure, “one of the greatest mathematicians of the first half of the twentieth century…No other mathematician could claim to have initiated more of the theories that are now being explored,” as Sir Michael Atiyah had put it once.

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## Skill Check XII: IITJEE Foundation Math

A) Evaluate the following fractions:

a) $2\frac{1}{7} + 3\frac{1}{2}+1$

b) $3\frac{1}{5} + 2\frac{1}{10}-\frac{1}{2}-\frac{1}{4}$

c) $\frac{7}{3} + \frac{11}{5}=2\frac{1}{15}$

d) $6-2\frac{1}{2} - 1\frac{2}{4}$

B) Evaluate the following fractions:

a) $1\frac{1}{2} \times 2\frac{1}{3} + 1\frac{1}{4}$

b) $3\frac{1}{3} \times 3 \frac{1}{4} \times 2 \frac{1}{7}$

c) $7\frac{1}{2} \div 6 \frac{2}{3}$

d) $12\frac{1}{3} \div 8 \frac{2}{9}$

C) Simplify:

a) $[44 \frac{1}{5} \times \frac{10}{34}] \div [\frac{14}{16} \times \frac{5}{12}]$

b) $[7 \frac{3}{9} \times 9 \frac{4}{5}] - [1\frac{2}{3} \div 8 \frac{1}{3}]$

c) $5.381 = 2.5 \hspace{0.1in} of \hspace{0.1in} [6.42-2.82]$

d) $4.396 + 32.06 - 0.7 + [2.52 + 1.2]$

D) In a class of 40 students, two fifth are girls. Each girl brings a ribbon of $3\frac{2}{4}$ m and each boy brings $2\frac{1}{4}$. What is the total length of ribbon collected by 40 students?

E) The cost of 7.25 kg oranges is Rs. 362.50 and the cost of 4.75 kg grapes is Rs. 191.25. Himani buys 6 kg oranges and 5 kg grapes. How much moeny has Himani spent?

Regards,
Nalin Pithwa

## Skill Check XI: IITJEE foundation math

1. $\frac{2}{5} - \frac{9}{10} \times \frac{4}{7} \div [1\frac{3}{7}-1\frac{3}{4} \times (\frac{4}{7} + \overline{\frac{6}{7}-\frac{2}{3}})]$
2. $3 + [2\frac{5}{6}-\frac{7}{15} \{ 5\frac{1}{2}-(1\frac{5}{6}+2\frac{1}{3}-3\frac{2}{3})\}]$
3. $2\frac{1}{6}-[1\frac{3}{5} + (1\frac{5}{6}+3\frac{1}{2}-2\frac{3}{5})] \times 12 \div [1\frac{2}{3}-\frac{7}{13} \hspace{0.1in} of \hspace{0.1in} (\frac{1}{3}+2\frac{1}{7})]$
4. $3.8 \hspace{0.1in} of \hspace{0.1in} [5.67 -2.2 \hspace{0.1in} of \hspace{0.1in} (4.66 - 1.4 \hspace{0.1in} of \hspace{0.1in} (3.2 - \overline{5.18-3.63})$
5. $1.4 + [1-0.34 \div \{4.5-(4.4 \hspace{0.1in} of \hspace{0.1in} 1.45) -2.6 \hspace{0.1in} of \hspace{0.1in} 1.05 \}]$
6. $\frac{6.5^{2} - 3.5^{2}}{1.26^{2}+2 \times 1.26 \times 2.74 +2.74^{2}}$
7. $\frac{1.385^{2}-2\times 1.385 \times 0.785 + 0.785^{2}}{3.25^{2}-2.75} \div \frac{1.2}{3.25-2.75}$

Regards,

Nalin Pithwa

## Skill Check X: IITJEE Foundation Math

I) Arrange the following decimal fractions in ascending order:

