Name the Unknown and the Problem Solves Itself

Give the unknown a letter, write everything else in terms of it, then turn the final sentence into an equation.

Five more than twice a number is 17 becomes $2x+5=17$, so $x=6$.

The translation is the whole difficulty; the algebra afterwards is easy. This page covers everything in the ICSE Class 7 Mathematics chapter's second part: translating statements, problems on numbers and ages, problems on money and geometry, and checking the solution.

Name the smallest or simplest unknown, state clearly what it stands for, and build the rest from it.

Common phrases translate directly:

- twice a number — $2x$
- five more than a number — $x+5$
- five less than a number — $x-5$
- one-third of a number — $\frac{x}{3}$
- the number increased by 7 — $x+7$
- 3 less than twice a number — $2x-3$

Worked example. When 7 is added to three times a number, the result is 34.

Let the number be $x$. Then

$$3x+7=34\Rightarrow 3x=27\Rightarrow x=9$$

Worked example. One-fourth of a number, decreased by 3, gives 5.

$$\frac{x}{4}-3=5\Rightarrow \frac{x}{4}=8\Rightarrow x=32$$

Two phrases are routinely mixed up, and they give different equations. Five less than a number is $x-5$, while a number less than five is $5-x$. Reading the order of the words carefully is what decides which one you write, so translate phrase by phrase rather than all at once.

Write consecutive quantities as $x$, $x+1$, $x+2$, and ages after $n$ years by adding $n$ to every person's age.

Consecutive numbers. The sum of three consecutive numbers is 51.

Let them be $x$, $x+1$ and $x+2$:

$$x+(x+1)+(x+2)=51\Rightarrow 3x+3=51\Rightarrow x=16$$

So the numbers are 16, 17 and 18, and $16+17+18=51$. Correct.

For consecutive even or odd numbers, the step is 2: $x$, $x+2$, $x+4$.

Ages. A father is three times as old as his son. In 12 years he will be twice as old as his son.

Let the son be $x$ years now, so the father is $3x$. In 12 years they will be $x+12$ and $3x+12$:

$$3x+12=2(x+12)\Rightarrow 3x+12=2x+24\Rightarrow x=12$$

So the son is 12 and the father 36. Check: in 12 years they will be 24 and 48, and 48 is twice 24. Correct.

The error that ruins age problems is adding the years to only one person. Twelve years pass for both, so both ages gain 12 — writing $3x+12=2x$ describes nothing real.

For coins and notes, multiply the number of each by its value. For geometry, use the standard formula as the equation.

Coins. A purse has 16 coins of ₹2 and ₹5, totalling ₹50. How many of each?

Let there be $x$ two-rupee coins, so there are $16-x$ five-rupee coins:

$$2x+5(16-x)=50\Rightarrow 2x+80-5x=50\Rightarrow -3x=-30\Rightarrow x=10$$

So 10 coins of ₹2 and 6 of ₹5. Check: $10+6=16$ coins, and $20+30=₹50$. Both conditions hold.

Perimeter. The length of a rectangle is 3 cm more than its breadth, and its perimeter is 26 cm.

Let the breadth be $b$, so the length is $b+3$:

$$2(b+3+b)=26\Rightarrow 4b+6=26\Rightarrow b=5$$

So breadth 5 cm, length 8 cm. Check: $2(8+5)=26$ cm. Correct.

Angles. Two complementary angles differ by 20°.

$$x+(x+20)=90\Rightarrow 2x=70\Rightarrow x=35$$

The angles are 35° and 55°, adding to 90°.

Notice what the coin problem needed: the value of a coin and the number of coins are different quantities. A ₹5 coin contributes 5 rupees, so the money equation multiplies them — using $x+(16-x)=50$ would be counting coins, not rupees.

Substitute back into the words of the problem, not just into your own equation — because an equation written wrongly will happily confirm a wrong answer.

Take the age problem. The answer was son 12, father 36. Now test each stated condition in turn:

- A father is three times as old as his son — is $36=3\times 12$? Yes.
- In 12 years he will be twice as old — in 12 years they are 48 and 24, and $48=2\times 24$? Yes.

Both conditions hold, so the answer is right.

Take the coin problem, answer 10 and 6:

- 16 coins in all — $10+6=16$. Yes.
- totalling ₹50 — $2(10)+5(6)=20+30=50$. Yes.

And a sense check costs nothing. An age cannot be negative, a number of coins cannot be a fraction, and a breadth cannot exceed the perimeter — so an answer of $b=-5$ or $x=4.5$ coins signals a setting-up error rather than an arithmetic one.

This is why checking against the words is stronger than checking the equation. If you had mistakenly written $3x+12=2x$, solving it gives $x=-12$ — an equation satisfied perfectly, and an age that cannot exist.