Real-World Applications പ്രയോഗങ്ങൾ
Two unknowns and two facts turn up everywhere – at the shop counter, in phone plans, at a ticket window, in a piggy bank, on a river. On every lab below, the sliders secretly choose the unknown numbers; the page then shows only the two facts you would be told, and solves them the way this chapter taught you. Change the sliders and watch the method catch the secret every time.
- Write each fact as an equation with two letters (page 5).
- Elimination: make one letter's coefficients equal, then subtract (page 7).
- Sum and difference: larger = (sum + difference) ÷ 2, smaller = (sum − difference) ÷ 2 (page 9).
- Check the answer in both facts, and make sure it makes sense (no negative prices, whole numbers of coins).
1. Two bills at the shop
You buy 3 soaps and 2 brushes, your friend buys 2 soaps and 3 brushes. You know only the two bills. Adding the bills gives 5 soaps and 5 brushes; subtracting them gives "1 soap − 1 brush". That is the sum and the difference of the two prices!
2. Which phone plan is cheaper? (graph view)
Plan A charges a small fixed amount plus a high rate per GB; plan B charges more fixed but less per GB. Each plan is an equation: cost = fixed + rate × GB. Drawn on a graph each is a straight line, and the two lines cross where the plans cost the same – that crossing point is the solution of the pair of equations. graph view · beyond the textbook
Left of the crossing the plan with the smaller fixed charge is cheaper; right of it the plan with the smaller rate wins.
3. Tickets at a school play
Adult tickets cost ₹50 and child tickets ₹20. The ticket counter knows how many tickets were sold and how much money came in. How many adults came?
4. Coins in a piggy bank
A piggy bank holds only ₹2 and ₹5 coins. You count the coins and add up their value. If every coin were a ₹2 coin, the value would be 2 × (number of coins); each ₹5 coin adds ₹3 more.
5. Blending tea
A shop mixes tea costing ₹200 a kilogram with tea costing ₹300 a kilogram to sell a blend at a chosen price. How much of each goes in?
6. A boat on a river
Going downstream the current helps: speed = boat + current. Going upstream it hinders: speed = boat − current. If you know both speeds, you know a sum and a difference.
7. Auto-rickshaw fares
An auto charges a fixed starting fare plus a rate for every kilometre. Two rides of different lengths reveal both: the extra kilometres account for the whole difference in fare.
8. Investing in two schemes
₹100000 is split between a scheme paying 8% a year and one paying 5%. The total interest (പലിശ) after a year tells how the money was split.
9. Ages that keep their distance
A father and son: you are told the sum of their ages and the difference. Sum and difference give both ages at once – and the difference never changes as the years pass, because both grow older together.
10. Fours and sixes
A batter hits only boundaries: fours and sixes. The scoreboard shows the number of boundaries and the runs from them. If all were fours, runs would be 4 × boundaries; every six adds 2 more.
11. Try these word problems
- Two unknown amounts, and two facts that link them – so two equations.
- "Totals" facts (bills, tickets, coins, runs) are eliminated by multiplying one equation and subtracting.
- "Sum and difference" facts (boats, ages) are solved by adding and subtracting the equations.
- Graphs show the same thing: the solution is where the two lines cross.
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