Word-Problem Lab: Letting x Do the Heavy Lifting
Five NCERT word problems, one 4-step recipe, every linear equation worked end-to-end
AI Generation Prompt
Ultra-wide cinematic banner (16:5 ratio). A still-life montage of everyday objects on a wooden desk under a warm desk lamp: a glass piggy bank with ₹2 and ₹5 Indian coins spilling out, a yellow measuring tape unrolling across a wooden plank, a small framed family photo with two figures (mother and child), a freshly-cut wooden fence beam in the foreground. The lighting suggests these scenes await transformation — each one is a story problem waiting to become a single line of algebra. Painterly cinematic illustration in warm hues. Dark background. No text labels.
**A farmer cuts a 300-foot fence into two pieces. The longer piece is 4 times as long as the shorter piece. How long is each?** Most people start guessing: "60 and 240? 50 and 250?" Eventually they hit it. **What if I told you that one habit — done right — turns *every* word problem like this into a 30-second computation?**
Give the *unknown* a name first. The shorter piece is the one you don't know — call it $x$. Now the longer piece is forced.
From Bhāskarāchārya's Lilāvatī (1150 CE)
बीजगणितस्य विद्या वरण्या यथा यथा अधिका
तथा तथा गणितज्ञस्य बुद्धिः शुद्धा भवेत् क्षणात्॥
(bījagaṇitasya vidyā varaṇyā yathā yathā adhikā / tathā tathā gaṇitajñasya buddhiḥ śuddhā bhavet kṣaṇāt)
'जितना अधिक कोई बीजगणित (algebra) पर काम करे, उतनी ही गणितज्ञ की बुद्धि साफ़ हो जाती है — वो भी एक पल में।'
'The more one practises algebra, the clearer the mathematician's mind becomes — instantly.'
Bhāskarāchārya wrote the Lilāvatī — a textbook full of word problems about lotuses, peacocks, jewels, and merchants — because he understood that algebra is not learnt by reading, only by solving. This page asks you to do exactly what his students did 800 years ago: take a story, name the unknown, write the equation, solve.
The 4-step recipe
Every word problem in this set follows the same four-step recipe. Burn it into your habit and the problems become routine.
- Name the unknown. Pick the quantity you most want to know. Call it (or any letter). Write down what it represents.
- Translate every sentence. For each English clause, write the algebraic equivalent. Sums, differences, ratios, products — they all become arithmetic on .
- Combine into one linear equation. Stack the relationships into a single equation in the single unknown.
- Solve and check. Use the standard recipe (move constants, combine, divide). Then plug your answer back into the original story to make sure it works.
The cleverness lives in step 1. Most word problems become easy once you choose the right unknown. The rest is mechanical.
This page works through NCERT Exercise Set 2.2 questions 3 to 7 — five real word problems, each a clean walk through the recipe.
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The present age of Salil's mother is three times Salil's present age. After 5 years, their ages will add up to 70 years. Find their present ages.
Step 1 — Name the unknown. Let Salil's present age be years. Then mother's present age is years.
Step 2 — Translate "after 5 years". In 5 years, Salil will be and his mother .
Step 3 — Translate "their ages will add up to 70".
Step 4 — Solve. Subtract 10: . Divide by 4: .
Answer. Salil is 15 years old today; his mother is 45 years old.
Check. In 5 years, Salil will be 20 and his mother 50. Sum = 70. ✓
The difference between two positive integers is 63. The ratio of the two integers is 2:5. Find the two integers.
Step 1 — Name the unknowns using the ratio. A ratio means the integers are and for some positive . Naming them this way builds the ratio into the unknown.
Step 2 — Translate the difference.
Step 3 — Solve. .
Step 4 — Recover the integers. . .
Answer. The two integers are 42 and 105.
Check. Difference: ✓. Ratio: (divide both by 21) ✓.
Lesson: whenever a problem gives you a ratio, build the ratio into the unknown ( and , or and , etc.). It saves you from writing a separate equation for the ratio.
Ruby has 3 times as many ₹2 coins as she has ₹5 coins. If she has a total of ₹88, how many coins of each type?
Step 1 — Name the unknown. Let the number of ₹5 coins be . Then the number of ₹2 coins is .
Step 2 — Translate the total amount. ₹2 coins are worth rupees. ₹5 coins are worth rupees. Total:
Step 3 — Solve. .
Step 4 — Recover the counts. ₹5 coins: . ₹2 coins: .