(i) 68.95, 6.985, 9.685, 86.59, 8.695

(ii) 1.36, 1.29, 1.48, 1.26, 1.38

(iii) 5.689, 5.869, 5.896, 5.698, 5.986

(iv) 1.1001, 1.0011, 1.111, 1.0101, 1.1011

(v) 7.8899, 7.9898, 7.9889, 7.9988, 7.8998

II) Add the following decimal fractions:

(i) 7.8 +3.2

(ii) 8.91 + 1.89

(iii) 2.3 + 2.65

(iv) 1.721 + 1.892

(v) 5.867+6.719+3.0018

(vi) 3.468+2.12+1.34464

III) Subtract the following decimal fractions:

(i) 6.9-2.9

(ii) 3.41 – 2.21

(iii) 5.4 – 3.22

(iv) 5.216 – 3.162

(v) 2.0-0.9963

(vi) 8.016-2.98694

IV) Multiply the following decimal fractions:

(i) $5.8 \times 2.5$

(ii) $4.64 \times 0.5$

(iii) $9.5 \times 0.04$

(iv) $2.13 \times 1.65$

(v) $5.68 \times 0.145$

(vi) $2.94 \times 0.3215$

V) Divide the following decimal fractions:

(i) $14.5 \div 2.9$

(ii) $2.1 \div 1.4$

(iii) $14.5 \div 4$

(iv) $19.68 \div 6.15$

(v) $2.028 \div 3.12$

(vi) $1.7019 \div 5.49$

VI) Find the HCF of the following decimal fractions:

(i) 0.72 and 4.8

(ii) 1.092 and 1.176

(iii) 0.21, 0.1925, and 0.175

(iv) 0.286, 0.3718, and 0.3146

VII) Find the LCM of the following decimal fractions:

(i) 4.5 and 0.42

(ii) 0.18 and 0.144

(iii) 2.52, 0.189 and 0.168

(iv) 0.112, 0.8, and 0.48

VIII) Find the greatest decimal fractions that can be divided by 0.33, 0.495 and 0.297 leaving exactly 0.15 as remainder.

IX) Simplify the following expressions:

(a) $(1.52 \hspace{0.1in} of \hspace{0.1in} 3.5 \div 0.8 -4.4) \div 2.5 =0.1$

(b) $(5.025 \div 2.5 + 1.49) \times 1.5 - 1.5 \times 2.5 + 1.5$

(c) $[4.5 +0.4 \{ 7.31 - (2.45 + 3.68 -1.32)\}] +1.1$

(d) $\frac{3.27 \times 0.77}{3.85 \times 2.18}$

(e) $\frac{1.82 \times 2.5}{3.25} - \frac{2.22}{1.85}$

(f) $\frac{0.47 \times 0.81 \times 0.63}{1.62 \times 0.85 \times 2.35} \div \frac{8}{5}$

(g) $\frac{6.84 \times 4.52}{1.808 \times 2.85} - \frac{1.62 \times 8.26}{2.36 \times 2.025}$

(h) $0.3 \div 0.1 - [0.3 + \{ 0.1 \times 0.3 - (0.3 + \overline{0.3-0.1} \hspace{0.1in} of \hspace{0.1in} 0.3)\}]$

X) Simplify the following expressions:

(i) $3.865^{2}-1.135^{2}$

(ii) $1.481^{2} + 2 \times 1.481 \times 0.519 + 0.519^{2}$

(iii) $4.694^{2} - 2 \times 4.694 \times 3.494 +3.494^{2}$

XI) Juno buys 1.00 kg of plasticine at Rs. 35.50 per kg and 0.35 litres of paint at Rs. 142.80 per litre. If he gives the shopkeeper a 100 rupee note, how much change should he get back?

XII) 0.125 part of a peace keeping force are doctors, 0.09375 are engineers, 0.03125 parts are cooks, and the rest are armed soldiers. If the peace-keeping force has 1152 members, how many armed soldiers are in it?

XIII) Gitanjali read 0.25 part of a book on the first day, 0.35 part on the second day, and 160 pages to finish the book on the third day. How many pages were there in the book?

