Expressions using Letter-Numbers · Lesson 2 of 11
Writing Expressions for Everyday Relationships
“Turn costs, lengths, and everyday instructions into expressions with clearly defined letters.”
• Translate more than, less than, and times into the correct operations. • Build expressions involving several different quantities. • Distinguish a fixed contribution from an amount repeated for each item. • Use expressions for money, combined lengths, and regular-shape perimeters. • Interpret an expression by describing a situation that matches it.
Translate the Meaning, Not Just the Words
An expression is a short set of instructions for calculating a quantity. To write it correctly, decide what you start with and what happens to it. Words such as “more”, “less”, and “times” describe different operations; the order of the words can also matter.
If x represents a number, five more than it is x + 5. Four less than it is x − 4: begin with x and remove four. “Four less than x” is not 4 − x, because that begins with four. For more complicated descriptions, work through the operations in the order suggested by the meaning.
| Description | Expression | How to read it |
|---|---|---|
| 5 more than a number x | x + 5 | Start with x, add 5. |
| 4 less than x | x − 4 | Start with x, subtract 4. |
| 2 less than 13 times x | 13 × x − 2 | Multiply x by 13, subtract 2. |
| 13 less than 2 times x | 2 × x − 13 | Multiply x by 2, subtract 13. |
Problem
Write and evaluate “2 less than 13 times a number” and “13 less than 2 times a number” when the number is 6.
- 1.Let x be the number. The first description gives 13 × x − 2; the second gives 2 × x − 13.
- 2.For x = 6, the first becomes 13 × 6 − 2 = 78 − 2 = 76.
- 3.The second becomes 2 × 6 − 13 = 12 − 13 = −1. Reading the word order carefully produces different expressions and different values.
One Cost for Each Kind of Purchase
When several kinds of items are bought, calculate each kind’s cost separately before adding. A coconut costs ₹35 and jaggery costs ₹60 per kilogram. The number of coconuts and the number of kilograms of jaggery may change independently, so we need two clearly defined letters.
Let c be the number of coconuts and j be the amount of jaggery in kilograms. Coconut cost is 35 × c rupees; jaggery cost is 60 × j rupees. The total is their sum. The multiplier 35 belongs to c because ₹35 is the price per coconut, and the multiplier 60 belongs to j because ₹60 is the price per kilogram.
Problem
Find the total cost for 10 coconuts and 5 kg of jaggery. Then find the total for 7 coconuts and 4 kg.
- 1.Use C = 35 × c + 60 × j, keeping each price attached to its own quantity.
- 2.For c = 10 and j = 5: C = 350 + 300 = ₹650.
- 3.For c = 7 and j = 4: C = 245 + 240 = ₹485. Changing the quantities changes the total without changing the formula.
The same idea works for notes of different values. Let x, y, and z count ₹100, ₹20, and ₹5 notes respectively. First convert each count to a money amount. Adding x + y + z gives the number of notes, whereas adding their values gives the amount of money.
Problem
Find the amount in 3 ₹100 notes, 5 ₹20 notes, and 6 ₹5 notes. Also express the value of 8 ₹100 notes, 4 ₹20 notes, and z ₹5 notes.
- 1.For the first collection, multiply each note value by its count: 3 × 100 + 5 × 20 + 6 × 5.
- 2.The total is 300 + 100 + 30 = ₹430.
- 3.For the second collection, the known part is 800 + 80 = ₹880. The z remaining notes contribute ₹5z, so the amount is 880 + 5z rupees.
Fixed Parts and Repeated Parts
Not every number in a situation is repeated for every item. A flour mill takes ten seconds to start, then eight seconds to grind each kilogram of grain. For one continuous run, the start-up time occurs once. Only the grinding time grows with the number of kilograms.
Problem
How long does one mill run take for 6 kg of grain? Explain why (10 + 8) × y is unsuitable.
- 1.Separate the fixed 10-second start-up from 8 seconds for each kilogram.
- 2.Grinding 6 kg takes 8 × 6 = 48 seconds. Include the one start-up: 10 + 48 = 58 seconds.
- 3.The expression (10 + 8) × y repeats the start-up for every kilogram. It would give 108 seconds for 6 kg, describing a different procedure.
A combined pipe length illustrates another fixed-plus-changing relationship. A 20 m pipe is joined to another pipe k metres long. Assuming the full lengths join without overlap, the combined length is 20 + k metres. There is no reason to multiply by k: the second pipe adds length rather than repeating the first pipe.
Geometry and Reading Expressions Backwards
A perimeter expression comes from counting all sides of a boundary. For regular figures, every side has the same length, so multiplication abbreviates repeated addition. “Regular” also means the angles are equal; the equal-side fact is the part used in these perimeter calculations.
| Figure | Side length | Perimeter |
|---|---|---|
| Triangle with equal sides | a | 3a |
| Regular pentagon | a | 5a |
| Regular hexagon | a | 6a |
Problem
A regular hexagon has side length 4 cm. Find its perimeter and explain the formula.
- 1.A hexagon has six sides, and regularity makes each side the same length.
- 2.With side length a, the boundary total is a + a + a + a + a + a = 6a.
- 3.Replace a by 4: 6 × 4 = 24 cm. The answer is a length, not an area.
You can reverse the process and create a situation for an expression. For example, 8x + 3y could be the cost of x pencils at ₹8 each and y erasers at ₹3 each. The expression 15j − 2k could describe a points game with j successful actions worth 15 points each and k unsuccessful actions costing 2 points each. Different sensible situations can produce the same expression, so always state what the letters and numbers mean.
Do not add the prices first and then multiply by the combined quantities unless every item really has that combined price. Also check units: a money expression gives rupees, a pipe expression gives metres, and a time expression gives seconds.
Check Your Understanding
Use the ideas from this lesson to choose an answer. Explain your choice to yourself before opening the explanations below.
Quiz
Which expression means “4 less than the number x”?
A 20 m pipe is joined without overlap to a k m pipe. What is the length?
A mill starts once in 10 seconds and needs 8 seconds per kg. Which expression models y kg?
A regular pentagon has side length a. What is its perimeter?
What does 100x + 20y + 5z represent when x, y, z count the corresponding notes?
Start with x and subtract four.
Practice Problems
- Write expressions for 5 more than x, 4 less than x, 2 less than 13 times x, and 13 less than 2 times x.
- Find the cost of 8 coconuts at ₹35 each and 9 kg jaggery at ₹60 per kg.
- A collection has value 6 × 100 + 4 × 20 + 3 × 5 = 695 rupees. State the number of each kind of note.
- Write the amount in x ₹100 notes, y ₹20 notes, and z ₹5 notes. What is the separate expression for the number of notes?
- Find the mill time for y = 4 kg using one start-up.
- Write perimeters for an equal-sided triangle, a regular pentagon, and a regular hexagon with side length a cm.
- Describe a situation for 8x + 3y and another for 15j − 2k. Define the letters.
- Jowar roti costs ₹30 per plate and pulao ₹20 per plate. Write the revenue from x roti plates and y pulao plates. Why is 50(x + y) wrong?
x + 5; x − 4; 13x − 2; 2x − 13.
Key Takeaways
• Read the operation and its order before choosing symbols. • “Less than x” means subtract from x. • Different quantities usually need separately defined letters. • Multiply each quantity by its own rate or value. • A fixed contribution is included once; a repeated contribution is multiplied. • Expressions need stated meanings and appropriate units.