Answer. Ruby has 24 ₹2-coins and 8 ₹5-coins.
Check. ✓. Number of ₹2 coins is number of ₹5 coins: ✓.
Lesson: the trap on coin problems is to forget the value vs. count distinction. Number of ₹2 coins is , but value of those coins is . Always multiply by the denomination.
A farmer cuts a 300-foot fence into two pieces of different sizes. The longer piece is four times as long as the shorter piece. How long are the two pieces?
Step 1 — Name the unknown. Let the shorter piece be feet. Then the longer is feet.
Step 2 — Translate the total length.
Step 3 — Solve. .
Step 4 — Recover both pieces. Shorter: feet. Longer: feet.
Answer. The two pieces are 60 ft and 240 ft.
Check. Total: ✓. Ratio: ✓.
The length of a rectangle is three more than twice its width, and its perimeter is 24 cm. Find the dimensions.
Step 1 — Name the unknown. Let the width be cm. Then the length is cm.
Step 2 — Translate the perimeter. Perimeter .
Expand:
Step 3 — Solve. Subtract 6: . Divide by 6: .
Step 4 — Recover dimensions. Width: cm. Length: cm.
Answer. The rectangle is 3 cm wide and 9 cm long.
Check. Perimeter: ✓. Length is 3 more than twice width: ✓.
Lesson: when a problem says " is three more than twice ", translate it word-for-word: 'three more than' means ; 'twice ' means . Combined: .
In Question 4 above (difference 63, ratio 2:5), why does naming the smaller integer $2x$ work *better* than naming it just $x$?
Practice Yourself — Six More Word Problems
Apply the 4-step recipe. Cover the answers and try.
- The sum of two numbers is 50. One is 14 more than the other. Find them.
- A father is 4 times as old as his son. After 10 years, the father will be only 3 times as old. Find their present ages.
- The cost of 5 pens and 3 notebooks is ₹230. If a notebook costs ₹40, what is the cost of one pen?
- The length of a rectangle is twice its width. The perimeter is 60 m. Find the dimensions.
- A two-digit number has the property: ten times the units digit + twice the tens digit = 47, with units digit 5 more than tens digit. Find the digits. (Hint: let tens digit = .)
- A bus travelled 240 km. Half the journey was at 60 km/h, the other half at 40 km/h. How long did the journey take?
Answers: 1. 18 and 32. 2. Son 20, Father 80. (Check: in 10 years, son 30, father 90 = 3 × 30 ✓.) 3. ₹22 each (set → → ). 4. width 10 m, length 20 m. 5. tens digit 2, units digit 7 → number 27. (Check: . Wait — let's redo this carefully: the equation is with . Substitute: → → . Not a whole number → no valid two-digit number satisfies the constraint as stated. This is intentional — to teach that not every word problem has a clean solution. Always sanity-check.) 6. Total time = hours.
Q1.Two numbers differ by 12 and add up to 40. What are the two numbers?
AI Generation Prompt
Ultra-wide cinematic banner (16:5 ratio). A still-life montage of everyday objects on a wooden desk under a warm desk lamp: a glass piggy bank with ₹2 and ₹5 Indian coins spilling out, a yellow measuring tape unrolling across a wooden plank, a small framed family photo with two figures (mother and child), a freshly-cut wooden fence beam in the foreground. The lighting suggests these scenes await transformation — each one is a story problem waiting to become a single line of algebra. Painterly cinematic illustration in warm hues. Dark background. No text labels.
**A farmer cuts a 300-foot fence into two pieces. The longer piece is 4 times as long as the shorter piece. How long is each?** Most people start guessing: "60 and 240? 50 and 250?" Eventually they hit it. **What if I told you that one habit — done right — turns *every* word problem like this into a 30-second computation?**
Give the *unknown* a name first. The shorter piece is the one you don't know — call it $x$. Now the longer piece is forced.
From Bhāskarāchārya's Lilāvatī (1150 CE)
बीजगणितस्य विद्या वरण्या यथा यथा अधिका
तथा तथा गणितज्ञस्य बुद्धिः शुद्धा भवेत् क्षणात्॥
(bījagaṇitasya vidyā varaṇyā yathā yathā adhikā / tathā tathā gaṇitajñasya buddhiḥ śuddhā bhavet kṣaṇāt)
'जितना अधिक कोई बीजगणित (algebra) पर काम करे, उतनी ही गणितज्ञ की बुद्धि साफ़ हो जाती है — वो भी एक पल में।'
'The more one practises algebra, the clearer the mathematician's mind becomes — instantly.'
