Mental Arithmetic and Algebra മനക്കണക്കും ബീജഗണിതവും
Many puzzles hide two unknown numbers. Sometimes a clever thought solves them in your head (മനക്കണക്ക്). Algebra (ബീജഗണിതം) does the same job by routine steps that work every time – with one letter, or with two letters and two equations. This section solves four puzzles all three ways, so you can see that the clever thought and the algebra are really the same idea.
- A letter can stand for an unknown number: "10 less than x" is x − 10, "5 times x" is 5x.
- To solve an equation, do the same thing to both sides: 2x − 10 = 100 ⇒ 2x = 110 ⇒ x = 55.
- Brackets: 2(x + 1) = 2x + 2. A fraction equal to ½ has its denominator twice its numerator.
1. A box of beads
The puzzle (page 7): A box holds 100 beads, black and white. There are 10 more black beads than white ones. How many of each?
🧠 In your head
Put the 10 extra black beads aside: 90 remain, half black and half white – 45 each. Put the 10 back: black 55, white 45.
✏️ One letter
✏️✏️ Two letters
2. A table and chairs
The puzzle (page 7): A table and a chair together cost ₹11000. A table and four chairs cost ₹14000. What does each cost?
One letter: let a chair cost x rupees. Then the table costs 11000 − x, and a table with four chairs costs (11000 − x) + 4x = 11000 + 3x = 14000, so 3x = 3000 and x = 1000.
Two letters: chair x, table y. The two facts become two equations: x + y = 11000 and 4x + y = 14000. From the first, y = 11000 − x. Put that in place of y in the second: 4x + (11000 − x) = 14000, that is 11000 + 3x = 14000 – exactly the one-letter equation. Replacing a letter by what it equals is called substitution.
Whatever the prices, the difference of the two totals pays for the extra chairs only. Divide by the number of extra chairs and the chair's price pops out.
3. Five times and a difference
The puzzle (page 8): Of two numbers, the larger is 5 times the smaller, and the larger minus the smaller is 32. Find them.
🧠 In your head
Five times a number, minus the number itself, is four times the number. Four times the smaller is 32, so the smaller is 8 and the larger is 40.
✏️ One letter
✏️✏️ Two letters
4. A fraction puzzle – where algebra wins
The puzzle (page 9): Adding 1 to the numerator (അംശം) of a fraction and simplifying gives ½. Adding 1 to the denominator (ഛേദം) instead gives ⅓. What is the fraction?
Try it in your head – it is hard! Taking just the numerator or just the denominator as x gets us nowhere. So take both: numerator x, denominator y.
Only one fraction passes both tests. Guessing could take a long time; the two equations find it in four lines.
When a puzzle has two unknowns, give each a letter and turn each fact into an equation. Two facts give a pair of equations (സമവാക്യജോടി). Then combine them so that only one letter is left – here by substitution: use one equation to say what one letter equals, and put that into the other.
Shopkeepers, accountants and engineers meet "two unknowns, two facts" all the time: two items with two bills, two speeds on a round trip, two ingredients in two mixtures. The mental shortcut works when the numbers are friendly; algebra works always – and it is exactly what spreadsheets and calculators do inside.
5. Check yourself
- Mental arithmetic: look for what the difference between two facts tells you (the extra 10 beads, the extra 3 chairs, 5 times minus once).
- Algebra with one letter: write the other unknown in terms of it (x − 10, 11000 − x, 5x) and solve one equation.
- Algebra with two letters: write two equations, then substitute one into the other to get a one-letter equation.
- The fraction puzzle: (x + 1)/y = ½ ⇒ y = 2(x + 1); x/(y + 1) = ⅓ ⇒ y + 1 = 3x ⇒ x = 3, y = 8: the fraction is 3/8.
Next: Practice set 1 – all eight questions solved with animations ▶