Class 7 Maths Chapter 4: Expressions using Letter-Numbers Notes

What is this chapter about?

Shabnam is 3 years older than Aftab. Whatever Aftab’s age is, we add 3 to get Shabnam’s. Writing a for Aftab’s age turns this into a short rule, s = a + 3, that works for every age. This chapter is about using letters in place of numbers to state such relations briefly, to work with them, and to explain why patterns hold.

Letter-numbers and formulas

Letters such as a and s that stand for numbers are letter-numbers, and expressions that contain them, such as a + 3, are algebraic expressions. A matchstick L needs 2 sticks, so n Ls need 2 × n. Ketaki’s laddu ingredients cost c × 35 + j × 60 rupees for c coconuts and j kg of jaggery. The perimeter of a square of side q is 4 × q. General relations written like this are called formulas. Read more in letter-numbers and algebraic expressions.

Shorter writing and finding values

The multiples of 4 are 4, 8, 12, …, so the nth term is 4 × n, which we shorten to 4n: the number comes first and the multiplication sign is left out. An algebraic expression takes a number value when every letter-number is replaced by a number. So 7k is 28 when k = 4, and 5m + 3 is 5 × 2 + 3 = 13 when m = 2. Signs and brackets need care: if a = −4, then 10 − a = 10 + 4 = 14.

Simplifying expressions

The rules for numbers work for letter-numbers too. The perimeter of a rectangle, l + b + l + b, rearranges to 2l + 2b, its simplified form; both give 14 when l = 3 and b = 4. Terms with the same letter-numbers, such as 5c, 3c and 10c, are like terms and add into one term, 18c, by the distributive property. Terms like 18c and 11d are unlike and cannot be combined. A minus sign before a bracket changes every sign inside it: (40x + 75y) − (6x + 10y) = 34x + 65y. Read simplifying algebraic expressions.

Equal or not?

Two expressions are equal when they give the same value for every value of their letter-numbers. The simplified and original forms always do. But 5u (5 times u) and 5 + u (5 more than u) give 10 and 7 when u = 2, so they are not equal; nor are 10y − 3 and 10(y − 3).

Patterns and relationships

Algebra helps find and explain patterns. A number machine that turns 5 and 2 into 8, and 8 and 1 into 15, follows the formula 2a − b. In a saree border with designs A, B, C repeating, the nth C is at position 3n, the nth B at 3n − 1 and the nth A at 3n − 2, so position 122 has B. In any 2 × 2 square of a calendar with top-left date a, the numbers are a, a + 1, a + 7 and a + 8, so both diagonal sums are 2a + 8: the pattern holds for every square. A row of matchstick triangles needs 2y + 1 sticks at step y. Read patterns and relationships with algebra.

Before we move on

Algebraic expressions model relations between quantities, follow the same rules as numbers, and say in a few symbols what would take long sentences in words. Answers to every question, both Mind the Mistake sets and all three Figure it Out sets are in the NCERT solutions for Chapter 4.