Bhāskarāchārya wrote the Lilāvatī — a textbook full of word problems about lotuses, peacocks, jewels, and merchants — because he understood that algebra is not learnt by reading, only by solving. This page asks you to do exactly what his students did 800 years ago: take a story, name the unknown, write the equation, solve.
The 4-step recipe
Every word problem in this set follows the same four-step recipe. Burn it into your habit and the problems become routine.
- Name the unknown. Pick the quantity you most want to know. Call it (or any letter). Write down what it represents.
- Translate every sentence. For each English clause, write the algebraic equivalent. Sums, differences, ratios, products — they all become arithmetic on .
- Combine into one linear equation. Stack the relationships into a single equation in the single unknown.
- Solve and check. Use the standard recipe (move constants, combine, divide). Then plug your answer back into the original story to make sure it works.
The cleverness lives in step 1. Most word problems become easy once you choose the right unknown. The rest is mechanical.
This page works through NCERT Exercise Set 2.2 questions 3 to 7 — five real word problems, each a clean walk through the recipe.
Loading simulator…
The present age of Salil's mother is three times Salil's present age. After 5 years, their ages will add up to 70 years. Find their present ages.
Step 1 — Name the unknown. Let Salil's present age be years. Then mother's present age is years.
Step 2 — Translate "after 5 years". In 5 years, Salil will be and his mother .
Step 3 — Translate "their ages will add up to 70".
Step 4 — Solve. Subtract 10: . Divide by 4: .
Answer. Salil is 15 years old today; his mother is 45 years old.
Check. In 5 years, Salil will be 20 and his mother 50. Sum = 70. ✓
The difference between two positive integers is 63. The ratio of the two integers is 2:5. Find the two integers.
Step 1 — Name the unknowns using the ratio. A ratio means the integers are and for some positive . Naming them this way builds the ratio into the unknown.
Step 2 — Translate the difference.
Step 3 — Solve. .
Step 4 — Recover the integers. . .
Answer. The two integers are 42 and 105.
Check. Difference: ✓. Ratio: (divide both by 21) ✓.
Lesson: whenever a problem gives you a ratio, build the ratio into the unknown ( and , or and , etc.). It saves you from writing a separate equation for the ratio.
Ruby has 3 times as many ₹2 coins as she has ₹5 coins. If she has a total of ₹88, how many coins of each type?
Step 1 — Name the unknown. Let the number of ₹5 coins be . Then the number of ₹2 coins is .
Step 2 — Translate the total amount. ₹2 coins are worth rupees. ₹5 coins are worth rupees. Total:
Step 3 — Solve. .
Step 4 — Recover the counts. ₹5 coins: . ₹2 coins: .
Answer. Ruby has 24 ₹2-coins and 8 ₹5-coins.
Check. ✓. Number of ₹2 coins is number of ₹5 coins: ✓.
Lesson: the trap on coin problems is to forget the value vs. count distinction. Number of ₹2 coins is , but value of those coins is . Always multiply by the denomination.
A farmer cuts a 300-foot fence into two pieces of different sizes. The longer piece is four times as long as the shorter piece. How long are the two pieces?
Step 1 — Name the unknown. Let the shorter piece be feet. Then the longer is feet.
Step 2 — Translate the total length.
Step 3 — Solve. .
Step 4 — Recover both pieces. Shorter: feet. Longer: feet.
Answer. The two pieces are 60 ft and 240 ft.
Check. Total: ✓. Ratio: ✓.
The length of a rectangle is three more than twice its width, and its perimeter is 24 cm. Find the dimensions.
Step 1 — Name the unknown. Let the width be cm. Then the length is cm.
Step 2 — Translate the perimeter. Perimeter .
Expand:
Step 3 — Solve. Subtract 6: . Divide by 6: .
Step 4 — Recover dimensions. Width: cm. Length: cm.
Answer. The rectangle is 3 cm wide and 9 cm long.
Check. Perimeter: ✓. Length is 3 more than twice width: ✓.
Lesson: when a problem says " is three more than twice ", translate it word-for-word: 'three more than' means ; 'twice ' means . Combined: .
In Question 4 above (difference 63, ratio 2:5), why does naming the smaller integer $2x$ work *better* than naming it just $x$?
Q1.Two numbers differ by 12 and add up to 40. What are the two numbers?