Letter-numbers and algebraic expressions

From words to letters

Shabnam is 3 years older than Aftab, so when Aftab is 10, Shabnam is 13; when Aftab is 18, Shabnam is 21. The rule in words is: Shabnam’s age = Aftab’s age + 3. Writing a for Aftab’s age and s for Shabnam’s turns it into s = a + 3. If a is 23, then s = 23 + 3 = 26. The rule can be turned round, too: Aftab is 3 years younger, so a = s − 3, and when Shabnam is 20, Aftab is 17.

Letters such as a and s that stand for numbers are called letter-numbers, and expressions containing them, such as a + 3, are algebraic expressions. Any letter would do; we choose ones that are easy to remember.

Writing formulas

Parthiv lays matchstick Ls side by side, and each L uses 2 sticks. So 5 Ls need 5 × 2 sticks, 45 Ls need 45 × 2, and n Ls need 2 × n. Ketaki buys coconuts at ₹35 each and jaggery at ₹60 a kg: c coconuts and j kg of jaggery cost c × 35 + j × 60 rupees, so 10 coconuts and 5 kg cost ₹650, and 8 coconuts and 9 kg cost ₹820. The perimeter of a square of side q is 4 × q. Relations like these, written with letter-numbers, are called formulas.

Formulas let us answer many questions at once. A triangle with all sides q has perimeter 3q, a regular pentagon 5q and a regular hexagon 6q. A pipe of 20 m joined to one of k m makes 20 + k metres. Phrases turn into expressions too: 5 more than a number x is x + 5, 4 less is x − 4, 2 less than 13 times it is 13x − 2, and 13 less than 2 times it is 2x − 13.

Leaving out the multiplication sign

The multiples of 4 form the sequence 4, 8, 12, 16, …. Its third term is 4 × 3 and its 29th term is 4 × 29, so its nth term is 4 × n. It is standard to drop the × sign and write 4n, with the number first. In the same way 5m + 3 means 5 × m + 3.

Finding the value

An algebraic expression takes a number value when each letter-number is replaced by a number. So 7k = 7 × 4 = 28 when k = 4, and 5m + 3 = 5 × 2 + 3 = 13 when m = 2. Arithmetic rules still apply: work out the terms, and brackets first. For 23 − 10 × 2, write the terms 23 and −10 × 2, so the value is 23 + (−20) = 3.

Common slips come from forgetting what the letters mean. If d = 6, then 3d is 3 × 6 = 18, not the two-digit number 36. If r = 8, 2r + 1 is 17, not 29. If s = 7, 3s − 2 is 21 − 2 = 19, not 3 × 5 = 15. If m = −6, then 3(m + 1) = 3 × (−5) = −15. If h = 5 and n = 6, then h − (3 − n) = 5 − (−3) = 8.

Why it matters

Formulas are used everywhere, from bills and recipes to perimeters and areas, because one expression works for every value. Next, see how to tidy expressions in simplifying algebraic expressions, and how they explain patterns in patterns and relationships with algebra. Answers to every question are in the NCERT solutions for Chapter 4; see all of the Chapter 4 notes.