XIV) If i metre is equal to 3.28084 feet, how many feet will 15 metres equal to?

XV) If 2.54 cm made an inch, how many inches will 60.96 cm make?

XVI) 0.4 part of a 9.3 g ornamental chain is made of gold. If the chain is cut into 6 equal pieces how much gold will be there in each piece?

XVII) The perimeter of an isosceles trapezium is 7.07 cm. If its unequal sides measure 1.85 cm, 2.32 cm, how much do its equal sides measure?

XVIII) Mrs Rita gives 0.025 part of her salary as pocket money to her son Vinod every month. Vinod spends 0.8 part of his pocket money and saves the rest. If he saves Rs. 277.50 in three months, how much does Mrs. Rita earn in a month?

XIX) The average earning of three family members A, B and C is Rs. 11240.25. One member B leaves for another town and a new family member D starts earning. The new average earning of A, C and D is now Rs. 10520.50 . If D is earning Rs. 9886.25, how much was B earning?

XX) A petrol pump attendant lowers a 5 m long dip-stick to check the oil-level in an underground tank. For every 0.1 m in height of oil on the dip-stick, there is 1200 liters of oil in the tank. If ).38 part of the dip-stick is wet with oil, how many litres of oil are there in the tank?

Regards,

Nalin Pithwa.

## Skill Check IX: IITJEE Foundation Math

A) Arrange all the fractions in descending order:

i) $2\frac{3}{7}, \frac{5}{8}, 1 \frac{6}{11}, 3 \frac{2}{5}, \frac{1}{2}$

ii) $\frac{3}{4}, \frac{1}{2}, \frac{5}{6}, \frac{7}{12}, \frac{2}{3}$

iii) $2\frac{1}{10}, 2\frac{1}{5}, 2\frac{4}{5}, 3\frac{1}{2}, \frac{14}{15}$

iv) $3\frac{2}{7}, 3\frac{1}{5}, 3\frac{5}{16}, 3\frac{4}{11}, 3\frac{3}{11}$

v) $\frac{6}{13}, \frac{4}{7}, \frac{5}{9}, \frac{11}{20}, \frac{12}{25}$

B) Arrange the following fractions in ascending order:

i) $3 \frac{1}{8}, \frac{1}{3}, 2\frac{3}{5}, 1\frac{4}{13}, \frac{5}{7}$

ii) $\frac{3}{5}, \frac{7}{10}, \frac{11}{15}, \frac{2}{3}, \frac{1}{2}$

iii) $\frac{24}{25}, 1\frac{5}{7}, 1 \frac{1}{7}, 1 \frac{1}{3}. 1 \frac{1}{5}$

iv) $\frac{24}{25},1 \frac{5}{7}, 1 \frac{1}{7}, 1 \frac{1}{3}, 1 \frac{1}{5}$

v) $1\frac{3}{7}, 1 \frac{2}{5}, 1\frac{4}{9}, 1\frac{5}{11}, 1\frac{7}{11}$

vi) $\frac{1}{2}, \frac{5}{11}, \frac{4}{9}, \frac{13}{25}$

III) Evaluate the following fractions:

(i) $\frac{3}{11} + \frac{2}{11} + \frac{5}{11}$

(ii) $\frac{2}{7} + \frac{2}{3}$

(iii) $\frac{3}{5} +1 \frac{2}{5}$

(iv) $6\frac{1}{2} + 1\frac{2}{3} + 1\frac{5}{6}$

(v) $\frac{2}{9} = \frac{3}{7} + 1\frac{1}{3}$

(vI) $\frac{4}{11} + 2\frac{1}{2} + 3\frac{1}{4} + 2\frac{5}{11} + \frac{7}{44}$

IV) Evaluate the following fractions:

(i) $\frac{8}{13} -\frac{3}{13}$

(ii) $\frac{5}{7} - \frac{1}{2}$

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

(iv) $\frac{7}{9} - \frac{3}{4}$

(v) $1\frac{3}{5} - \frac{4}{7}$

(vi) $2\frac{2}{6} - 1\frac{3}{5}$

V) Evaluate the following fractions:

