Expressions Using Letter-Numbers
Learn how to use letters in math to describe patterns and relationships that work for any number.
Age Problems That Repeat
Shabnam is 3 years older than Aftab. When Aftab is 10, Shabnam is 13. When Aftab is 18, Shabnam is 21. When Aftab is 25, Shabnam is 28.
We keep adding 3 to Aftab's age to get Shabnam's age. But writing this out every time is tedious! What if we use a letter to represent Aftab's age, and write the rule once? Then we can use it forever!
That's what letter-numbers (or variables) do: they let us express general rules that work for any number.
Imagine a recipe: "Add 2 cups of flour." The recipe works whether you're making bread for 1 person or 10 people—you still add 2 cups. Now imagine saying: "Add 2 cups of flour for every n people." Here, n is a placeholder: it can be any number. The rule is general and flexible, like the recipe that works for any number of people!
Spotting the Relationship
Shabnam's age = Aftab's age + 3. This relationship always holds, no matter what Aftab's age is.
We could write out examples forever:
- If Aftab is 5, Shabnam is 8
- If Aftab is 15, Shabnam is 18
- If Aftab is 25, Shabnam is 28
But there's a pattern! Shabnam's age is always 3 more than Aftab's.
Introducing Letter-Numbers
Instead of writing out every example, we use a letter to represent the variable (the changing quantity):
Let a = Aftab's age (in years)
Let s = Shabnam's age (in years)
Then: s = a + 3
This single equation captures the relationship for all possible ages!
Letters are convenient and conventional in math. 'a' for Aftab, 's' for Shabnam. We could use any letter—it's just notation. The power is in the idea of using a symbol to represent a variable quantity.
Using Letter-Numbers to Find Specific Values
If Aftab is 23 years old, what is Shabnam's age?
Replace a with 23 in the expression s = a + 3:
s = 23 + 3 = 26
Shabnam is 26 years old.
We "substitute" the specific value (23) for the letter (a). This converts the general relationship into a specific answer.
Algebraic Expressions Are General Formulas
An algebraic expression is a math phrase that contains one or more letter-numbers.
Examples:
a + 3(Shabnam's age given Aftab's)2n(twice a number n)c × 35 + j × 60(total cost of c coconuts and j kg of jaggery)
Each is a template or formula that generates answers for different inputs.
A Concrete Pattern: Matchsticks
Parthiv arranges matchsticks into Ls. Each L uses 2 matchsticks.
- 1 L needs 2 matchsticks
- 2 Ls need 4 matchsticks
- 3 Ls need 6 matchsticks
- 5 Ls need 10 matchsticks
- 45 Ls need 90 matchsticks
Pattern: Number of matchsticks = 2 × Number of Ls
Generalizing with a Letter
Let n = number of Ls
Then: Number of matchsticks = 2 × n or simply 2n
This single expression works for any number of Ls!
Instead of saying "2 × 45" when asked about 45 Ls, we have a general formula. We can apply it immediately to any problem without explaining the pattern each time.
Expressions with Multiple Operations
Example: Ketaki buys coconuts (₹35 each) and jaggery (₹60 per kg).
Let c = number of coconuts, j = kilograms of jaggery
Cost expression: 35c + 60j
If she buys 7 coconuts and 4 kg jaggery:
35(7) + 60(4) = 245 + 240 = ₹485
In 35c + 60j, we have two terms: 35c and 60j. Each term is a product (or could be a single letter). We add them together. This is the same term structure we learned in Chapter 2!
Writing Expressions from Words
Problem: "5 more than a number"
Solution: Let x = the number. Expression: x + 5
Problem: "3 times a number, minus 2"
Solution: Let y = the number. Expression: 3y – 2
Problem: "Half of a number"
Solution: Let z = the number. Expression: z/2 or z ÷ 2
The Perimeter of a Square
A square has 4 equal sides. If each side has length s, then the perimeter is 4s.
Example: If s = 7 cm, perimeter = 4(7) = 28 cm.
The formula P = 4s works for ANY square, no matter its size!
Extending to Other Shapes
Triangle with all sides equal: Let t = side length. Perimeter = 3t
Regular pentagon (all sides equal): Let p = side length. Perimeter = 5p
Regular hexagon (all sides equal): Let h = side length. Perimeter = 6h
Perimeter = (number of sides) × (side length). This works for any regular polygon! Formulas capture universal truths about shapes.
