Chapter 4 — Expressions using Letter Numbers

Chapter 4 of Class 7 Ganita Prakash, 'Expressions Using Letter-Numbers', introduces algebraic expressions — a concise way of expressing mathematical relationships and patterns using letters (letter-numbers) in place of specific numbers. Students learn to write, evaluate, and simplify these expressions and use them to describe and prove patterns.

Letters such as a, n, or x that are used to represent numbers in mathematical expressions are called letter-numbers. For example, if a represents Aftab's age, then Shabnam's age (who is 3 years older) can be written as a + 3.

A mathematical expression containing letter-numbers is called an algebraic expression. For example, a + 3 (Shabnam's age when Aftab's age is a) and 2n (the number of matchsticks needed for n L-shapes) are both algebraic expressions.

As a standard practice, the multiplication sign between a number and a letter-number is skipped — 4 × n is written as 4n. The number is written first, followed by the letter(s). So 7k means 7 × k and 5m means 5 × m.

Sets of terms that involve the same letter-numbers are called like terms — for example, 5c, 3c, and 10c are like terms. Terms with different letter-numbers, such as 18c and 11d, are called unlike terms. Like terms can be added together and simplified into a single term; unlike terms cannot.

Like terms are combined using the distributive property. For example, 5c + 3c + 10c = (5 + 3 + 10)c = 18c. This works because adding 5 times c, then 3 times c, then 10 times c is the same as adding c a total of 18 times.

When a negative sign is outside brackets, every term inside changes sign when the brackets are opened. For example, (40x + 75y) − (6x + 10y) becomes 40x + 75y − 6x − 10y, which simplifies to 34x + 65y.

No. 5u means 5 times the number u, while 5 + u means 5 more than u. These are different operations and give different values for most values of u, so the two expressions are not equal. Two expressions are equal only if they take the same value for every possible replacement of the letter-number.

In any 2 × 2 square on a calendar, if the top-left number is a, the other three numbers are a + 1, a + 7, and a + 8. The two diagonal sums are a + (a + 8) = 2a + 8 and (a + 1) + (a + 7) = 2a + 8. Since both simplify to 2a + 8 for any value of a, the diagonal sums are always equal — this proves the pattern holds for every possible 2 × 2 square, not just specific examples.

For the triangle matchstick pattern in the chapter, each new step adds 2 matchsticks. Step 1 has 3 matchsticks, Step 2 has 5, and so on. The expression for the number of matchsticks at Step y is 3 + 2 × (y − 1), which simplifies to 2y + 1. This formula lets you find the number of matchsticks at any step without drawing and counting.

Mathematical relations expressed using letter-numbers are often called formulas. For example, the perimeter of a square with side length q is given by the formula 4q. Formulas allow a general rule to be stated once and applied to any specific value.

In a repeating saree border pattern with designs A, B, C repeating in order, Design C appears at positions that are multiples of 3, so its nth occurrence is at position 3n. Design B appears one position before C, so its nth occurrence is at 3n − 1. Design A appears two positions before C, so its nth occurrence is at 3n − 2.

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