(i) $\frac{3}{4} \times \frac{1}{2}$

(ii) $\frac{6}{11} \times 1\frac{2}{9}$

(iii) $\frac{2}{5} \times \frac{15}{16} \times \frac{8}{9}$

(iv) $\frac{6}{7} \times 3\frac{1}{2} \times 2\frac{1}{3}$

(v) $1\frac{5}{7} \times 2\frac{1}{10} \times 6\frac{1}{4}$

(vi) $5 \frac{1}{4} \times 3 \frac{1}{7} \times 2 \frac{2}{11}$

VI) Evaluate the following fractions:

(i) $\frac{3}{8} \div 1\frac{1}{2}$

(ii) $\frac{1}{4} \div \frac{1}{2}$

(iii) $\frac{3}{5} \div \frac{5}{3}$

(iv) $6\frac{3}{7} \div 1\frac{2}{7}$

(v) $3\frac{4}{7} \div \frac{5}{7}$

(vi) $4\frac{2}{3} \div \frac{4}{9}$

VII) Find the HCF of the following fractions:

(i) $\frac{1}{3}$ and $\frac{1}{2}$

(ii) $\frac{3}{4}$ and $\frac{2}{5}$

(iii) $\frac{3}{7}$ and $1\frac{5}{7}$

(iv) $\frac{15}{22}$ and $\frac{10}{11}$

(v) $1\frac{5}{7}$ and $1\frac{1}{35}$ and $2\frac{2}{5}$

(vi) $\frac{15}{16}, \frac{21}{40}$ and $\frac{9}{20}$

IX) Find the LCM of the following fractions:

(i) $\frac{1}{4}$ and $\frac{2}{3}$

(ii) $\frac{2}{3}$ and $\frac{4}{5}$

(iii) $\frac{2}{7}$ and $\frac{5}{14}$

(iv) $\frac{6}{11}$ and $\frac{9}{11}$

(v) $\frac{6}{5}, \frac{3}{5}, \frac{3}{4}, \frac{1}{3}$

(vi) $1\frac{1}{6}, 1\frac{5}{9}, \frac{21}{24}, 1\frac{9}{12}$

IX) Find the greatest fraction that divides $\frac{1}{6}$ and $2\frac{1}{2}$ exactly and also find the smallest fraction that can be divided by the given fractions.

X) Simplify the following expressions:

(i) $2\frac{3}{5} - [ \frac{2}{3} + \{ 2\frac{1}{3}-(1\frac{1}{2}- \overline{\frac{3}{5}-\frac{1}{2}})\}]$

(ii) $\frac{5}{8} \div 1 \frac{3}{7} \times \frac{2}{7} \div (3\frac{1}{6}-2\frac{1}{2})$

(iii) $\frac{7}{11} of (1\frac{3}{5}-\overline{1\frac{2}{5}-\frac{3}{7}})$

(iv) $7(\frac{3}{8} \div \frac{1}{4} - \frac{3}{7} - 1\frac{3}{4} of \frac{6}{7} \div 1\frac{1}{2})$

(v) $1\frac{4}{7} - \frac{6}{7}[1\frac{2}{3}-\frac{3}{4} \{ \frac{2}{3} \div (\frac{5}{9}-\frac{1}{3})\}]$

(vi) $\frac{1\frac{1}{2}}{1\frac{13}{14}} - \frac{1\frac{2}{11}}{1\frac{17}{22}}$

(vii) $\frac{2\div 1\frac{5}{7}-1\frac{3}{4}}{3\div 2\frac{1}{2}-\frac{4}{15}} + \frac{3 \hspace{0.1in} of \hspace{0.1in} 1\frac{1}{4}-3\frac{1}{8}}{2 \hspace{0.1in} of \hspace{0.1in} 2 \frac{2}{5}-4\frac{1}{5}}$

XI) In a village consisting of 150 females and 100 males, $\frac{1}{15}$ of all females and $\frac{1}{10}$ of all males are graduates. What fraction of all the villagers are graduates?