Writing Formulas from Real Situations
Example: Munirathna has a 20 m pipe. He wants to add another pipe of length k meters. What's the combined length?
Formula: Combined length = 20 + k
If he adds a 5 m pipe: 20 + 5 = 25 m
If he adds a 12 m pipe: 20 + 12 = 32 m
One formula captures all scenarios!
Order of Operations with Letter-Numbers
When we write algebraic expressions, the same order of operations applies! A grain mill takes 10 seconds to start. Then it grinds 1 kg in 8 seconds. How long to grind y kg?
Correct expression: 10 + 8y
Not: (10 + 8) × y = 18y (this would mean we wait 10 + 8 = 18 seconds, then multiply by y, which is wrong!)
Why? By order of operations, 8y is a single term (multiplication happens first). So 10 + 8y means: wait 10 seconds (startup), then grind for 8y seconds (8 seconds per kg, times y kg).
Students sometimes think: "I have a + and a × in the expression, so I must decide which to do first." Not true! The order of operations is fixed: multiplication/division before addition/subtraction. Letters don't change this rule.
Evaluating Expressions: Substitution and Calculation
Once we have a general formula, we can substitute specific values and compute answers.
Example 1: Cost of Items. Expression for total cost: c × 35 + j × 60 (c coconuts at ₹35, j kg jaggery at ₹60). Find cost of 8 coconuts and 9 kg jaggery: substitute c = 8 and j = 9: 8 × 35 + 9 × 60 = 280 + 540 = ₹820
Example 2: Distance Over Time. Expression for distance traveled: d = s × t (speed s, time t). Find distance at 60 km/h for 3 hours: substitute s = 60 and t = 3: d = 60 × 3 = 180 km
1) Write the expression. 2) Replace each letter with its given value. 3) Calculate following order of operations. 4) State your answer with units (rupees, km, etc.)
Socratic Sandbox — Test Your Understanding
1. If the expression for Shabnam's age is s = a + 3, and Aftab is 30, what is Shabnam's age?
Reveal Hint
Replace the letter 'a' with 30 in the expression.
Reveal Answer
s = 30 + 3 = 33. Shabnam is 33 years old.
2. Write the expression for "twice a number, plus 5"
Reveal Hint
Choose a letter for the unknown number. "Twice" means multiply by 2.
Reveal Answer
Let n = the number. Expression: 2n + 5
3. Why is the expression for total cost of c coconuts and j kg jaggery written as 35c + 60j instead of 35 + 60 + c + j?
Reveal Hint
What does each number (35 and 60) represent? How many coconuts and jaggery are there?
Reveal Answer
35c means "35 rupees per coconut times c coconuts" = total cost of coconuts. 60j means "60 rupees per kg times j kg" = total cost of jaggery. If we wrote 35 + 60 + c + j, we'd be adding prices to quantities, which doesn't make sense! The expression must show the total cost calculation correctly.
4. Why is 10 + 8y different from (10 + 8) × y when solving the grain mill problem?
Reveal Hint
In the first expression, which operation happens first (by order of operations)? In the second, the brackets change what happens first. Does it make sense physically?
Reveal Answer
In 10 + 8y, multiplication happens first: we calculate 8y (grinding time), then add 10 (startup time). In (10 + 8) × y, brackets force us to add 10 + 8 first, then multiply by y—which doesn't fit the real scenario. We wait 10 seconds once (not per kg), then grind 8y seconds total. So 10 + 8y is correct!
5. Write a formula for the perimeter of a regular hexagon with side length h. Then find the perimeter if h = 5 cm.
Reveal Hint
A hexagon has 6 sides. If each is length h, what's the total perimeter?
Reveal Answer
Formula: P = 6h. If h = 5 cm, then P = 6 × 5 = 30 cm.
6. Krithika has some notes: x notes of ₹100, y notes of ₹20, and z notes of ₹5. Write an expression for her total money. If x = 3, y = 5, z = 6, how much does she have?
Reveal Hint
Each note type contributes: (number of notes) × (value of note). Add all contributions.
Reveal Answer
Expression: 100x + 20y + 5z. With x = 3, y = 5, z = 6: 100(3) + 20(5) + 5(6) = 300 + 100 + 30 = ₹430