Building Expressions from Real Problems
A garden, a bent wire, and the moment one variable becomes special
AI Generation Prompt
Ultra-wide cinematic banner (16:5 ratio). A peaceful Indian rural garden at golden hour, viewed from a low angle. The garden is rectangular, with a tall wooden picket fence running along the two long sides and a thinner wire fence on the two short sides. Inside, neat rows of vegetables and yellow marigolds stretch into the soft golden distance. A small white farmhouse and a few trees sit in the background. The atmosphere conveys: every fence, every plot, every measurement is secretly an algebraic expression. Painterly cinematic illustration in the style of warm rural Indian art. Dark background. No text, no labels.
A wire of length 20 cm is bent into a rectangle. Different bends give different rectangles — long-and-thin, almost-square, etc. Can you write a single expression that gives the area of any such rectangle, depending on how you bend the wire?
If you call the length , the width is forced — the four sides must add up to 20.
From Mahāvīra's Gaṇita-sāra-saṅgraha (c. 850 CE) — On Practical Calculation
क्षेत्रस्य परिमाणं वा मूल्यं वा निश्चयेन ज्ञेयम्
अक्षरैरेव संख्यया च — एतद् गणितस्य सार्थकता॥
(kṣetrasya parimāṇaṃ vā mūlyaṃ vā niścayena jñeyam / akṣarair eva saṃkhyayā ca — etad gaṇitasya sārthakatā)
'किसी ज़मीन का नाप हो या कीमत — दोनों को अक्षरों और संख्याओं से ठीक-ठीक जाना जा सकता है। यही गणित का मतलब है।'
'Whether it is the measurement of land or its price — both can be known precisely through letters and numbers. This is the meaning of mathematics.'
Mahāvīra, a Jain mathematician, wrote the Gaṇita-sāra-saṅgraha — one of the great Indian medieval algebra texts. He showed how everyday land-measurement, fencing, and pricing problems all become a few lines of algebra. The garden problem on this page is exactly the kind of thing his students worked through 1,200 years ago.
A garden of length and width
A rectangular garden of length metres and width metres needs three things:
- A wire fence along the two length sides — costs ₹100 per metre.
- A wooden fence along the two width sides — costs ₹80 per metre.
- Special seeds to be sown throughout the garden — costs ₹50 per square metre of area.
Let's compute the total cost step by step:
- Length of wire fencing needed = metres. Cost = .
- Length of wooden fencing needed = metres. Cost = .
- Area to be sown = square metres. Seed cost = .
Adding them up:
This expression has three terms: , , and . Two variables: and . No constant (because every bit of the cost depends on the size of the garden).
Notice the term — it has both variables multiplied together. We'll meet such terms again when you formally study quadratic and higher-degree polynomials in later chapters.
AI Generation Prompt
Painted illustration of a rectangular vegetable garden viewed slightly from above. Two long sides have a tall wooden picket fence; two short sides have a thinner wire mesh fence. Length of long sides labelled 'l metres' twice (once on each long side, with a thin red double-headed arrow); width of short sides labelled 'w metres' twice. Inside the garden: neat rows of yellow marigolds and small green vegetable plants. Background shows a small white farmhouse and trees, soft blue sky with white clouds. Style: warm hand-painted Indian textbook illustration. Dark background, orange accent labels, clean technical illustration style.
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For the cost expression , identify: (a) the number of terms, (b) the variables, (c) the coefficient of , (d) is there a constant?
From two variables to one — bending a wire
Now consider a different problem.
A wire of length 20 cm is bent into a rectangle. The wire goes around the rectangle's perimeter, so:
Dividing by 2: length + width = . So if the length is cm, the width must be cm. The area is:
Notice what just changed: this expression has only one variable, . The garden expression had two ( and ). The shop expression on Page 1 had two ( and ). This one has one.
Single-variable expressions are special enough to deserve their own name. We'll meet that name on the next page: polynomials.
For the area expression , identify: (a) the number of terms, (b) the variable, (c) the coefficient of , (d) the coefficient of .
Use to find the area of the rectangle when the length is cm. Then find the area when cm. What do you notice?
Write an algebraic expression for the total cost of an auto-rickshaw ride: a base fare of ₹40 plus ₹15 per kilometre. Then evaluate the cost for a 7 km ride.
Compare three expressions: (A) from the shop, (B) from the garden, (C) from the bent wire. Which one is fundamentally different from the other two, and why?
Practice Yourself — Translate the Story into Algebra
For each scenario, write an algebraic expression. Cover the answers and try.
- A taxi charges ₹50 base fare plus ₹12 per km. Distance = km.
- A mobile data plan: ₹200 fixed + ₹2 per MB used. MB used = .
- A rectangular room of length m and width m needs paint costing ₹15 per square metre.
- A piggy bank has ₹2 coins and ₹5 coins. What is the total amount in rupees?