XII) $\frac{7}{11}$ of all the money in Mr Ghoshn’s bank account is Rs 98000/-. How much money does Mr Ghoshn have in his bank account?

XIII) A $116\frac{2}{3}$ m long cable is cut into equal pieces measuring $8\frac{1}{3}$ m each. How many such small pieces are there?

XIV) $\frac{1}{6}$ of a ship’s crew are deck officers, $\frac{1}{4}$ are engineers and stewards, and the rest are sailors. If there are 48 crew members in all, how many sailors are on board the ship?

XV) $\frac{2}{7}$ part of a road was paved on the first day, $\frac{1}{5}$ part on the second day, and $\frac{1}{3}$ part was paved on the third day. If $443\frac{1}{3}$ m was paved on the fourth day to complete the road, what is the total length of the road paved?

XVI) The perimeter of an isosceles trapezium measures $13\frac{7}{30}$ cm. If its unequal sides measure $3\frac{2}{5}$ cm and $5\frac{1}{6}$ cm, find the measure of its equal sides.

XVII) A father and his two sons construct a house for INR 525,000. The elder son contributes $\frac{3}{5}$ of his father’s contributions while the younger son contributes $\frac{1}{2}$ of his father’s contributions. How much do the three contribute individually?

XVIII) Ramu inherited $\frac{2}{9}$ of the money his grandfather left behind while his cousin Rakesh’s share was $\frac{1}{7}$. If Ramu’s share was Rs 60000/- more than Rakesh’s share, find how much money their grandfather left behind.

XIX) A, B and C receive a total of Rs 2016 as monthly allowance from their dad such that C gets $\frac{1}{2}$ of what A gets, and B gets $1\frac{2}{3}$ times C’s share. How much money do the three brothers get individually?

XX) $\frac{1}{4}$ studentts of a school come by school bus while $\frac{2}{5}$ students ride a bicycle to school. All the other students walk to school, of which $\frac{1}{3}$ walk on their own and the rest are escorted by an elder. If 196 students come to school walking on their own, how many students study in that school?

Regards,

Nalin Pithwa

## Skill Check VIII: IITJEE Foundation Math

I. Find the prime factorization of the following numbers: (a) 420 (b) 995 (c) 1224 (d) 8712

II. Find the HCF or GCD of the following numbers : (a) 170 and 340 (b) 3535, 9191 and 9896 (c) 1064, 4560, and 3004 (d) 80010, 71160, 62100 and 11520.

III. Find the LCM of the following numbers: (i) 56, 84, and 77 (ii) 495, 990, and 1962 (iii) 1674, 1716, and 2532 (iv) 5220, 1860, 3870, and 2034

IV. Find the greatest 5-digit number that is exactly divisible by 135, 225, and 405.

V. The length, breadth and length of a room are 1750 cm, 7050 cm, and 4025 cm respectively. Find the length of the longest tape which can measure the three dimensions of the room exactly.

VI. Two buses start off together from the terminal at 6 am on different routes. A round trip by one bus takes 36 minutes while the other bus takes 48 minutes. If the buses keep making round trips without stopping at all, how many times will their drivers meet at the terminus till they stop work at 6 pm ?

Regards.

Nalin Pithwa

## Skill Check VII: IITJEE Foundation Math

A) Find the LCM of the following numbers by the prime factorization method: (a) 24, 36 and 72 (b) 84 and 112 (c) 144 and 192 (d) 624 and 520 (e) 225 and 270 (f) 1008 and 1512 (g) 2310, 1540, and 770 (h) 840, 504, and 672 (i) 528, 396, and 352 (j) 6552, 4368, and 9828.

