Author Archives: Nalin Pithwa

I am a DSP Engineer and a mathematician working in closely related areas of DSP, Digital Control, Digital Comm and Error Control Coding. I have a passion for both Pure and Applied Mathematics.

Wisdom of mathematician Lloyd Shapley, game theorist

It is very hard to be a good mathematician unless you can hold a single thought in your mind. — Lloyd Shapley, game theorist.

broken systems

covid corona from a systems viewpoint

Wisdom of John Nash, Jr. Nobel Laureate, Abel Laureate genius mathematician

‘The only thing more important the power of the human brain is the courage of the human heart.” — John Nash, Jr., Nobel Laureate, Abel Laureate genius mathematician

Come, my friends with Alfred, Lord Tennyson

‘Tis not too late to seek a newer world.

Push off, and sitting well in order smite

The surrouding furrows; for my purpose holds

To sail beyond the sunset, and the baths

Of all the western stars, until I die.

It may be that the gulls will wash us down:

It may be we shall touch the Happly Isles,

And see the great Achilles, whom we knew.

Though much is taken, much abides; and though

We are not now that strength which in old days

Moved earth and heaven; that which we are, we are;

One equal temper of heroic hearts,

Made weak by time and fate, but strong in will

To strive, to seek, to find, and not to yield.

—- Alfred, Lord Tennyson.

A word about Hermann Weyl

“It is a crying shame that Weyl is leaving Zurich. He is a great master.” 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.


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?

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}


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?


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?


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 ?


Nalin Pithwa