- A train moves at 80 km/h for hours. Distance covered (km)?
- A pen costs ₹15. You buy pens and pay ₹100, getting some change back. Express the change.
- A square has side length cm. Express its perimeter.
- A square has side length cm. Express its area. (Is this a polynomial in one variable?)
Answers: 1. · 2. · 3. · 4. · 5. · 6. · 7. · 8. — yes, a polynomial in one variable, but not linear (it has , so it's quadratic).
Q1.How many variables does the expression have?
AI Generation Prompt
Ultra-wide cinematic banner (16:5 ratio). A peaceful Indian rural garden at golden hour, viewed from a low angle. The garden is rectangular, with a tall wooden picket fence running along the two long sides and a thinner wire fence on the two short sides. Inside, neat rows of vegetables and yellow marigolds stretch into the soft golden distance. A small white farmhouse and a few trees sit in the background. The atmosphere conveys: every fence, every plot, every measurement is secretly an algebraic expression. Painterly cinematic illustration in the style of warm rural Indian art. Dark background. No text, no labels.
A wire of length 20 cm is bent into a rectangle. Different bends give different rectangles — long-and-thin, almost-square, etc. Can you write a single expression that gives the area of any such rectangle, depending on how you bend the wire?
If you call the length , the width is forced — the four sides must add up to 20.
From Mahāvīra's Gaṇita-sāra-saṅgraha (c. 850 CE) — On Practical Calculation
क्षेत्रस्य परिमाणं वा मूल्यं वा निश्चयेन ज्ञेयम्
अक्षरैरेव संख्यया च — एतद् गणितस्य सार्थकता॥
(kṣetrasya parimāṇaṃ vā mūlyaṃ vā niścayena jñeyam / akṣarair eva saṃkhyayā ca — etad gaṇitasya sārthakatā)
'किसी ज़मीन का नाप हो या कीमत — दोनों को अक्षरों और संख्याओं से ठीक-ठीक जाना जा सकता है। यही गणित का मतलब है।'
'Whether it is the measurement of land or its price — both can be known precisely through letters and numbers. This is the meaning of mathematics.'
Mahāvīra, a Jain mathematician, wrote the Gaṇita-sāra-saṅgraha — one of the great Indian medieval algebra texts. He showed how everyday land-measurement, fencing, and pricing problems all become a few lines of algebra. The garden problem on this page is exactly the kind of thing his students worked through 1,200 years ago.
A garden of length and width
A rectangular garden of length metres and width metres needs three things:
- A wire fence along the two length sides — costs ₹100 per metre.
- A wooden fence along the two width sides — costs ₹80 per metre.
- Special seeds to be sown throughout the garden — costs ₹50 per square metre of area.
Let's compute the total cost step by step:
- Length of wire fencing needed = metres. Cost = .
- Length of wooden fencing needed = metres. Cost = .
- Area to be sown = square metres. Seed cost = .
Adding them up:
This expression has three terms: , , and . Two variables: and . No constant (because every bit of the cost depends on the size of the garden).
Notice the term — it has both variables multiplied together. We'll meet such terms again when you formally study quadratic and higher-degree polynomials in later chapters.
AI Generation Prompt
Painted illustration of a rectangular vegetable garden viewed slightly from above. Two long sides have a tall wooden picket fence; two short sides have a thinner wire mesh fence. Length of long sides labelled 'l metres' twice (once on each long side, with a thin red double-headed arrow); width of short sides labelled 'w metres' twice. Inside the garden: neat rows of yellow marigolds and small green vegetable plants. Background shows a small white farmhouse and trees, soft blue sky with white clouds. Style: warm hand-painted Indian textbook illustration. Dark background, orange accent labels, clean technical illustration style.
Loading simulator…
For the cost expression , identify: (a) the number of terms, (b) the variables, (c) the coefficient of , (d) is there a constant?
From two variables to one — bending a wire
Now consider a different problem.
A wire of length 20 cm is bent into a rectangle. The wire goes around the rectangle's perimeter, so:
Dividing by 2: length + width = . So if the length is cm, the width must be cm. The area is:
Notice what just changed: this expression has only one variable, . The garden expression had two ( and ). The shop expression on Page 1 had two ( and ). This one has one.
Single-variable expressions are special enough to deserve their own name. We'll meet that name on the next page: polynomials.
For the area expression , identify: (a) the number of terms, (b) the variable, (c) the coefficient of , (d) the coefficient of .
Use to find the area of the rectangle when the length is cm. Then find the area when cm. What do you notice?
Write an algebraic expression for the total cost of an auto-rickshaw ride: a base fare of ₹40 plus ₹15 per kilometre. Then evaluate the cost for a 7 km ride.
Compare three expressions: (A) from the shop, (B) from the garden, (C) from the bent wire. Which one is fundamentally different from the other two, and why?
Q1.How many variables does the expression have?