B) Find the LCM of the following numbers by the common division method: (a) 336 and 224 (b) 840 and1260 (c) 630 and 840 and 504 (d) 864, 1296, and 576 (e) 144, 216, and 384 (f) 1764, 1176, and 2352 (g) 260, 390, 156, and 104 (h) 1170, 780, 1755, and 2340 (i) 2520, 1680, 3780, and 3024 (j) 2730, 1950, 3822, and 1820.

C) Find the smallest number that is exactly divisible by 2016 and 3024.

D) Find the greatest 5-digit number that is exactly divisible by 420, 490, and 280.

E) Find the smallest 6-digit number which, when divided by 96, 144, 72, and 192, leaves exactly 8 as a remainder.

F) The LCM of two coprime numbers is 70560. If one of the numbers is 245, find the other number.

G) The LCM of 42 and another number is 168. If the HCF of the two numbers is 14, find the other number.

H) Four bells begin to toll together. The bells tolls after 8, 10, 12, and 15 seconds, respectively. After how long will all four bells toll together?

I) A toy soldier salutes after taking 14 steps while another salutes after every 21 steps. If both toy soldiers take a step every second, how long will it take for the toy soldiers to salute together five times after starting off together?

Regards,

Nalin Pithwa

## Skill Check VI: IITJEE Foundation Math

A) Find all the factors of the following numbers: 42, 66, 88, 180, 810.

B) Use prime tree factorization to find the factors of : 1122, 2211, 2121, 8181, 8000.

C) Find the HCF of the following numbers by prime factorization: (a) 88 and 99 (b) 84 and 108 (c) 80 and 96 (d) 208 and 234

D) Find the HCF or GCD of the following numbers by Euclid’s Long Division Method: (a) 432, 540, and 648 (b) 408, 476, and 510 (c) 1350 and 1800 (d) 3600 and 5400 (e) 7560 and 8820 (f) 7920 and 8910 (g) 14112 and 12936 (h) 25740 and 24024 (i) 108, 288, and 360 (j) 1056, 1584, and 2178

E) Find the HCF or GCD of the following numbers by Euclid’s Long Division Method: (a) 1701, 1575, and 2016 (b) 4680, 4160, and 5200 (c) 3168. 3432, and 3696 (d) 4752, 5184, and 5616 (e) 8640, 10368, 12096 (f) 9072, 8400, 9744

F) Find the greatest number that divides 10368, 9504 and 11232 exactly leaving no remainders.

G) Find the greatest number that divides 7355, 8580, and 9805 leaving exactly 5 as a remainder in each case.

H) Find the greatest number that divides 9243 and 12325 leaving exactly 3 and 5 as remainders respectively.

I) What would be the length of the the longest tape that can be used to measure the length and breadth of an auditorium 204 feet wide and 486 feet long in an exact number of times.

J) A big cardboard picture 126 cm wide and 135 cm long is to be cut up into square pieces to create a jigsaw puzzle. How many small pieces would go on to make the jigsaw puzzle if each piece is to be equal and of the maximum possible size?

K) Square placards need to be cut out from a rectangular piece of cardboard 60 inches wide and 72 inches long. What is the maximum number of equal sized placards of the biggest possible size that can be cut out? What would be the length of each placard?

L) Three ribbons, 171 cm, 185 cm, and 199 cm long are to be cut into equal pieces of maximum possible length, leaving bits of ribbons 3 cm long from each. What would be the length of each piece of ribbon and how many such pieces can one get ?

M) The capacities of two emtpy water tanks are 504 litres and 490 litres. What would be the maximum capacity of a bucket that can be used an exact number of times to fill the tanks? How many bickets full of water will be needed?

Regards,

Nalin Pithwa

## Skill Check V: IITJEE Foundation Math

I. Consider the following relationship amongst various number systems: $\mathcal{N} \subset \mathcal{W} \subset \mathcal{Z} \subset \mathcal{Q} \subset \mathcal{R}$.

