Tiles and a party

Forming and solving equations gives us the ability to find solutions to many problems in our lives. Here are two.

Example 7: Ranjana creates a sequence of arrangements with square tiles. Can she extend it and make an arrangement using 100 tiles, and if so, at which step? The pattern can be seen in different ways. Method 1 sees Step k as k + k + k + 1 = 3k + 1. Method 2 sees it as k + (2k + 1) = 3k + 1. Different pictures, the same expression — which is itself worth noticing.

So Step k needs 3k + 1 tiles, and the question becomes the equation 3k + 1 = 100. Solve it and you will know whether 100 tiles can be used, and where.

Example 8: Madhubanti organises a party. Each plate of snacks costs ₹25, and the shop charges an extra FIXED ₹50 for delivery. There are 5 in her family including herself, she may spend ₹500, and everyone at the party gets one plate. How many friends can she invite?

Fatima works it out by arithmetic. Out of ₹500, subtract the fixed delivery of ₹50, leaving ₹450. The question is now 'how many plates at ₹25 can be bought for ₹450?' — and 450 ÷ 25 = 18. Counting her 5 family members, she can invite 18 − 5 = 13 friends.

Mahesh uses an unknown for the TOTAL number of people, p. The cost is 25p + 50, and it must be 500, so 25p + 50 = 500. Subtract 50: 25p = 450. Divide by 25: p = 18. Eighteen people, of whom 5 are family, so 13 friends.

Srikanth chooses his unknown differently — f, the number of FRIENDS. Then there are f + 5 people, so the snack cost is 25(f + 5), and after delivery there is ₹450 for snacks: 25(f + 5) = 450. Divide by 25: f + 5 = 18. Subtract 5: f = 13. Three routes, one answer — and the lesson is that CHOOSING what the letter stands for changes the equation but not the truth.