Algebra Play

Find the dates if the final answers are the following: (i) 1269 (ii) 394 (iii) 296

Without building the entire pyramid, find the number in the topmost row given the bottom row in each of these cases. Bottom row: 4, 13, 8, 7, 11, 3, 10, 14, 25

Write an expression for the topmost row of a pyramid with 4 rows in terms of the values in the bottom row.

What can you say about the numbers in the pyramid and the number at the top in the following cases? (i) The first four Virahānka-Fibonacci numbers are written in the bottom row of a four row pyramid. (ii) The first 29 Virahānka-Fibonacci numbers are written in the bottom row of a 29 row pyramid.

If the bottom row of an n row pyramid contains the first n Virahānka-Fibonacci numbers, what can we say about the numbers in the pyramid? What can we say about the number at the top?

Fill the digits 1, 3, and 7 in □□ × □ to make the largest product possible.

Fill the digits 3, 5, and 9 in □□ × □ to make the largest product possible.

Choose a 2-digit number of different digits and don’t reveal it! Reverse the digits to get another number. Find their difference. Divide the result by 9. In the trick given above, what is the quotient when you divide by 9? Is there a relationship between the two numbers and the quotient?

In the trick given above, instead of finding the difference of the two 2-digit numbers, find their sum. What will happen? Observe that all these numbers are divisible by 11. Is this always true? Can we justify this claim using algebra?

Consider any 3-digit number, say abc (100a + 10b + c). Make two other 3-digit numbers from these digits by cycling these digits around, yielding bca and cab. Now add the three numbers. Using algebra, justify that the sum is always divisible by 37. Will it also always be divisible by 3?

Consider any 3-digit number, say abc. Make it a 6-digit number by repeating the digits, that is abcabc. Divide this number by 7, then by 11, and finally by 13. What do you get? Try this with other numbers. Figure out why it works.

A farm has some horses and hens. The total number of heads of these animals is 55 and the total number of legs is 150. How many horses and how many hens are on the farm?

A mother is 5 times her daughter's age. In 6 years' time, the mother will be 3 times her daughter's age. How old is the daughter now?

Two friends, Gauri and Naina, are cowherds. One day, they pass each other on the road with their cows. Gauri says to Naina, 'You have twice as many cows as I do'. Naina says, 'That's true, but if I gave you three of my cows, we would each have the same number of cows'. How many cows do Gauri and Naina have?

I run a small dosa cart and my expenses are as follows: Rent for the dosa cart is ₹5000 per day. The cost of making one dosa (including all the ingredients and fuel) is ₹10. (i) If I can sell 100 dosas a day, what should be the selling price of my dosa to make a profit of ₹2000? (ii) If my customers are willing to pay only ₹50 for a dosa, how many dosas should I aim to sell in a day to make a profit of ₹2000?

Evaluate the following sequence of fractions: 1/3, (1+3)/(5+7), (1+3+5)/(7+9+11). What do you observe? Can you explain why this happens?

Karim and the Genie: Karim was taking a nap under a tree. He had a dream about a magical lamp and a genie... [Story about doubling coins and paying 8 coins each round] (i) How many coins did Karim initially have? (ii) For what cost per round should Karim agree to the deal, if he wants to increase the number of coins he has? (iii) Through its magical powers, the genie knows the number of coins that Karim has. How should the genie set the cost per round so that it gets all of Karim's coins?