Word problems , translating English into equations

The most testable application of this chapter is the word problem: an English description hiding two linear equations. Once you have the equations, any of the methods (substitution, elimination, even graph) will solve them. The art is the translation.

A four-step plan

Read carefully, identifying the two unknowns. Give them names like $x$ and $y$ .
Set up two equations from the conditions in the problem. Most well-posed word problems give exactly two pieces of independent information.
Solve by substitution or elimination.
Interpret the answer: convert back to the units / meanings the problem uses, and check for consistency (no negative ages, no fractional people, etc.).

Common templates

Ages. "Five years ago, $A$ was four times as old as $B$ " $\Rightarrow$ $A - 5 = 4(B - 5)$ . "Ten years hence, $A$ will be twice $B$ " $\Rightarrow$ $A + 10 = 2(B + 10)$ .

Two-digit numbers. Let the tens digit be $x$ and units digit be $y$ . The number is $10 x + y$ . The reversed number is $10 y + x$ .

Fractions. "When $1$ is added to the numerator and $1$ subtracted from the denominator, the fraction becomes $1/2$ " , let the fraction be $x/y$ .

Speed–time–distance. Distance $=$ speed $\times$ time. With a stream/current of speed $u$ : upstream speed $= v - u$ , downstream $= v + u$ .

Work. If $A$ alone takes $a$ days and $B$ alone $b$ days, together they take $\dfrac{ab}{a + b}$ days. Word problems usually give the combined work time and a relation between $a, b$ .

Mixtures. " $3$ kg of tea at ₹ $100$ /kg and $2$ kg of tea at ₹ $x$ /kg gives a mix at ₹ $y$ /kg" , track total cost and total weight separately.

A general template

If two variables $x$ and $y$ are linked by two independent conditions, set them up as a pair of linear equations and solve. The hardest part is correctly writing the equations; we'll practise on the standard examples below.

Worked examples

Example 1 (ages). The present ages of a father and his son are in the ratio $7 : 3$ . After $10$ years, the ratio becomes $2 : 1$ . Find their present ages.

Let father $= 7 x$ , son $= 3 x$ (years). $\frac{7 x + 10}{3 x + 10} = \frac{2}{1} \Rightarrow 7 x + 10 = 2(3 x + 10) = 6 x + 20 \Rightarrow x = 10.$ Father $= 70$ years, son $= 30$ years.

Example 2 (two-digit number). A two-digit number is such that the sum of its digits is $9$ . When $27$ is added to the number, the digits are reversed. Find the number.

Let tens digit $= x$ , units digit $= y$ . Number $= 10 x + y$ .

Conditions: $x + y = 9$ and $10 x + y + 27 = 10 y + x$ , i.e., $9 x - 9 y = -27 \Rightarrow x - y = -3$ .

Adding to $x + y = 9$ : $2 x = 6 \Rightarrow x = 3$ , $y = 6$ . Number $= 36$ .

Check: $3 + 6 = 9$ ✓; $36 + 27 = 63$ which is digit-reversed. ✓

Example 3 (boats and current). A boat goes $30$ km upstream and $44$ km downstream in $10$ hours. In $13$ hours it can go $40$ km upstream and $55$ km downstream. Find the speed of the boat in still water and the speed of the stream.

Let boat speed $= u$ km/h, stream speed $= v$ km/h. Upstream $= u - v$ , downstream $= u + v$ .

$\frac{30}{u - v} + \frac{44}{u + v} = 10, \qquad \frac{40}{u - v} + \frac{55}{u + v} = 13.$ Let $p = 1/(u - v), q = 1/(u + v)$ . System: $30 p + 44 q = 10$ , $40 p + 55 q = 13$ .

Multiply first by $4$ and second by $3$ : $120 p + 176 q = 40$ , $120 p + 165 q = 39$ . Subtract: $11 q = 1 \Rightarrow q = 1/11$ , so $u + v = 11$ . From first: $30 p + 4 = 10 \Rightarrow p = 1/5$ , so $u - v = 5$ .

