Practice Sets 2 & 3 – Animated Solutions ചിന്തിക്കാം
Five questions from the "think" boxes after the pens-and-pencils and the water-vessels examples. In every one, two facts become two equations; then we make one letter's coefficients equal, subtract (or add), and that letter disappears. Try each in your notebook first – the hint is there if you are stuck – then press Next ▶ to watch the solution grow.
- Same coefficient? Subtract the equations: 2x + 5y − (2x + 3y) = 2y.
- Different coefficients? Multiply one or both equations first, so that one letter has the same coefficient in both.
- Swapped coefficients (2 and 3, 3 and 2)? Adding the equations gives a multiple of x + y; subtracting gives x − y.
- Always check by putting the answers back into both facts.
Practice set 2 · Question 1 – pens and pencils
Both bills have the same 3 pencils. So what does the extra ₹45 pay for?
When two bills share exactly the same items except one kind, the difference of the bills is the price of the extra items only. Shopkeepers check bills this way every day.
Practice set 2 · Question 2 – the stretched rectangle
Half the perimeter is length + breadth. Write l + b = 13 and 2l + 3b = 31. Doubling the first equation makes the l's match.
Never forget to halve the perimeter: a rectangle's perimeter is 2(l + b), so "perimeter 26" means l + b = 13.
Practice set 3 · Question 1 – oranges and apples
The numbers 2 and 3 are swapped between the two bills. Put both bills together: how many kg of each fruit is that?
The usual method (multiply the first by 3 and the second by 2, then subtract) also works. But when the coefficients are swapped, adding and subtracting is much faster – look for it!
Practice set 3 · Question 2 – one wire, two shapes
Work in centimetres: 1 m = 100 cm. The square uses 4 sides and the triangle 3 sides, so 4s + 3t = 100. The other fact is 3s + 2t = 71.
The question asks where to cut, not the sides – so the last step turns the sides back into the lengths of the two pieces.
Practice set 3 · Question 3 – Rahim and Ramu
Let their ages now be R and m. Four years ago they were R − 4 and m − 4; in two years R + 2 and m + 2.
In every age puzzle the difference of ages never changes (here 12 years). It is a handy check on your answer.
Check yourself
- Same items in both bills except one kind ⇒ subtract the bills.
- Perimeter facts: halve first (l + b = half the perimeter), then match coefficients.
- Swapped coefficients ⇒ add (gives x + y) and subtract (gives x − y).
- Ages: write the past and future ages with the same letters, then solve.
Next: Section 3 – sum and difference, and the special cases ▶