Write the following numbers in the smallest set or subset in the above relationship:

(a) 8 (b) -8 (c) +478 (d) -2191 (e) -21.91 (f) +3.6 (g) 0 (h) +4.6 (i) $-6.\dot{7}$ (j) 8.292992999…(k) $\frac{3}{8}$ (l) $\frac{8}{2}$ (m) $0 \frac{0}{7}$ (n) $-3\frac{1}{5}$ (o) $\frac{22}{33}$ (p) $\sqrt{64}$ (r) $\sqrt{6.4}$ (s) $2+\sqrt{3}$ (t) $6\sqrt{4}$ (u) $4\sqrt{6}$

II. If $\frac{22}{7} = 3.1428571...$ is $\frac{22}{7}$ an irrational number?

III. Fill in the boxes with the correct real numbers in the following statements: (a) $2\sqrt{7}+\sqrt{7} = \Box+ 2\sqrt{7}$ (b) $3.\dot{8} + 4.65 = 4.65 + \Box$ (c) $\Box + 29 = 29 + 5\sqrt{10}$ (d) $3.\dot{9} + (4.69 +2.12) = (\Box + 4.69) + 2.12$ (e) $(\frac{7}{8} + \frac{3}{7})+\frac{6}{5} = (\frac{6}{5} + \frac{3}{7}) + \Box$ (f) $3\sqrt{2}(\sqrt{3}+2\sqrt{5}) = (3\sqrt{2}+2\sqrt{5}) + (\Box \times \Box)$ (g) $1\frac{3}{7} (2\frac{7}{11} + \Box) = (1\frac{3}{7} \times 1 \frac{8}{9}) + (1\frac{3}{7} + 2\frac{7}{11})$ (h) $2\frac{1}{3} + \Box =0$ (xi) $\frac{7}{-8} \times \Box = 1$ (i) $-7.35 + \Box = 0$

IV. Find the answers to the following expressions by using the properties of addition and multiplication of real numbers:

Before that, we can recapitulate the relevant properties here :

Properties of Real Numbers:
Closure Property: The sum, difference, product,or quotient of two real numbers is a real number.

Commutative Property of Addition and Multiplication:

A change in the order of addition or multiplication of two real numbers does not change their respective sum or product. (a) $x+y = y+x$ (b) $x \times y = y \times x$

Associative Property of Addition and Multiplication:

A change in the grouping of three real numbers while adding or multiplying does not change their respective sum or product :

$(a+b)+c = (a+b)+c$ and $a \times (b \times c) = (a \times b) \times c$

Distributive Property of Multiplication over Addition:

When a real number is multiplied by the sum of two or more real numbers, the product is the same as the sum of the individual products of the real number and each addend.

$m(a+b) = ma+mb$. Clearly, multiplication has “distributed” over addition.

Identity Property of Real Numbers
The addition of zero or the multiplication with one does not change a real number. That is,

$a+0=0+a=a$ and $a \times 1 = a = 1 \times a$

Inverse Property of Real Numbers

• Corresponding to every real number, there exists another real number of opposite sign such that the sum of the two real numbers is zero: $a+ a^{'}=0$ such that $a^{'}=-a$
• Corresponding to every (non-zero) real number, there exists a real number, known as its reciprocal, such that the product of the two real numbers is 1. That is, $a \times \frac{1}{a} = 1$, where $r \neq 0$.

Now, in the questions below, identify the relevant properties:

(a) $283 +(717 + 386)$

(b) $(2154 - 1689) + 1689$

(c) $3.18 + (6.82+1.35)$

(d) $(6.784-3.297) + 3.297$

(e) $\frac{7}{13} + (\frac{6}{13}-1)$

(e) $0.25 \times (4.17 -0.17)$

(f) $(6.6 \times 6.6) + (6.6 \times 3.4)$

(g) $(\frac{2}{3} \times 5) - (\frac{2}{3} \times 2)$

(h) $(6.\dot{8} \times 5) - (6.\dot{8} \times 4)$

(i) $\frac{6}{7} \times \frac{7}{6} \times \frac{6}{7}$

V) Which of the following operations on irrational numbers are correct?