Hence $u = 8, v = 3$ . Boat speed $= 8$ km/h, stream $= 3$ km/h.

Example 4 (fraction). A fraction becomes $1/3$ when $1$ is subtracted from the numerator and it becomes $1/4$ when $8$ is added to its denominator. Find the fraction.

Let the fraction be $x/y$ . Then $(x - 1)/y = 1/3$ and $x/(y + 8) = 1/4$ .

From the first: $3(x - 1) = y \Rightarrow y = 3 x - 3$ . From the second: $4 x = y + 8 \Rightarrow y = 4 x - 8$ .

Equate: $3 x - 3 = 4 x - 8 \Rightarrow x = 5$ . So $y = 12$ . Fraction $= 5/12$ .

Example 5 (work). $2$ men and $7$ boys can do a piece of work in $4$ days, while $4$ men and $4$ boys do the same work in $3$ days. How long would $1$ man alone take, and $1$ boy alone?

Let one man do $m$ units of work per day, one boy do $b$ units. Total work $= 1$ .

$(2 m + 7 b) \cdot 4 = 1$ and $(4 m + 4 b) \cdot 3 = 1$ , i.e., $8 m + 28 b = 1$ and $12 m + 12 b = 1$ .

Subtract suitable multiples. Multiply first by $3$ and second by $2$ : $24 m + 84 b = 3$ and $24 m + 24 b = 2$ . Subtract: $60 b = 1 \Rightarrow b = 1/60$ , so a boy alone does $1/60$ of the work per day , $60$ days for the boy. From $12 m + 12/60 = 1$ : $12 m = 4/5 \Rightarrow m = 1/15$ , so a man takes $15$ days alone.

Try it yourself

The sum of the digits of a two-digit number is $12$ . The number obtained by interchanging the digits exceeds the original by $18$ . Find the number.
A man rows downstream $20$ km in $2$ hours and upstream $4$ km in $2$ hours. Find his rowing speed in still water and the stream speed.
The cost of $2$ kg apples and $1$ kg grapes is ₹ $160$ . After a month the cost is ₹ $300$ for $4$ kg apples and $2$ kg grapes. (Same prices.) Find the cost per kg.
Five years ago Nuri was thrice as old as Sonu. Ten years later Nuri will be twice as old as Sonu. How old are they now?
A railway half-ticket costs half the full fare and the reservation charge is the same on both. Two reservation tickets from station $A$ to $B$ cost ₹ $2530$ (one full and one half). Find the full first-class fare from $A$ to $B$ if reservation charge is ₹ $10$ .
The denominator of a fraction is $1$ more than twice the numerator. If the sum of the fraction and its reciprocal is $2 \tfrac{16}{21}$ , find the fraction.
A father is $3$ times as old as his son. In $12$ years' time he will be twice his son's age. Find their present ages.
In a rectangle, if length is increased by $2$ m and breadth decreased by $2$ m, the area decreases by $28$ m². If length is decreased by $1$ m and breadth increased by $2$ m, area increases by $33$ m². Find the dimensions.
The sum of the two-digit number $10x + y$ and the number formed by reversing its digits is $99$ . Find $x + y$ . (One equation; comment on why this doesn't pin down the number.)
Two pipes can fill a tank in $20$ min and $30$ min respectively. If both are open, how long to fill the tank?

Pitfalls / Insight

Define variables clearly with units. "Let $x$ be the speed" is not enough , "Let $x$ km/h be the speed of the boat in still water" is.
Check that the answer makes sense. Negative ages, fractional people, or speeds exceeding light should make you re-check.
In digit problems, the digits are integers between $0$ and $9$ ; the tens digit must be at least $1$ .

Insight. Almost every word problem of this chapter follows the pattern two unknowns, two conditions. If you can spot the unknowns and write the conditions, the algebra is routine.