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A shopkeeper's notes read: '4 kg of rice and 3 kg of lentils cost ₹500' and, on another page, '8 kg of rice and 6 kg of lentils cost ₹1000'. Can the price of each item be worked out from these two notes?

Class 10Pair of Linear Equations in Two VariablesAlgebra — Systems of Equations
  • ANo — the second note just repeats the first, so infinitely many price pairs fit
  • BYes — rice costs ₹80 per kg and lentils ₹60 per kg
  • CNo — the two notes contradict each other, so no prices fit at all
  • DYes — there is exactly one pair of prices

Step-by-Step Solution

Ratios: 4/8 = 3/6 = 500/1000 = 1/2, so a₁/a₂ = b₁/b₂ = c₁/c₂ and the two equations are the SAME line — the second note is simply the first one doubled. A dependent pair has infinitely many solutions, so the individual prices cannot be pinned down. Saying the notes 'contradict each other' confuses a dependent (coincident) pair with an inconsistent (parallel) one.
Answer: A) No — the second note just repeats the first, so infinitely many price pairs fit

Key idea: Word problems and translation

Most Class 10 questions on this topic are word problems. The skill is in TRANSLATION:

1. Identify the two unknowns; assign variables (e.g., let = age of A, = age of B).
2. Write two equations from the two given pieces of information.
3. Solve by your preferred method.

**Common types**:
- **Age problems**: 'Father is twice as old as son. Five years ago father was three times. Find ages.'
- **Number problems**: 'A two-digit number plus its reverse is 99...'
- **Distance/time/speed**: 'Boat in still water speeds, downstream/upstream rates...'
- **Cost problems**: 'Cost of pencils and pens.'
- **Fraction problems**: 'A fraction becomes 1/2 when 1 is added to numerator and 2 is subtracted from denominator...'

Always CHECK by substituting solution back into the original word problem (not just the equations).

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