(a) $6\sqrt{5} - 4\sqrt{3}=2\sqrt{2}$

(b) $\sqrt{7} \times \sqrt{7} = 7$

(c) $3 \sqrt{3} + 3 \sqrt{3} = 6 \sqrt{3}$

(d) $\sqrt{7} \times \sqrt{7} = 49$

(e) $\sqrt{7} + \sqrt{2} = \sqrt{9}$

(f) $2 \sqrt{8} \times 3\sqrt{2} =24$

(g) $8\sqrt{2} + 8 \sqrt{2} =32$

(h) $2\sqrt{3}= 3\sqrt{6} = \frac{2}{3\sqrt{2}}$

(i) $5+\sqrt{3} = 5\sqrt{3}$

(j) $3\sqrt{20} \div 3\sqrt{5}=2$

VI) Find the rationalizing factors of the following irrational numbers:

(a) $\sqrt{10}$

(b) $\sqrt{7}$

(c) $2\sqrt{5}$

(d) $3\sqrt{7}$

(e) $-2\sqrt{8}$

(f) $-6\sqrt{7}$

(g) $\frac{1}{\sqrt{2}}$

(h) $\frac{2}{\sqrt{3}}$

(i) $2\sqrt{3}=4\sqrt{3}$

(j) $7\sqrt{5} - 2\sqrt{5}$

(k) $1+\sqrt{2}$

(l) $3-\sqrt{5}$

(m) $3\sqrt{2}+6$

(n) $4\sqrt{7} + 6\sqrt{2}$

(o) $3\sqrt{6}-2\sqrt{3}$

VII) Rationalize the denominators of the following numbers:

(a) $\frac{1}{\sqrt{3}}$

(b) $\frac{3}{\sqrt{3}}$

(c) $\frac{3}{\sqrt{5}}$

(d) $\frac{8}{\sqrt{6}}$

(e) $\frac{3}{2\sqrt{5}}$

(f) $\frac{\sqrt{5}}{\sqrt{7}}$

(g) $\frac{3\sqrt{3}}{3\sqrt{5}}$

(h) $\frac{3}{\sqrt{5}-sqrt{3}}$

(i) $\frac{5}{\sqrt{3}+\sqrt{2}}$

(j) $\frac{17}{4\sqrt{6}+3\sqrt{5}}$

(k) $\frac{3}{3+\sqrt{3}}$

(l) $\frac{11}{3\sqrt{5}-2\sqrt{3}}$

(m) $\frac{\sqrt{5}}{3\sqrt{5}-3\sqrt{2}}$

(n) $\frac{\sqrt{3}+1}{\sqrt{3}-1}$

(o) $\frac{\sqrt{5}-sqrt{2}}{\sqrt{5}+\sqrt{2}}$

VIII. Find the additive inverse of each of the following irrational numbers:

(i) $\sqrt{7}$ (ii) $3\sqrt{5}$ (iii) $-6\sqrt{7}$ (iv) $5+\sqrt{7}$ (v) $3\sqrt{7} - 2\sqrt{8}$

IX. Find the multiplicative inverse of each of the following irrational numbers:

(i) $\sqrt{6}$ (ii) $\frac{1}{2\sqrt{7}}$ (iii) $\frac{3\sqrt{8}}{2\sqrt{7}}$ (iv) $\frac{4}{3+\sqrt{2}}$ (v) $\frac{2\sqrt{5}+3\sqrt{6}}{5\sqrt{8}-4\sqrt{7}}$

X. Illustrate the closure property of addition of real numbers using the irrational numbers $\sqrt{5}$ and $2\sqrt{5}$.

XI. Illustrate that the closure property does not apply on subtraction of real numbers using two rational numbers: $2\frac{1}{7}$ and $-3\frac{2}{5}$.

XII. Illustrate the distributive property of multiplication over addition of real numbers using three irrational numbers: $3\sqrt{7}$, $-2\sqrt{7}$ and $\sqrt{7}$.

Regards,

Nalin Pithwa